![Page 1: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/1.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 20071 / 69
Topological Quantum Computationwith non-Abelian anyons in the
Pfaffian quantum Hall state
Lachezar S. GeorgievInstitute for Nuclear Research and Nuclear Energy,
Sofia, BulgariaBulgarian Academy of Sciences
(since 1869)
Phys. Rev. B 74 (2006) 235112,hep-th/0611340
L. Georgiev [INRNE-Sofia] Theory of Elementary Particles Support: EUCLID, NCSR-BG
![Page 2: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/2.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 20072 / 69
OUTLINE
![Page 3: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/3.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 20072 / 69
OUTLINE
� Quantum Computation in general:
![Page 4: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/4.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 20072 / 69
OUTLINE
� Quantum Computation in general:
+ Qubits, quantum gates, decoherence
![Page 5: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/5.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 20072 / 69
OUTLINE
� Quantum Computation in general:
+ Qubits, quantum gates, decoherence+ Discrete universal set of gates
![Page 6: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/6.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 20072 / 69
OUTLINE
� Quantum Computation in general:
+ Qubits, quantum gates, decoherence+ Discrete universal set of gates
� Topological Quantum Computation:
![Page 7: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/7.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 20072 / 69
OUTLINE
� Quantum Computation in general:
+ Qubits, quantum gates, decoherence+ Discrete universal set of gates
� Topological Quantum Computation:
+ Braiding non-Abelian quasiparticles as gates
![Page 8: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/8.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 20072 / 69
OUTLINE
� Quantum Computation in general:
+ Qubits, quantum gates, decoherence+ Discrete universal set of gates
� Topological Quantum Computation:
+ Braiding non-Abelian quasiparticles as gates+ Topological protection
![Page 9: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/9.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 20072 / 69
OUTLINE
� Quantum Computation in general:
+ Qubits, quantum gates, decoherence+ Discrete universal set of gates
� Topological Quantum Computation:
+ Braiding non-Abelian quasiparticles as gates+ Topological protection
L. Georgiev [INRNE-Sofia] Theory of Elementary Particles Support: EUCLID, NCSR-BG
![Page 10: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/10.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 20073 / 69
� TQC scheme of Das Sarma–Freedman–Nayak:
![Page 11: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/11.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 20073 / 69
� TQC scheme of Das Sarma–Freedman–Nayak:
+ Qubit construction, state initialization, measurement
![Page 12: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/12.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 20073 / 69
� TQC scheme of Das Sarma–Freedman–Nayak:
+ Qubit construction, state initialization, measurement+ Computational basis
![Page 13: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/13.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 20073 / 69
� TQC scheme of Das Sarma–Freedman–Nayak:
+ Qubit construction, state initialization, measurement+ Computational basis+ Quantum logical NOT gate
![Page 14: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/14.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 20073 / 69
� TQC scheme of Das Sarma–Freedman–Nayak:
+ Qubit construction, state initialization, measurement+ Computational basis+ Quantum logical NOT gate
� Braid-group extension:
![Page 15: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/15.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 20073 / 69
� TQC scheme of Das Sarma–Freedman–Nayak:
+ Qubit construction, state initialization, measurement+ Computational basis+ Quantum logical NOT gate
� Braid-group extension:
+ Exchange matrices for 4 quasiholes
![Page 16: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/16.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 20073 / 69
� TQC scheme of Das Sarma–Freedman–Nayak:
+ Qubit construction, state initialization, measurement+ Computational basis+ Quantum logical NOT gate
� Braid-group extension:
+ Exchange matrices for 4 quasiholes+ Finite representation of the braid group B4
![Page 17: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/17.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 20073 / 69
� TQC scheme of Das Sarma–Freedman–Nayak:
+ Qubit construction, state initialization, measurement+ Computational basis+ Quantum logical NOT gate
� Braid-group extension:
+ Exchange matrices for 4 quasiholes+ Finite representation of the braid group B4
+ Single-qubit gates
![Page 18: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/18.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 20073 / 69
� TQC scheme of Das Sarma–Freedman–Nayak:
+ Qubit construction, state initialization, measurement+ Computational basis+ Quantum logical NOT gate
� Braid-group extension:
+ Exchange matrices for 4 quasiholes+ Finite representation of the braid group B4
+ Single-qubit gates
� Two-qubit construction and two-qubit gates:
![Page 19: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/19.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 20073 / 69
� TQC scheme of Das Sarma–Freedman–Nayak:
+ Qubit construction, state initialization, measurement+ Computational basis+ Quantum logical NOT gate
� Braid-group extension:
+ Exchange matrices for 4 quasiholes+ Finite representation of the braid group B4
+ Single-qubit gates
� Two-qubit construction and two-qubit gates:
+ Exchange matrices for 6 quasiholes
![Page 20: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/20.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 20073 / 69
� TQC scheme of Das Sarma–Freedman–Nayak:
+ Qubit construction, state initialization, measurement+ Computational basis+ Quantum logical NOT gate
� Braid-group extension:
+ Exchange matrices for 4 quasiholes+ Finite representation of the braid group B4
+ Single-qubit gates
� Two-qubit construction and two-qubit gates:
+ Exchange matrices for 6 quasiholes
L. Georgiev [INRNE-Sofia] Theory of Elementary Particles Support: EUCLID, NCSR-BG
![Page 21: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/21.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 20074 / 69
+ Single-qubit gates in the two-qubit basis
![Page 22: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/22.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 20074 / 69
+ Single-qubit gates in the two-qubit basis+ Controlled-Z and Controlled-NOT gates
![Page 23: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/23.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 20074 / 69
+ Single-qubit gates in the two-qubit basis+ Controlled-Z and Controlled-NOT gates
� Three qubits and three-qubit gates:
![Page 24: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/24.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 20074 / 69
+ Single-qubit gates in the two-qubit basis+ Controlled-Z and Controlled-NOT gates
� Three qubits and three-qubit gates:
+ Exchange matrices for 8 quasiholes
![Page 25: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/25.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 20074 / 69
+ Single-qubit gates in the two-qubit basis+ Controlled-Z and Controlled-NOT gates
� Three qubits and three-qubit gates:
+ Exchange matrices for 8 quasiholes+ 1-qubit and 2-qubit gates for 3-qubits
![Page 26: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/26.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 20074 / 69
+ Single-qubit gates in the two-qubit basis+ Controlled-Z and Controlled-NOT gates
� Three qubits and three-qubit gates:
+ Exchange matrices for 8 quasiholes+ 1-qubit and 2-qubit gates for 3-qubits+ Toffoli gate by Controlled-S and CNOT
![Page 27: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/27.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 20074 / 69
+ Single-qubit gates in the two-qubit basis+ Controlled-Z and Controlled-NOT gates
� Three qubits and three-qubit gates:
+ Exchange matrices for 8 quasiholes+ 1-qubit and 2-qubit gates for 3-qubits+ Toffoli gate by Controlled-S and CNOT
� Open problems:
![Page 28: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/28.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 20074 / 69
+ Single-qubit gates in the two-qubit basis+ Controlled-Z and Controlled-NOT gates
� Three qubits and three-qubit gates:
+ Exchange matrices for 8 quasiholes+ 1-qubit and 2-qubit gates for 3-qubits+ Toffoli gate by Controlled-S and CNOT
� Open problems:
� Acknowledegments:Ivan Todorov, Ady Stern, Valentina Petkova, Chetan
L. Georgiev [INRNE-Sofia] Theory of Elementary Particles Support: EUCLID, NCSR-BG
![Page 29: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/29.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 20075 / 69
Nayak, Lyudmil Hadjiivanov, Michael Geller and PreslavKonstantinov
![Page 30: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/30.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 20075 / 69
Nayak, Lyudmil Hadjiivanov, Michael Geller and PreslavKonstantinov
L. Georgiev [INRNE-Sofia] Theory of Elementary Particles Support: EUCLID, NCSR-BG
![Page 31: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/31.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 20076 / 69
Quantum computation in general
� Qubit construction:
![Page 32: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/32.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 20076 / 69
Quantum computation in general
� Qubit construction: j i D ˛j0i Cˇj1i
![Page 33: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/33.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 20076 / 69
Quantum computation in general
� Qubit construction: j i D ˛j0i Cˇj1i, j˛j2 C jˇj2 D 1
![Page 34: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/34.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 20076 / 69
Quantum computation in general
� Qubit construction: j i D ˛j0i Cˇj1i, j˛j2 C jˇj2 D 1
� State Preparation:
![Page 35: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/35.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 20076 / 69
Quantum computation in general
� Qubit construction: j i D ˛j0i Cˇj1i, j˛j2 C jˇj2 D 1
� State Preparation: define j0i, j1i
![Page 36: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/36.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 20076 / 69
Quantum computation in general
� Qubit construction: j i D ˛j0i Cˇj1i, j˛j2 C jˇj2 D 1
� State Preparation: define j0i, j1i
and construct j i D j0i or j i D j1i
![Page 37: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/37.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 20076 / 69
Quantum computation in general
� Qubit construction: j i D ˛j0i Cˇj1i, j˛j2 C jˇj2 D 1
� State Preparation: define j0i, j1i
and construct j i D j0i or j i D j1i
� Measurement:
![Page 38: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/38.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 20076 / 69
Quantum computation in general
� Qubit construction: j i D ˛j0i Cˇj1i, j˛j2 C jˇj2 D 1
� State Preparation: define j0i, j1i
and construct j i D j0i or j i D j1i
� Measurement: standard orthogonal projectors family
![Page 39: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/39.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 20076 / 69
Quantum computation in general
� Qubit construction: j i D ˛j0i Cˇj1i, j˛j2 C jˇj2 D 1
� State Preparation: define j0i, j1i
and construct j i D j0i or j i D j1i
� Measurement: standard orthogonal projectors family
� Multiple-qubit construction:
![Page 40: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/40.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 20076 / 69
Quantum computation in general
� Qubit construction: j i D ˛j0i Cˇj1i, j˛j2 C jˇj2 D 1
� State Preparation: define j0i, j1i
and construct j i D j0i or j i D j1i
� Measurement: standard orthogonal projectors family
� Multiple-qubit construction: tensor products;
![Page 41: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/41.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 20076 / 69
Quantum computation in general
� Qubit construction: j i D ˛j0i Cˇj1i, j˛j2 C jˇj2 D 1
� State Preparation: define j0i, j1i
and construct j i D j0i or j i D j1i
� Measurement: standard orthogonal projectors family
� Multiple-qubit construction: tensor products;Hn D ˝nC2
![Page 42: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/42.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 20076 / 69
Quantum computation in general
� Qubit construction: j i D ˛j0i Cˇj1i, j˛j2 C jˇj2 D 1
� State Preparation: define j0i, j1i
and construct j i D j0i or j i D j1i
� Measurement: standard orthogonal projectors family
� Multiple-qubit construction: tensor products;Hn D ˝nC2 ' C2n
![Page 43: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/43.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 20076 / 69
Quantum computation in general
� Qubit construction: j i D ˛j0i Cˇj1i, j˛j2 C jˇj2 D 1
� State Preparation: define j0i, j1i
and construct j i D j0i or j i D j1i
� Measurement: standard orthogonal projectors family
� Multiple-qubit construction: tensor products;Hn D ˝nC2 ' C2n
(projective)
![Page 44: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/44.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 20076 / 69
Quantum computation in general
� Qubit construction: j i D ˛j0i Cˇj1i, j˛j2 C jˇj2 D 1
� State Preparation: define j0i, j1i
and construct j i D j0i or j i D j1i
� Measurement: standard orthogonal projectors family
� Multiple-qubit construction: tensor products;Hn D ˝nC2 ' C2n
(projective)
� Initialization:
![Page 45: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/45.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 20076 / 69
Quantum computation in general
� Qubit construction: j i D ˛j0i Cˇj1i, j˛j2 C jˇj2 D 1
� State Preparation: define j0i, j1i
and construct j i D j0i or j i D j1i
� Measurement: standard orthogonal projectors family
� Multiple-qubit construction: tensor products;Hn D ˝nC2 ' C2n
(projective)
� Initialization: 010011000:::01
![Page 46: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/46.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 20076 / 69
Quantum computation in general
� Qubit construction: j i D ˛j0i Cˇj1i, j˛j2 C jˇj2 D 1
� State Preparation: define j0i, j1i
and construct j i D j0i or j i D j1i
� Measurement: standard orthogonal projectors family
� Multiple-qubit construction: tensor products;Hn D ˝nC2 ' C2n
(projective)
� Initialization: 010011000:::01QC! j010011000:::01i
![Page 47: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/47.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 20076 / 69
Quantum computation in general
� Qubit construction: j i D ˛j0i Cˇj1i, j˛j2 C jˇj2 D 1
� State Preparation: define j0i, j1i
and construct j i D j0i or j i D j1i
� Measurement: standard orthogonal projectors family
� Multiple-qubit construction: tensor products;Hn D ˝nC2 ' C2n
(projective)
� Initialization: 010011000:::01QC! j010011000:::01i
� Quantum gates: unitary operators over Hn
![Page 48: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/48.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 20076 / 69
Quantum computation in general
� Qubit construction: j i D ˛j0i Cˇj1i, j˛j2 C jˇj2 D 1
� State Preparation: define j0i, j1i
and construct j i D j0i or j i D j1i
� Measurement: standard orthogonal projectors family
� Multiple-qubit construction: tensor products;Hn D ˝nC2 ' C2n
(projective)
� Initialization: 010011000:::01QC! j010011000:::01i
� Quantum gates: unitary operators over Hn
L. Georgiev [INRNE-Sofia] Theory of Elementary Particles Support: EUCLID, NCSR-BG
![Page 49: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/49.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 20077 / 69
� Exponential speed-up:
![Page 50: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/50.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 20077 / 69
� Exponential speed-up:entanglement ! quantum parallelism:
![Page 51: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/51.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 20077 / 69
� Exponential speed-up:entanglement ! quantum parallelism:compute all values of f .x/ for x D 010011000:::01 in just onerun (global properties)
![Page 52: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/52.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 20077 / 69
� Exponential speed-up:entanglement ! quantum parallelism:compute all values of f .x/ for x D 010011000:::01 in just onerun (global properties)
� Discrete Universal set of quantum gates:
![Page 53: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/53.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 20077 / 69
� Exponential speed-up:entanglement ! quantum parallelism:compute all values of f .x/ for x D 010011000:::01 in just onerun (global properties)
� Discrete Universal set of quantum gates: any unitaryoperator can be approximated by products of 4 universalgates: H , S , CNOT and T
![Page 54: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/54.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 20077 / 69
� Exponential speed-up:entanglement ! quantum parallelism:compute all values of f .x/ for x D 010011000:::01 in just onerun (global properties)
� Discrete Universal set of quantum gates: any unitaryoperator can be approximated by products of 4 universalgates: H , S , CNOT and T
with arbitrary precision in
E.U;V / � maxj i
jjU � V j ijj
L. Georgiev [INRNE-Sofia] Theory of Elementary Particles Support: EUCLID, NCSR-BG
![Page 55: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/55.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 20078 / 69
� Explicit gate realization in the basis fj0i; j1ig:
![Page 56: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/56.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 20078 / 69
� Explicit gate realization in the basis fj0i; j1ig:
H D1
p2
�1 1
1 �1
�; S D
�1 0
0 i
�; T D
�1 0
0 ei�=4
�
![Page 57: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/57.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 20078 / 69
� Explicit gate realization in the basis fj0i; j1ig:
H D1
p2
�1 1
1 �1
�; S D
�1 0
0 i
�; T D
�1 0
0 ei�=4
�
CNOT D
26641 0 0 0
0 1 0 0
0 0 0 1
0 0 1 0
3775 ; in the basis fj00i; j01i; j10i; j11ig
![Page 58: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/58.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 20078 / 69
� Explicit gate realization in the basis fj0i; j1ig:
H D1
p2
�1 1
1 �1
�; S D
�1 0
0 i
�; T D
�1 0
0 ei�=4
�
CNOT D
26641 0 0 0
0 1 0 0
0 0 0 1
0 0 1 0
3775 ; in the basis fj00i; j01i; j10i; j11ig
L. Georgiev [INRNE-Sofia] Theory of Elementary Particles Support: EUCLID, NCSR-BG
![Page 59: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/59.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 20079 / 69
� Decoherence and noise: unavoidable
![Page 60: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/60.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 20079 / 69
� Decoherence and noise: unavoidable due to localinteractions destroying coherent phenomena and flippingj0i $ j1i
![Page 61: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/61.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 20079 / 69
� Decoherence and noise: unavoidable due to localinteractions destroying coherent phenomena and flippingj0i $ j1i
� Quantum Error-Correcting algorithms:
![Page 62: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/62.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 20079 / 69
� Decoherence and noise: unavoidable due to localinteractions destroying coherent phenomena and flippingj0i $ j1i
� Quantum Error-Correcting algorithms:
Nice if the threshold theorem indeed applies
![Page 63: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/63.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 20079 / 69
� Decoherence and noise: unavoidable due to localinteractions destroying coherent phenomena and flippingj0i $ j1i
� Quantum Error-Correcting algorithms:
Nice if the threshold theorem indeed applies
Not a big help on the scale of 500 qubits(huge overhead)
L. Georgiev [INRNE-Sofia] Theory of Elementary Particles Support: EUCLID, NCSR-BG
![Page 64: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/64.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200710 / 69
What is Topological Quantum Computation?
� Noise and decoherence due to local interactions
![Page 65: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/65.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200710 / 69
What is Topological Quantum Computation?
� Noise and decoherence due to local interactions
� Avoid them by global quantum information encoding:
![Page 66: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/66.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200710 / 69
What is Topological Quantum Computation?
� Noise and decoherence due to local interactions
� Avoid them by global quantum information encoding:i.e., encoding quantum information non-locally
![Page 67: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/67.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200710 / 69
What is Topological Quantum Computation?
� Noise and decoherence due to local interactions
� Avoid them by global quantum information encoding:i.e., encoding quantum information non-locally
� Use topological characteristics of the system:
![Page 68: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/68.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200710 / 69
What is Topological Quantum Computation?
� Noise and decoherence due to local interactions
� Avoid them by global quantum information encoding:i.e., encoding quantum information non-locally
� Use topological characteristics of the system:e.g., homotopy classes of exchange paths.
![Page 69: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/69.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200710 / 69
What is Topological Quantum Computation?
� Noise and decoherence due to local interactions
� Avoid them by global quantum information encoding:i.e., encoding quantum information non-locally
� Use topological characteristics of the system:e.g., homotopy classes of exchange paths.
� Topological protection of qubit operations:
![Page 70: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/70.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200710 / 69
What is Topological Quantum Computation?
� Noise and decoherence due to local interactions
� Avoid them by global quantum information encoding:i.e., encoding quantum information non-locally
� Use topological characteristics of the system:e.g., homotopy classes of exchange paths.
� Topological protection of qubit operations: quantuminformation is inaccessible to local interactions, becausethey cannot distinguish between j0i and j1i and hencecannot lead to decoherence and noise
![Page 71: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/71.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200710 / 69
What is Topological Quantum Computation?
� Noise and decoherence due to local interactions
� Avoid them by global quantum information encoding:i.e., encoding quantum information non-locally
� Use topological characteristics of the system:e.g., homotopy classes of exchange paths.
� Topological protection of qubit operations: quantuminformation is inaccessible to local interactions, becausethey cannot distinguish between j0i and j1i and hencecannot lead to decoherence and noise
L. Georgiev [INRNE-Sofia] Theory of Elementary Particles Support: EUCLID, NCSR-BG
![Page 72: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/72.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200711 / 69
� Quantum gates: change homotopy classes
![Page 73: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/73.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200711 / 69
� Quantum gates: change homotopy classesimplemented by braid operations(exchanges of quasiparticles in the plane)
![Page 74: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/74.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200711 / 69
� Quantum gates: change homotopy classesimplemented by braid operations(exchanges of quasiparticles in the plane)
� Degeneracy needed in the plane:
![Page 75: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/75.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200711 / 69
� Quantum gates: change homotopy classesimplemented by braid operations(exchanges of quasiparticles in the plane)
� Degeneracy needed in the plane:non-Abelian quasiparticles(plektons: higher dimensional representations of the Braidgroup)
![Page 76: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/76.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200711 / 69
� Quantum gates: change homotopy classesimplemented by braid operations(exchanges of quasiparticles in the plane)
� Degeneracy needed in the plane:non-Abelian quasiparticles(plektons: higher dimensional representations of the Braidgroup)
L. Georgiev [INRNE-Sofia] Theory of Elementary Particles Support: EUCLID, NCSR-BG
![Page 77: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/77.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200712 / 69
The Pfaffian quantum Hall state:
� Most promising non-Abelian candidate:
![Page 78: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/78.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200712 / 69
The Pfaffian quantum Hall state:
� Most promising non-Abelian candidate: fract. quantumHall state with filling factor � D 5=2 (second Landau level)
![Page 79: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/79.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200712 / 69
The Pfaffian quantum Hall state:
� Most promising non-Abelian candidate: fract. quantumHall state with filling factor � D 5=2 (second Landau level)
� Routinely observed in ultrahigh-mobility samples [Pan etal., Xia et al., Eisenstein et al., Choi et al.]
![Page 80: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/80.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200712 / 69
The Pfaffian quantum Hall state:
� Most promising non-Abelian candidate: fract. quantumHall state with filling factor � D 5=2 (second Landau level)
� Routinely observed in ultrahigh-mobility samples [Pan etal., Xia et al., Eisenstein et al., Choi et al.]
� In the universality class of the Moore–Read state(Pfaffian CFT: du.1/ � Ising )
![Page 81: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/81.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200712 / 69
The Pfaffian quantum Hall state:
� Most promising non-Abelian candidate: fract. quantumHall state with filling factor � D 5=2 (second Landau level)
� Routinely observed in ultrahigh-mobility samples [Pan etal., Xia et al., Eisenstein et al., Choi et al.]
� In the universality class of the Moore–Read state(Pfaffian CFT: du.1/ � Ising )
� non-Abelian statistics might be easier to be observed thanthe Abelian one [Stern-Halperin, Kitaev et al.]
L. Georgiev [INRNE-Sofia] Theory of Elementary Particles Support: EUCLID, NCSR-BG
![Page 82: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/82.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200713 / 69
Topological protection in the Pfaffian FQH state
� Residual noise and decoherence:
![Page 83: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/83.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200713 / 69
Topological protection in the Pfaffian FQH state
� Residual noise and decoherence: due to thermallyactivated q.p.–q.h. pairs (executing uncontrolled braidings)
![Page 84: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/84.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200713 / 69
Topological protection in the Pfaffian FQH state
� Residual noise and decoherence: due to thermallyactivated q.p.–q.h. pairs (executing uncontrolled braidings)
� Exponentially suppressed at low temperature (T � 5
mK) by the bulk energy gap (�=kB � 500 mK) [Das Sarma]
![Page 85: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/85.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200713 / 69
Topological protection in the Pfaffian FQH state
� Residual noise and decoherence: due to thermallyactivated q.p.–q.h. pairs (executing uncontrolled braidings)
� Exponentially suppressed at low temperature (T � 5
mK) by the bulk energy gap (�=kB � 500 mK) [Das Sarma]
Error rate '
�kBT
�
�exp
���
kBT
�< 10�30
)
![Page 86: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/86.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200713 / 69
Topological protection in the Pfaffian FQH state
� Residual noise and decoherence: due to thermallyactivated q.p.–q.h. pairs (executing uncontrolled braidings)
� Exponentially suppressed at low temperature (T � 5
mK) by the bulk energy gap (�=kB � 500 mK) [Das Sarma]
Error rate '
�kBT
�
�exp
���
kBT
�< 10�30
)
� Unpresedented precision of quantum informationprocessing
L. Georgiev [INRNE-Sofia] Theory of Elementary Particles Support: EUCLID, NCSR-BG
![Page 87: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/87.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200714 / 69
DISCLAIMER: Note for CFT haters:� CFT: effective field theory at low T and TD limit (N ! 1)
![Page 88: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/88.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200714 / 69
DISCLAIMER: Note for CFT haters:� CFT: effective field theory at low T and TD limit (N ! 1)
� but: efficiently captures all universal properties
![Page 89: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/89.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200714 / 69
DISCLAIMER: Note for CFT haters:� CFT: effective field theory at low T and TD limit (N ! 1)
� but: efficiently captures all universal propertiesespecially topological ones
![Page 90: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/90.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200714 / 69
DISCLAIMER: Note for CFT haters:� CFT: effective field theory at low T and TD limit (N ! 1)
� but: efficiently captures all universal propertiesespecially topological ones
� e.g.: electric charge and quantum statistics of localizedexcitations
![Page 91: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/91.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200714 / 69
DISCLAIMER: Note for CFT haters:� CFT: effective field theory at low T and TD limit (N ! 1)
� but: efficiently captures all universal propertiesespecially topological ones
� e.g.: electric charge and quantum statistics of localizedexcitations
� Promise: We shall only use universal quantities for TQC!
L. Georgiev [INRNE-Sofia] Theory of Elementary Particles Support: EUCLID, NCSR-BG
![Page 92: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/92.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200715 / 69
TQC scheme of Das Sarma et al.� Main idea: use 4 q.h. Pfaffian wave functions as qubit(2-dim. space) N electrons (holes) at z1; : : : ; zN and 4quasiholes at �1; : : : ; �4
4qh.�1; �2; �3; �4I z1; : : : ; zN /defD
h qh.�1/ qh.�2/ qh.�3/ qh.�4/
NYiD1
hole.zi/i
![Page 93: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/93.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200715 / 69
TQC scheme of Das Sarma et al.� Main idea: use 4 q.h. Pfaffian wave functions as qubit(2-dim. space) N electrons (holes) at z1; : : : ; zN and 4quasiholes at �1; : : : ; �4
4qh.�1; �2; �3; �4I z1; : : : ; zN /defD
h qh.�1/ qh.�2/ qh.�3/ qh.�4/
NYiD1
hole.zi/i
hole.z/ D .z/ : eip
2�.z/ : and qh.�/ D �.�/ : ei 1
2p
2�.�/
:;
L. Georgiev [INRNE-Sofia] Theory of Elementary Particles Support: EUCLID, NCSR-BG
![Page 94: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/94.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200716 / 69
� Explicit result [Nayak–Wilczek]: (�ab D �a � �b)
4qh.�1; �2; �3; �4I z1; : : : ; zN / D .0/4qh C
.1/4qh;
![Page 95: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/95.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200716 / 69
� Explicit result [Nayak–Wilczek]: (�ab D �a � �b)
4qh.�1; �2; �3; �4I z1; : : : ; zN / D .0/4qh C
.1/4qh;
.0;1/4qh D
.�13�24/14p
1 ˙p
x
�.13/.24/ ˙
px .14/.23/
�; x �
�14�23
�13�24
![Page 96: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/96.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200716 / 69
� Explicit result [Nayak–Wilczek]: (�ab D �a � �b)
4qh.�1; �2; �3; �4I z1; : : : ; zN / D .0/4qh C
.1/4qh;
.0;1/4qh D
.�13�24/14p
1 ˙p
x
�.13/.24/ ˙
px .14/.23/
�; x �
�14�23
�13�24
.ab/.cd/ D Pf
.zi � �a/.zi � �b/.zj � �c/.zj � �d/C .i $ j /
zi � zj
!�
�
Y1�i<j �N
.zi � zj /2; .a < b; c < d/ s:v:
L. Georgiev [INRNE-Sofia] Theory of Elementary Particles Support: EUCLID, NCSR-BG
![Page 97: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/97.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200717 / 69
The NOT gate of Das Sarma et al.:
� Claim [LG]: Mach–Zehnder interference iffˇ̌̌�12�34�13�24
ˇ̌̌< 1
![Page 98: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/98.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200717 / 69
The NOT gate of Das Sarma et al.:
� Claim [LG]: Mach–Zehnder interference iffˇ̌̌�12�34�13�24
ˇ̌̌< 1
� NOT gate: take �3 along a closed loop around �2, i.e.,
NOT W �23 ! limt!1�
e2� i t�23; where t 2 Œ0; 1/
![Page 99: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/99.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200717 / 69
The NOT gate of Das Sarma et al.:
� Claim [LG]: Mach–Zehnder interference iffˇ̌̌�12�34�13�24
ˇ̌̌< 1
� NOT gate: take �3 along a closed loop around �2, i.e.,
NOT W �23 ! limt!1�
e2� i t�23; where t 2 Œ0; 1/
Then [Das Sarma et al.] where qubit D .�1; �2/
j0i ! j1i and j1i ! j0i
![Page 100: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/100.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200717 / 69
The NOT gate of Das Sarma et al.:
� Claim [LG]: Mach–Zehnder interference iffˇ̌̌�12�34�13�24
ˇ̌̌< 1
� NOT gate: take �3 along a closed loop around �2, i.e.,
NOT W �23 ! limt!1�
e2� i t�23; where t 2 Œ0; 1/
Then [Das Sarma et al.] where qubit D .�1; �2/
j0i ! j1i and j1i ! j0i
� N.B.: The NOT gate is a monodromy transformation
L. Georgiev [INRNE-Sofia] Theory of Elementary Particles Support: EUCLID, NCSR-BG
![Page 101: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/101.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200718 / 69
Braiding: Extend TQC scheme of Das Sarma
� Computational basis (reminder):
j0i; j1i $ .0;1/4qh D
.�13�24/14p
1 ˙p
x
�.13/.24/ ˙
px .14/.23/
�.13/.24/ and .14/.23/ .single � valued/; x �
�14�23
�13�24
![Page 102: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/102.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200718 / 69
Braiding: Extend TQC scheme of Das Sarma
� Computational basis (reminder):
j0i; j1i $ .0;1/4qh D
.�13�24/14p
1 ˙p
x
�.13/.24/ ˙
px .14/.23/
�.13/.24/ and .14/.23/ .single � valued/; x �
�14�23
�13�24
� Fermion parity: double degenrated Ramond sector �˙
Œ 0; F �C D 0; 20 D
1
2; 2
F D 1 ) F�˙ F D ˙�˙
L. Georgiev [INRNE-Sofia] Theory of Elementary Particles Support: EUCLID, NCSR-BG
![Page 103: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/103.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200719 / 69
� Physical quasihole: Modular invariance ) GSOprojection ) non-Abelian statistics
�.�/ D�C.�/C ��.�/
p2
) � � � D I C
� Parity conservation: h�e1�e2�e3�e4i ¤ 0 , e1e2 D e3e4
![Page 104: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/104.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200719 / 69
� Physical quasihole: Modular invariance ) GSOprojection ) non-Abelian statistics
�.�/ D�C.�/C ��.�/
p2
) � � � D I C
� Parity conservation: h�e1�e2�e3�e4i ¤ 0 , e1e2 D e3e4
� Computational basis: CFT desciption
j0i � h�C�C�C�Ci; j1i � h�C���C��i;
![Page 105: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/105.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200719 / 69
� Physical quasihole: Modular invariance ) GSOprojection ) non-Abelian statistics
�.�/ D�C.�/C ��.�/
p2
) � � � D I C
� Parity conservation: h�e1�e2�e3�e4i ¤ 0 , e1e2 D e3e4
� Computational basis: CFT desciption
j0i � h�C�C�C�Ci; j1i � h�C���C��i;
� Space dimension = correlation functions multiplicity:
L. Georgiev [INRNE-Sofia] Theory of Elementary Particles Support: EUCLID, NCSR-BG
![Page 106: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/106.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200720 / 69
conservation ) qubit = .e1; e2/ where ei D ˙1
![Page 107: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/107.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200720 / 69
conservation ) qubit = .e1; e2/ where ei D ˙1
� Ising OPE: ) .C;C/ ' .�;�/
![Page 108: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/108.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200720 / 69
conservation ) qubit = .e1; e2/ where ei D ˙1
� Ising OPE: ) .C;C/ ' .�;�/ and .C;�/ ' .�;C/
![Page 109: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/109.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200720 / 69
conservation ) qubit = .e1; e2/ where ei D ˙1
� Ising OPE: ) .C;C/ ' .�;�/ and .C;�/ ' .�;C/
) dim H4� D 2.
![Page 110: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/110.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200720 / 69
conservation ) qubit = .e1; e2/ where ei D ˙1
� Ising OPE: ) .C;C/ ' .�;�/ and .C;�/ ' .�;C/
) dim H4� D 2. In general:
![Page 111: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/111.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200720 / 69
conservation ) qubit = .e1; e2/ where ei D ˙1
� Ising OPE: ) .C;C/ ' .�;�/ and .C;�/ ' .�;C/
) dim H4� D 2. In general: dim H2n� D 2n�1
L. Georgiev [INRNE-Sofia] Theory of Elementary Particles Support: EUCLID, NCSR-BG
![Page 112: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/112.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200721 / 69
� Braiding as analytic continuation: (0 � t � 1)
�01 D
�1 C �2
2C ei� t�1 � �2
2; �0
2 D�1 C �2
2� ei� t�1 � �2
2;
2η 1η
221 ηη +
tiπηη e221−
12R
L. Georgiev [INRNE-Sofia] Theory of Elementary Particles Support: EUCLID, NCSR-BG
![Page 113: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/113.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200722 / 69
1η
2η
3η′
4η′
![Page 114: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/114.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200722 / 69
1η
2η
3η′
4η′
1η
2η
3η′
4η′
![Page 115: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/115.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200722 / 69
1η
2η
3η′
4η′
1η
2η
3η′
4η′
1η
2η
3η′
4η′
![Page 116: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/116.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200722 / 69
1η
2η
3η′
4η′
1η
2η
3η′
4η′
1η
2η
3η′
4η′
1η
2η
3η′
4η′
![Page 117: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/117.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200722 / 69
1η
2η
3η′
4η′
1η
2η
3η′
4η′
1η
2η
3η′
4η′
1η
2η
3η′
4η′
1η
2η
3η′
4η′
![Page 118: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/118.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200722 / 69
1η
2η
3η′
4η′
1η
2η
3η′
4η′
1η
2η
3η′
4η′
1η
2η
3η′
4η′
1η
2η
3η′
4η′
1η
2η
3η′
4η′
![Page 119: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/119.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200722 / 69
1η
2η
3η′
4η′
1η
2η
3η′
4η′
1η
2η
3η′
4η′
1η
2η
3η′
4η′
1η
2η
3η′
4η′
1η
2η
3η′
4η′
1η
2η
3η′
4η′
![Page 120: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/120.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200722 / 69
1η
2η
3η′
4η′
1η
2η
3η′
4η′
1η
2η
3η′
4η′
1η
2η
3η′
4η′
1η
2η
3η′
4η′
1η
2η
3η′
4η′
1η
2η
3η′
4η′
1η
2η
3η′
4η′
![Page 121: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/121.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200722 / 69
1η
2η
3η′
4η′
1η
2η
3η′
4η′
1η
2η
3η′
4η′
1η
2η
3η′
4η′
1η
2η
3η′
4η′
1η
2η
3η′
4η′
1η
2η
3η′
4η′
1η
2η
3η′
4η′
1η
2η
3η′
4η′
![Page 122: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/122.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200722 / 69
1η
2η
3η′
4η′
1η
2η
3η′
4η′
1η
2η
3η′
4η′
1η
2η
3η′
4η′
1η
2η
3η′
4η′
1η
2η
3η′
4η′
1η
2η
3η′
4η′
1η
2η
3η′
4η′
1η
2η
3η′
4η′
1η
2η
3η′
4η′
![Page 123: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/123.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200722 / 69
1η
2η
3η′
4η′
1η
2η
3η′
4η′
1η
2η
3η′
4η′
1η
2η
3η′
4η′
1η
2η
3η′
4η′
1η
2η
3η′
4η′
1η
2η
3η′
4η′
1η
2η
3η′
4η′
1η
2η
3η′
4η′
1η
2η
3η′
4η′
1η
2η
3η′
4η′
L. Georgiev [INRNE-Sofia] Theory of Elementary Particles Support: EUCLID, NCSR-BG
![Page 124: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/124.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200723 / 69
Exchange matrices for 4 Pfaffian quasiholes
� Use analytic continuation properties of Ising 4-pointfunctions
p1 ˙
px with x D
�14�23�13�24
![Page 125: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/125.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200723 / 69
Exchange matrices for 4 Pfaffian quasiholes
� Use analytic continuation properties of Ising 4-pointfunctions
p1 ˙
px with x D
�14�23�13�24
� The single-valued functions .ab;cd/ also contributebecause the braid transformations exchange points
![Page 126: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/126.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200723 / 69
Exchange matrices for 4 Pfaffian quasiholes
� Use analytic continuation properties of Ising 4-pointfunctions
p1 ˙
px with x D
�14�23�13�24
� The single-valued functions .ab;cd/ also contributebecause the braid transformations exchange points
� Finite 2-dimensional representation of the braid group B4
R.4/12
D R.4/34
D
�1 0
0 i
�; R
.4/23
Dei �
4
p2
�1 �i
�i 1
�L. Georgiev [INRNE-Sofia] Theory of Elementary Particles Support: EUCLID, NCSR-BG
![Page 127: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/127.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200724 / 69
Details:
![Page 128: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/128.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200724 / 69
Details: consider �1 ! �2, �2 ! �1
![Page 129: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/129.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200724 / 69
Details: consider �1 ! �2, �2 ! �1 so that x ! 1=x
![Page 130: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/130.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200724 / 69
Details: consider �1 ! �2, �2 ! �1 so that x ! 1=x and.14/.23/ $ .13/.24/
![Page 131: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/131.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200724 / 69
Details: consider �1 ! �2, �2 ! �1 so that x ! 1=x and.14/.23/ $ .13/.24/
R.4/12.0;1/4qh D
.�23�14/14r
1 ˙
q1x
�.14/.23/ ˙
1p
x.13/.24/
�:
![Page 132: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/132.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200724 / 69
Details: consider �1 ! �2, �2 ! �1 so that x ! 1=x and.14/.23/ $ .13/.24/
R.4/12.0;1/4qh D
.�23�14/14r
1 ˙
q1x
�.14/.23/ ˙
1p
x.13/.24/
�:
� Express �23�14 D �13�24 x
![Page 133: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/133.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200724 / 69
Details: consider �1 ! �2, �2 ! �1 so that x ! 1=x and.14/.23/ $ .13/.24/
R.4/12.0;1/4qh D
.�23�14/14r
1 ˙
q1x
�.14/.23/ ˙
1p
x.13/.24/
�:
� Express �23�14 D �13�24 x
� Take outq
˙p
1=x from the denominator
![Page 134: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/134.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200724 / 69
Details: consider �1 ! �2, �2 ! �1 so that x ! 1=x and.14/.23/ $ .13/.24/
R.4/12.0;1/4qh D
.�23�14/14r
1 ˙
q1x
�.14/.23/ ˙
1p
x.13/.24/
�:
� Express �23�14 D �13�24 x
� Take outq
˙p
1=x from the denominator
� Sign ambiguity resloved either by fusion consistency or byinvoking Yang–Baxter equations
L. Georgiev [INRNE-Sofia] Theory of Elementary Particles Support: EUCLID, NCSR-BG
![Page 135: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/135.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200725 / 69
Next, we first exchange �1 with �3 to find R.4/13
and then use
R.4/23
D R.4/12
R.4/13
�R.4/12
��1
![Page 136: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/136.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200725 / 69
Next, we first exchange �1 with �3 to find R.4/13
and then use
R.4/23
D R.4/12
R.4/13
�R.4/12
��1
�1 ! �3; �3 ! �1
![Page 137: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/137.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200725 / 69
Next, we first exchange �1 with �3 to find R.4/13
and then use
R.4/23
D R.4/12
R.4/13
�R.4/12
��1
�1 ! �3; �3 ! �1; �13 ! ei��13;
![Page 138: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/138.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200725 / 69
Next, we first exchange �1 with �3 to find R.4/13
and then use
R.4/23
D R.4/12
R.4/13
�R.4/12
��1
�1 ! �3; �3 ! �1; �13 ! ei��13; x ! Qx D 1 � x
and exchanges the single-valued functions .ab/.cd/
.13/.24/ ! .13/.24/; while .14/.23/ ! .12/.34/:
![Page 139: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/139.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200725 / 69
Next, we first exchange �1 with �3 to find R.4/13
and then use
R.4/23
D R.4/12
R.4/13
�R.4/12
��1
�1 ! �3; �3 ! �1; �13 ! ei��13; x ! Qx D 1 � x
and exchanges the single-valued functions .ab/.cd/
.13/.24/ ! .13/.24/; while .14/.23/ ! .12/.34/:
L. Georgiev [INRNE-Sofia] Theory of Elementary Particles Support: EUCLID, NCSR-BG
![Page 140: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/140.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200726 / 69
Then use the identity [Nayak–Wilczek]
.1 � x/.12/.34/ D .13/.24/ � x .14/.23/;
![Page 141: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/141.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200726 / 69
Then use the identity [Nayak–Wilczek]
.1 � x/.12/.34/ D .13/.24/ � x .14/.23/; as well asq1 C
p1 � Qx ˙
q1 �
p1 � Qx D ˙
p2
q1 ˙
pQx;
![Page 142: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/142.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200726 / 69
Then use the identity [Nayak–Wilczek]
.1 � x/.12/.34/ D .13/.24/ � x .14/.23/; as well asq1 C
p1 � Qx ˙
q1 �
p1 � Qx D ˙
p2
q1 ˙
pQx;
to find
R.4/13
".0/4qh
.1/4qh
#D
ei �4
p2
�1 1
�1 1
�".0/4qh
.1/4qh
#:
![Page 143: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/143.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200726 / 69
Then use the identity [Nayak–Wilczek]
.1 � x/.12/.34/ D .13/.24/ � x .14/.23/; as well asq1 C
p1 � Qx ˙
q1 �
p1 � Qx D ˙
p2
q1 ˙
pQx;
to find
R.4/13
".0/4qh
.1/4qh
#D
ei �4
p2
�1 1
�1 1
�".0/4qh
.1/4qh
#:
fix sign ambiguity by�R.4/23
�2DNOT
![Page 144: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/144.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200726 / 69
Then use the identity [Nayak–Wilczek]
.1 � x/.12/.34/ D .13/.24/ � x .14/.23/; as well asq1 C
p1 � Qx ˙
q1 �
p1 � Qx D ˙
p2
q1 ˙
pQx;
to find
R.4/13
".0/4qh
.1/4qh
#D
ei �4
p2
�1 1
�1 1
�".0/4qh
.1/4qh
#:
fix sign ambiguity by�R.4/23
�2DNOT
L. Georgiev [INRNE-Sofia] Theory of Elementary Particles Support: EUCLID, NCSR-BG
![Page 145: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/145.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200727 / 69
Finite monodromy group generated by �1 and �3�R.4/12
�2D
�1 0
0 �1
�;
�R.4/23
�2D
�0 1
1 0
�;
isomorphic to the symmetry group of the square (8elements) typically given in the 2D rep. as
D4 � f˙I2; ˙�1; ˙i�2; ˙�3g;
![Page 146: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/146.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200727 / 69
Finite monodromy group generated by �1 and �3�R.4/12
�2D
�1 0
0 �1
�;
�R.4/23
�2D
�0 1
1 0
�;
isomorphic to the symmetry group of the square (8elements) typically given in the 2D rep. as
D4 � f˙I2; ˙�1; ˙i�2; ˙�3g;
Monodromy group is normal subgroup of braid group and
Bn=Mn ' Sn
![Page 147: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/147.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200727 / 69
Finite monodromy group generated by �1 and �3�R.4/12
�2D
�1 0
0 �1
�;
�R.4/23
�2D
�0 1
1 0
�;
isomorphic to the symmetry group of the square (8elements) typically given in the 2D rep. as
D4 � f˙I2; ˙�1; ˙i�2; ˙�3g;
Monodromy group is normal subgroup of braid group and
Bn=Mn ' Sn
Finite representation of Mn ” finite repr. of Bn
![Page 148: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/148.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200727 / 69
Finite monodromy group generated by �1 and �3�R.4/12
�2D
�1 0
0 �1
�;
�R.4/23
�2D
�0 1
1 0
�;
isomorphic to the symmetry group of the square (8elements) typically given in the 2D rep. as
D4 � f˙I2; ˙�1; ˙i�2; ˙�3g;
Monodromy group is normal subgroup of braid group and
Bn=Mn ' Sn
Finite representation of Mn ” finite repr. of Bn
L. Georgiev [INRNE-Sofia] Theory of Elementary Particles Support: EUCLID, NCSR-BG
![Page 149: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/149.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200728 / 69
� Braid group (Artin) relations: Bi D Ri;iC1 2 Bn
BiBj D Bj Bi ; for ji � j j � 2
BiBiC1Bi D BiC1BiBiC1; i D 1; : : : ; n � 1
![Page 150: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/150.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200728 / 69
� Braid group (Artin) relations: Bi D Ri;iC1 2 Bn
BiBj D Bj Bi ; for ji � j j � 2
BiBiC1Bi D BiC1BiBiC1; i D 1; : : : ; n � 1
� Projective representation: for braid group on the sphere
B1B2 � � � Bn�2B2n�1Bn�2 � � � B2B1 D I:
is only up to phase
![Page 151: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/151.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200728 / 69
� Braid group (Artin) relations: Bi D Ri;iC1 2 Bn
BiBj D Bj Bi ; for ji � j j � 2
BiBiC1Bi D BiC1BiBiC1; i D 1; : : : ; n � 1
� Projective representation: for braid group on the sphere
B1B2 � � � Bn�2B2n�1Bn�2 � � � B2B1 D I:
is only up to phase
Dimino’s algorithm: given the generators of a finite group,enumerate the group elements [Lou Dimino, Bell Labs.1971]
L. Georgiev [INRNE-Sofia] Theory of Elementary Particles Support: EUCLID, NCSR-BG
![Page 152: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/152.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200729 / 69
For the group B4 Dimino’s algorithm gives [Maple withmodules]
jImage .B4/ j D 96:
![Page 153: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/153.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200729 / 69
For the group B4 Dimino’s algorithm gives [Maple withmodules]
jImage .B4/ j D 96:
For general 2n � 6 [Read]
jImage .B2n/ j D
(22n�1.2n/! for n D even22n.2n/! for n D odd
:
![Page 154: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/154.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200729 / 69
For the group B4 Dimino’s algorithm gives [Maple withmodules]
jImage .B4/ j D 96:
For general 2n � 6 [Read]
jImage .B2n/ j D
(22n�1.2n/! for n D even22n.2n/! for n D odd
:
� Too bad for TQC:
![Page 155: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/155.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200729 / 69
For the group B4 Dimino’s algorithm gives [Maple withmodules]
jImage .B4/ j D 96:
For general 2n � 6 [Read]
jImage .B2n/ j D
(22n�1.2n/! for n D even22n.2n/! for n D odd
:
� Too bad for TQC: for universal TQC we need Image .B2n/
to be dense in the unitary group
L. Georgiev [INRNE-Sofia] Theory of Elementary Particles Support: EUCLID, NCSR-BG
![Page 156: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/156.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200730 / 69
Single-qubit gates: The Pauli X gate
The NOT gate [Das Sarma et al.] X � R223
D
�0 1
1 0
�
43
21
X
L. Georgiev [INRNE-Sofia] Theory of Elementary Particles Support: EUCLID, NCSR-BG
![Page 157: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/157.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200731 / 69
Single-qubit gates: the Hadamard gate
H ' R212R13 D R12R23R12 D
ei �4
p2
�1 1
1 �1
�:
43
2
1
H
L. Georgiev [INRNE-Sofia] Theory of Elementary Particles Support: EUCLID, NCSR-BG
![Page 158: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/158.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200732 / 69
Single-qubit gates: the Pauli Y gate
(defined in QC without i ) Y � R�112
R223
R12 D
�0 1
�1 0
�
43
21
Y
L. Georgiev [INRNE-Sofia] Theory of Elementary Particles Support: EUCLID, NCSR-BG
![Page 159: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/159.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200733 / 69
Single-qubit gates: the Pauli Z gate
Z � R212
D R234
D
�1 0
0 �1
�
43
21
Z
L. Georgiev [INRNE-Sofia] Theory of Elementary Particles Support: EUCLID, NCSR-BG
![Page 160: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/160.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200734 / 69
Single-qubit gates: the phase gate S
S � R12 D R34 D
�1 0
0 i
�
43
21
S
L. Georgiev [INRNE-Sofia] Theory of Elementary Particles Support: EUCLID, NCSR-BG
![Page 161: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/161.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200735 / 69
Construct the �=8 gate T by braiding? No Way!
![Page 162: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/162.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200735 / 69
Construct the �=8 gate T by braiding? No Way!
T D
�1 0
0 ei�=4
�) det T D ei�=4;
![Page 163: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/163.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200735 / 69
Construct the �=8 gate T by braiding? No Way!
T D
�1 0
0 ei�=4
�) det T D ei�=4;
� Generators: det R.4/a;aC1
D i
� ) T is not expressible by products of R.4/a;aC1
:
![Page 164: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/164.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200735 / 69
Construct the �=8 gate T by braiding? No Way!
T D
�1 0
0 ei�=4
�) det T D ei�=4;
� Generators: det R.4/a;aC1
D i
� ) T is not expressible by products of R.4/a;aC1
:
� ) No universality!
L. Georgiev [INRNE-Sofia] Theory of Elementary Particles Support: EUCLID, NCSR-BG
![Page 165: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/165.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200736 / 69
Two-qubits construction and two-qubit gatesdim H2n D 2n�1;
![Page 166: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/166.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200736 / 69
Two-qubits construction and two-qubit gatesdim H2n D 2n�1; j0i ' �C�C � I, j1i ' �C�� �
![Page 167: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/167.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200736 / 69
Two-qubits construction and two-qubit gatesdim H2n D 2n�1; j0i ' �C�C � I, j1i ' �C�� �
1η 2η 3η 4η 5η 6η
qubit 1 qubit 2
![Page 168: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/168.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200736 / 69
Two-qubits construction and two-qubit gatesdim H2n D 2n�1; j0i ' �C�C � I, j1i ' �C�� �
1η 2η 3η 4η 5η 6η
qubit 1 qubit 2
j00i � h�C�C�C�C�C�Ci; j01i � h�C�C�C���C��i
j10i � h�C���C���C�Ci; j11i � h�C���C�C�C��i
L. Georgiev [INRNE-Sofia] Theory of Elementary Particles Support: EUCLID, NCSR-BG
![Page 169: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/169.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200737 / 69
Exchange matrices for 6-quasiholesCompute R
.6/12
:
![Page 170: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/170.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200737 / 69
Exchange matrices for 6-quasiholesCompute R
.6/12
: first fuse �5 ! �6
![Page 171: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/171.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200737 / 69
Exchange matrices for 6-quasiholesCompute R
.6/12
: first fuse �5 ! �6
j00i !�5!�6
h�C�C�C�Ci; j01i !�5!�6
h�C�C�C�� i
j10i !�5!�6
h�C���C��i; j11i !�5!�6
h�C���C�C i
![Page 172: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/172.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200737 / 69
Exchange matrices for 6-quasiholesCompute R
.6/12
: first fuse �5 ! �6
j00i !�5!�6
h�C�C�C�Ci; j01i !�5!�6
h�C�C�C�� i
j10i !�5!�6
h�C���C��i; j11i !�5!�6
h�C���C�C i
Then �1 $ �2 is represented by R.4/12
![Page 173: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/173.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200737 / 69
Exchange matrices for 6-quasiholesCompute R
.6/12
: first fuse �5 ! �6
j00i !�5!�6
h�C�C�C�Ci; j01i !�5!�6
h�C�C�C�� i
j10i !�5!�6
h�C���C��i; j11i !�5!�6
h�C���C�C i
Then �1 $ �2 is represented by R.4/12
R.6/12
D
26641 0 0 0
0 1 0 0
0 0 i 0
0 0 0 i
3775 D R.4/12
˝ I2:
L. Georgiev [INRNE-Sofia] Theory of Elementary Particles Support: EUCLID, NCSR-BG
![Page 174: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/174.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200738 / 69
Compute R.6/23
:
![Page 175: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/175.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200738 / 69
Compute R.6/23
: fuse �5 ! �6
R.6/23
Dei �
4
p2
26641 0 �i 0
0 1 0 �i
�i 0 1 0
0 �i 0 1
3775 D R.4/23
˝ I2:
![Page 176: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/176.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200738 / 69
Compute R.6/23
: fuse �5 ! �6
R.6/23
Dei �
4
p2
26641 0 �i 0
0 1 0 �i
�i 0 1 0
0 �i 0 1
3775 D R.4/23
˝ I2:
In the same way, by fusing first �1 ! �2, we obtain
![Page 177: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/177.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200738 / 69
Compute R.6/23
: fuse �5 ! �6
R.6/23
Dei �
4
p2
26641 0 �i 0
0 1 0 �i
�i 0 1 0
0 �i 0 1
3775 D R.4/23
˝ I2:
In the same way, by fusing first �1 ! �2, we obtain
R.6/45
Dei �
4
p2
26641 �i 0 0
�i 1 0 0
0 0 1 �i
0 0 �i 1
3775 D I2 ˝ R.4/23; and
L. Georgiev [INRNE-Sofia] Theory of Elementary Particles Support: EUCLID, NCSR-BG
![Page 178: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/178.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200739 / 69
R.6/56
D
26641 0 0 0
0 i 0 0
0 0 1 0
0 0 0 i
3775 D I2 ˝ R.4/34
![Page 179: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/179.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200739 / 69
R.6/56
D
26641 0 0 0
0 i 0 0
0 0 1 0
0 0 0 i
3775 D I2 ˝ R.4/34
� Doesn’t work for R.6/34
(not factorizable)
![Page 180: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/180.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200739 / 69
R.6/56
D
26641 0 0 0
0 i 0 0
0 0 1 0
0 0 0 i
3775 D I2 ˝ R.4/34
� Doesn’t work for R.6/34
(not factorizable)
� However: R.6/34
diagonal due to NS-sector supersel. rule
![Page 181: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/181.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200739 / 69
R.6/56
D
26641 0 0 0
0 i 0 0
0 0 1 0
0 0 0 i
3775 D I2 ˝ R.4/34
� Doesn’t work for R.6/34
(not factorizable)
� However: R.6/34
diagonal due to NS-sector supersel. rule
� ) Obtain R.6/34
directly from the OPE
j00i !�3!�4
��1=834
h�C�C�C�Ci; j01i !�3!�4
�3=834
h�C�C �C��i
j10i !�3!�4
�3=834
h�C�� �C�Ci; j11i !�3!�4
��1=834
h�C���C��i
L. Georgiev [INRNE-Sofia] Theory of Elementary Particles Support: EUCLID, NCSR-BG
![Page 182: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/182.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200740 / 69
�3 $ �4 ) R.6/34
D
26641 0 0 0
0 i 0 0
0 0 i 0
0 0 0 1
3775
![Page 183: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/183.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200740 / 69
�3 $ �4 ) R.6/34
D
26641 0 0 0
0 i 0 0
0 0 i 0
0 0 0 1
3775� Remember: Not all 6-q.h. elementary exchanges aretensor products of the 4-q.h. ones!
![Page 184: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/184.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200740 / 69
�3 $ �4 ) R.6/34
D
26641 0 0 0
0 i 0 0
0 0 i 0
0 0 0 1
3775� Remember: Not all 6-q.h. elementary exchanges aretensor products of the 4-q.h. ones!
� Important: creating entanglement purely by braiding
![Page 185: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/185.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200740 / 69
�3 $ �4 ) R.6/34
D
26641 0 0 0
0 i 0 0
0 0 i 0
0 0 0 1
3775� Remember: Not all 6-q.h. elementary exchanges aretensor products of the 4-q.h. ones!
� Important: creating entanglement purely by braiding
� Explain this in the universal R matrix approach
![Page 186: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/186.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200740 / 69
�3 $ �4 ) R.6/34
D
26641 0 0 0
0 i 0 0
0 0 i 0
0 0 0 1
3775� Remember: Not all 6-q.h. elementary exchanges aretensor products of the 4-q.h. ones!
� Important: creating entanglement purely by braiding
� Explain this in the universal R matrix approach
� Dimino: jImage .B6/ j D 46080, jImage .M6/ j D 32
L. Georgiev [INRNE-Sofia] Theory of Elementary Particles Support: EUCLID, NCSR-BG
![Page 187: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/187.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200741 / 69
Single-qubit gates in the two-qubit basis� Hadamard gates acting on first and second qubits:
H1 ' H ˝ I2 D
�R.6/12
��1 �R.6/23
��1 �R.6/12
��1D
De�i �
4
p2
26641 0 1 0
0 1 0 1
1 0 �1 0
0 1 0 �1
3775 ;L. Georgiev [INRNE-Sofia] Theory of Elementary Particles Support: EUCLID, NCSR-BG
![Page 188: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/188.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200742 / 69
H2 ' I2 ˝ H D R.6/56
R.6/45
R.6/56
Dei �
4
p2
26641 1 0 0
1 �1 0 0
0 0 1 1
0 0 1 �1
3775 :
![Page 189: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/189.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200742 / 69
H2 ' I2 ˝ H D R.6/56
R.6/45
R.6/56
Dei �
4
p2
26641 1 0 0
1 �1 0 0
0 0 1 1
0 0 1 �1
3775 :� Phase gates on first and second qubits:
S1 D S ˝ I2 D R.6/12
![Page 190: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/190.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200742 / 69
H2 ' I2 ˝ H D R.6/56
R.6/45
R.6/56
Dei �
4
p2
26641 1 0 0
1 �1 0 0
0 0 1 1
0 0 1 �1
3775 :� Phase gates on first and second qubits:
S1 D S ˝ I2 D R.6/12
and S2 D I2 ˝ S D R.6/56:
L. Georgiev [INRNE-Sofia] Theory of Elementary Particles Support: EUCLID, NCSR-BG
![Page 191: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/191.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200743 / 69
Controlled-Z and Controlled-NOT gates
� General construction: [Nielsen–Chuang]
CZ D ei �4 ei �
4Z1Z2e�i �
4Z1e�i �
4Z2;
where Z1 D Z ˝ I2 and Z2 D I2 ˝ Z
![Page 192: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/192.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200743 / 69
Controlled-Z and Controlled-NOT gates
� General construction: [Nielsen–Chuang]
CZ D ei �4 ei �
4Z1Z2e�i �
4Z1e�i �
4Z2;
where Z1 D Z ˝ I2 and Z2 D I2 ˝ Z
� Recall:
e˙i �4A
D cos��
4
�I2 ˙ iA sin
��4
�; for A2
D I2
L. Georgiev [INRNE-Sofia] Theory of Elementary Particles Support: EUCLID, NCSR-BG
![Page 193: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/193.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200744 / 69
� Observe:
ei �4Z1Z2 D ei �
4
�R.6/34
�|;
e�i �4Z1 D e�i �
4 R.6/12; e�i �
4Z2 D e�i �
4 R.6/56
![Page 194: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/194.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200744 / 69
� Observe:
ei �4Z1Z2 D ei �
4
�R.6/34
�|;
e�i �4Z1 D e�i �
4 R.6/12; e�i �
4Z2 D e�i �
4 R.6/56
� Therefore:
CZ D R.6/12
�R.6/34
��1R.6/56
D
26641 0 0 0
0 1 0 0
0 0 1 0
0 0 0 �1
3775
L. Georgiev [INRNE-Sofia] Theory of Elementary Particles Support: EUCLID, NCSR-BG
![Page 195: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/195.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200745 / 69
� Braid diagram for CZ:
1
2
3
4
56
Z
![Page 196: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/196.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200745 / 69
� Braid diagram for CZ:
1
2
3
4
56
Z
� Amazing: 100 % topological protection for CZ !
![Page 197: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/197.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200745 / 69
� Braid diagram for CZ:
1
2
3
4
56
Z
� Amazing: 100 % topological protection for CZ !� How can this be?
![Page 198: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/198.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200745 / 69
� Braid diagram for CZ:
1
2
3
4
56
Z
� Amazing: 100 % topological protection for CZ !� How can this be?
� Topological entanglement built into R.6/34
L. Georgiev [INRNE-Sofia] Theory of Elementary Particles Support: EUCLID, NCSR-BG
![Page 199: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/199.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200746 / 69
� CNOT is now straightforward
CNOT D H2 CZ H2 D R56R45R�156 R�1
34 R12R45R56
'
26641 0 0 0
0 1 0 0
0 0 0 1
0 0 1 0
3775
L. Georgiev [INRNE-Sofia] Theory of Elementary Particles Support: EUCLID, NCSR-BG
![Page 200: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/200.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200747 / 69
� Alternatively
CNOT D R�134 R45R34R12R56R45R�1
34 :
1
2
3
4
56
� N.B.: 100 % topological protection for CNOT� Just 7 elementary braids!
L. Georgiev [INRNE-Sofia] Theory of Elementary Particles Support: EUCLID, NCSR-BG
![Page 201: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/201.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200748 / 69
The two-qubit SWAP gate: deceptively simple
Exchange two neighbouring qubits (answer to S. Bravyi)
![Page 202: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/202.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200748 / 69
The two-qubit SWAP gate: deceptively simple
Exchange two neighbouring qubits (answer to S. Bravyi)
1η 2η 3η 4η 5η 6η��������������
![Page 203: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/203.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200748 / 69
The two-qubit SWAP gate: deceptively simple
Exchange two neighbouring qubits (answer to S. Bravyi)
1η 2η 3η 4η 5η 6η��������������
1η 2η 3η 4η 5η 6η��������������
23R
![Page 204: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/204.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200748 / 69
The two-qubit SWAP gate: deceptively simple
Exchange two neighbouring qubits (answer to S. Bravyi)
1η 2η 3η 4η 5η 6η��������������
1η 2η 3η 4η 5η 6η��������������
23R
1η 2η 3η 4η 5η 6η��������������
23R12R
![Page 205: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/205.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200748 / 69
The two-qubit SWAP gate: deceptively simple
Exchange two neighbouring qubits (answer to S. Bravyi)
1η 2η 3η 4η 5η 6η��������������
1η 2η 3η 4η 5η 6η��������������
23R
1η 2η 3η 4η 5η 6η��������������
23R12R
1η 2η3η 4η 5η 6η��������������
23R12R
![Page 206: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/206.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200748 / 69
The two-qubit SWAP gate: deceptively simple
Exchange two neighbouring qubits (answer to S. Bravyi)
1η 2η 3η 4η 5η 6η��������������
1η 2η 3η 4η 5η 6η��������������
23R
1η 2η 3η 4η 5η 6η��������������
23R12R
1η 2η3η 4η 5η 6η��������������
23R12R
1η 2η3η 4η 5η 6η��������������
![Page 207: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/207.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200748 / 69
The two-qubit SWAP gate: deceptively simple
Exchange two neighbouring qubits (answer to S. Bravyi)
1η 2η 3η 4η 5η 6η��������������
1η 2η 3η 4η 5η 6η��������������
23R
1η 2η 3η 4η 5η 6η��������������
23R12R
1η 2η3η 4η 5η 6η��������������
23R12R
1η 2η3η 4η 5η 6η��������������
1η 2η3η 4η 5η 6η��������������
34R
![Page 208: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/208.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200748 / 69
The two-qubit SWAP gate: deceptively simple
Exchange two neighbouring qubits (answer to S. Bravyi)
1η 2η 3η 4η 5η 6η��������������
1η 2η 3η 4η 5η 6η��������������
23R
1η 2η 3η 4η 5η 6η��������������
23R12R
1η 2η3η 4η 5η 6η��������������
23R12R
1η 2η3η 4η 5η 6η��������������
1η 2η3η 4η 5η 6η��������������
34R
1η 2η3η 4η 5η 6η��������������
34R23R
![Page 209: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/209.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200748 / 69
The two-qubit SWAP gate: deceptively simple
Exchange two neighbouring qubits (answer to S. Bravyi)
1η 2η 3η 4η 5η 6η��������������
1η 2η 3η 4η 5η 6η��������������
23R
1η 2η 3η 4η 5η 6η��������������
23R12R
1η 2η3η 4η 5η 6η��������������
23R12R
1η 2η3η 4η 5η 6η��������������
1η 2η3η 4η 5η 6η��������������
34R
1η 2η3η 4η 5η 6η��������������
34R23R
1η 2η3η 4η 5η 6η��������������
34R23R
![Page 210: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/210.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200748 / 69
The two-qubit SWAP gate: deceptively simple
Exchange two neighbouring qubits (answer to S. Bravyi)
1η 2η 3η 4η 5η 6η��������������
1η 2η 3η 4η 5η 6η��������������
23R
1η 2η 3η 4η 5η 6η��������������
23R12R
1η 2η3η 4η 5η 6η��������������
23R12R
1η 2η3η 4η 5η 6η��������������
1η 2η3η 4η 5η 6η��������������
34R
1η 2η3η 4η 5η 6η��������������
34R23R
1η 2η3η 4η 5η 6η��������������
34R23R
1η 2η3η 4η 5η 6η��������������
![Page 211: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/211.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200748 / 69
The two-qubit SWAP gate: deceptively simple
Exchange two neighbouring qubits (answer to S. Bravyi)
1η 2η 3η 4η 5η 6η��������������
1η 2η 3η 4η 5η 6η��������������
23R
1η 2η 3η 4η 5η 6η��������������
23R12R
1η 2η3η 4η 5η 6η��������������
23R12R
1η 2η3η 4η 5η 6η��������������
1η 2η3η 4η 5η 6η��������������
34R
1η 2η3η 4η 5η 6η��������������
34R23R
1η 2η3η 4η 5η 6η��������������
34R23R
1η 2η3η 4η 5η 6η��������������
1η 2η3η 4η 5η 6η������� �������
![Page 212: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/212.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200748 / 69
The two-qubit SWAP gate: deceptively simple
Exchange two neighbouring qubits (answer to S. Bravyi)
1η 2η 3η 4η 5η 6η��������������
1η 2η 3η 4η 5η 6η��������������
23R
1η 2η 3η 4η 5η 6η��������������
23R12R
1η 2η3η 4η 5η 6η��������������
23R12R
1η 2η3η 4η 5η 6η��������������
1η 2η3η 4η 5η 6η��������������
34R
1η 2η3η 4η 5η 6η��������������
34R23R
1η 2η3η 4η 5η 6η��������������
34R23R
1η 2η3η 4η 5η 6η��������������
1η 2η3η 4η 5η 6η������� �������
1η 2η3η 4η 5η 6η������� �������
23123423 RRRRU =
L. Georgiev [INRNE-Sofia] Theory of Elementary Particles Support: EUCLID, NCSR-BG
![Page 213: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/213.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200749 / 69
U.1;2/$.3;4/ D R23R34R12R23 D i
26641 0 0 0
0 0 1 0
0 1 0 0
0 0 0 �1
3775
![Page 214: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/214.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200749 / 69
U.1;2/$.3;4/ D R23R34R12R23 D i
26641 0 0 0
0 0 1 0
0 1 0 0
0 0 0 �1
3775� Problem: extra "-" in j11i ! �j11i
![Page 215: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/215.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200749 / 69
U.1;2/$.3;4/ D R23R34R12R23 D i
26641 0 0 0
0 0 1 0
0 1 0 0
0 0 0 �1
3775� Problem: extra "-" in j11i ! �j11i (top. entanglement)
![Page 216: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/216.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200749 / 69
U.1;2/$.3;4/ D R23R34R12R23 D i
26641 0 0 0
0 0 1 0
0 1 0 0
0 0 0 �1
3775� Problem: extra "-" in j11i ! �j11i (top. entanglement)� A way out: use CZ D diag.1; 1; 1;�1/
![Page 217: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/217.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200749 / 69
U.1;2/$.3;4/ D R23R34R12R23 D i
26641 0 0 0
0 0 1 0
0 1 0 0
0 0 0 �1
3775� Problem: extra "-" in j11i ! �j11i (top. entanglement)� A way out: use CZ D diag.1; 1; 1;�1/ D R12R34R�1
56
![Page 218: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/218.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200749 / 69
U.1;2/$.3;4/ D R23R34R12R23 D i
26641 0 0 0
0 0 1 0
0 1 0 0
0 0 0 �1
3775� Problem: extra "-" in j11i ! �j11i (top. entanglement)� A way out: use CZ D diag.1; 1; 1;�1/ D R12R34R�1
56
SWAP D
�R12R34R�1
56
�R23R34R12R23 D i
26641 0 0 0
0 0 1 0
0 1 0 0
0 0 0 1
3775L. Georgiev [INRNE-Sofia] Theory of Elementary Particles Support: EUCLID, NCSR-BG
![Page 219: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/219.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200750 / 69
Three qubits and three-qubit gates:
1η 2η 3η 4η 5η 6η
qubit 1 qubit 2 qubit 3
8η7η
� Fermion parity conservation:
e3e4 D e1e2e5e6e7e8 ) #states D 8:
L. Georgiev [INRNE-Sofia] Theory of Elementary Particles Support: EUCLID, NCSR-BG
![Page 220: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/220.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200751 / 69
� Computational basis:
j000i � h�C�C�C�C�C�C�C�Ci;
j001i � h�C�C�C���C�C�C��i;
j010i � h�C�C�C���C���C�Ci;
j011i � h�C�C�C�C�C���C��i;
j100i � h�C���C���C�C�C�Ci;
j101i � h�C���C�C�C�C�C��i;
j110i � h�C���C�C�C���C�Ci;
j111i � h�C���C���C���C��i:
L. Georgiev [INRNE-Sofia] Theory of Elementary Particles Support: EUCLID, NCSR-BG
![Page 221: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/221.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200752 / 69
Exchange matrices for 8 quasiholes
� Recursive: obtain from R.6/a;aC1
� Fuse �7 ! �8 or �5 ! �6 or �3 ! �4 etc.
R.8/12
D diag.1; 1; 1; 1; i; i; i; i/ D R.6/12
˝ I2;
L. Georgiev [INRNE-Sofia] Theory of Elementary Particles Support: EUCLID, NCSR-BG
![Page 222: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/222.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200753 / 69
R.8/23
Dei �
4
p2
2666666666664
1 0 0 0 �i 0 0 0
0 1 0 0 0 �i 0 0
0 0 1 0 0 0 �i 0
0 0 0 1 0 0 0 �i
�i 0 0 0 1 0 0 0
0 �i 0 0 0 1 0 0
0 0 �i 0 0 0 1 0
0 0 0 �i 0 0 0 1
3777777777775D R
.6/23
˝ I2;
R.8/34
D diag.1; i; i; 1; i; 1; 1; i/;
L. Georgiev [INRNE-Sofia] Theory of Elementary Particles Support: EUCLID, NCSR-BG
![Page 223: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/223.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200754 / 69
R.8/45
Dei �
4
p2
2666666666664
1 0 �i 0 0 0 0 0
0 1 0 �i 0 0 0 0
�i 0 1 0 0 0 0 0
0 �i 0 1 0 0 0 0
0 0 0 0 1 0 �i 0
0 0 0 0 0 1 0 �i
0 0 0 0 �i 0 1 0
0 0 0 0 0 �i 0 1
3777777777775D R
.6/45
˝ I2;
R.8/56
D diag.1; 1; i; i; 1; 1; i; i/ D R.6/56
˝ I2;
L. Georgiev [INRNE-Sofia] Theory of Elementary Particles Support: EUCLID, NCSR-BG
![Page 224: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/224.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200755 / 69
R.8/67
Dei �
4
p2
2666666666664
1 0 0 �i 0 0 0 0
0 1 �i 0 0 0 0 0
0 �i 1 0 0 0 0 0
�i 0 0 1 0 0 0 0
0 0 0 0 1 0 0 �i
0 0 0 0 0 1 �i 0
0 0 0 0 0 �i 1 0
0 0 0 0 �i 0 0 1
3777777777775;
R.8/78
D diag.1; i; 1; i; 1; i; 1; i/ D I2 ˝ R.6/56:
� Artin relations for B8
![Page 225: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/225.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200755 / 69
R.8/67
Dei �
4
p2
2666666666664
1 0 0 �i 0 0 0 0
0 1 �i 0 0 0 0 0
0 �i 1 0 0 0 0 0
�i 0 0 1 0 0 0 0
0 0 0 0 1 0 0 �i
0 0 0 0 0 1 �i 0
0 0 0 0 0 �i 1 0
0 0 0 0 �i 0 0 1
3777777777775;
R.8/78
D diag.1; i; 1; i; 1; i; 1; i/ D I2 ˝ R.6/56:
� Artin relations for B8
jImage.B8/j D 5160960; jImage.M8/j D 128
L. Georgiev [INRNE-Sofia] Theory of Elementary Particles Support: EUCLID, NCSR-BG
![Page 226: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/226.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200756 / 69
The universal R matrix for Uq.sl.2//
� n-qubit spaces: 2.n C 1/ Ising anyons realize spinor IR’sof SO.2n C 2/ (of dimension 2n)
![Page 227: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/227.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200756 / 69
The universal R matrix for Uq.sl.2//
� n-qubit spaces: 2.n C 1/ Ising anyons realize spinor IR’sof SO.2n C 2/ (of dimension 2n)
� Braid matrices in terms of R for Uq.sl.2// with q D e�i �4
R.2nC2/k;kC1
D I ˝ � � � ˝ I„ ƒ‚ …k�1
˝R ˝ I ˝ � � � ˝ I
![Page 228: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/228.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200756 / 69
The universal R matrix for Uq.sl.2//
� n-qubit spaces: 2.n C 1/ Ising anyons realize spinor IR’sof SO.2n C 2/ (of dimension 2n)
� Braid matrices in terms of R for Uq.sl.2// with q D e�i �4
R.2nC2/k;kC1
D I ˝ � � � ˝ I„ ƒ‚ …k�1
˝R ˝ I ˝ � � � ˝ I
Spinor representation of the R matrix for the PfaffianFQH state (Ising model) [Nayak-Wilczek, Slingerland-Bais,Ivanov]:
L. Georgiev [INRNE-Sofia] Theory of Elementary Particles Support: EUCLID, NCSR-BG
![Page 229: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/229.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200757 / 69
� Case k odd: R acts only on the k-th factor
R D
�1 0
0 i
�
![Page 230: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/230.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200757 / 69
� Case k odd: R acts only on the k-th factor
R D
�1 0
0 i
�� Case k even: R acts on the k-th and .k C 1/-th factor
R D1
p2
26641 0 0 �i
0 1 �i 0
0 �i 1 0
�i 0 0 1
3775
L. Georgiev [INRNE-Sofia] Theory of Elementary Particles Support: EUCLID, NCSR-BG
![Page 231: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/231.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200758 / 69
N.B.: Project fj000i; j001i; j010i; j011i; j100i; j101i; j110i; j111i}to the two-qubit basis fj000i; j011i; j101i; j110ig
P2 WD
2666666666666664
1 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 1 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 1 0 0
0 0 0 0 0 0 1 0
0 0 0 0 0 0 0 0
3777777777777775L. Georgiev [INRNE-Sofia] Theory of Elementary Particles Support: EUCLID, NCSR-BG
![Page 232: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/232.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200759 / 69
� Factorizability lost after projection
R.6/34
D P2
�I2 ˝
�1 0
0 i
�˝ I2
�P2 D
26641 0 0 0
0 i 0 0
0 0 i 0
0 0 0 1
3775(N.B.: delete all null rows and columns)
![Page 233: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/233.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200759 / 69
� Factorizability lost after projection
R.6/34
D P2
�I2 ˝
�1 0
0 i
�˝ I2
�P2 D
26641 0 0 0
0 i 0 0
0 0 i 0
0 0 0 1
3775(N.B.: delete all null rows and columns)
� Artin’s relation follow from Yang–Baxter eqs
![Page 234: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/234.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200759 / 69
� Factorizability lost after projection
R.6/34
D P2
�I2 ˝
�1 0
0 i
�˝ I2
�P2 D
26641 0 0 0
0 i 0 0
0 0 i 0
0 0 0 1
3775(N.B.: delete all null rows and columns)
� Artin’s relation follow from Yang–Baxter eqs only iff theprojection commutes with I ˝ � � � ˝ I ˝ R ˝ I ˝ � � � ˝ I
![Page 235: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/235.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200759 / 69
� Factorizability lost after projection
R.6/34
D P2
�I2 ˝
�1 0
0 i
�˝ I2
�P2 D
26641 0 0 0
0 i 0 0
0 0 i 0
0 0 0 1
3775(N.B.: delete all null rows and columns)
� Artin’s relation follow from Yang–Baxter eqs only iff theprojection commutes with I ˝ � � � ˝ I ˝ R ˝ I ˝ � � � ˝ I
� Use universal R matrices for TQC [Kauffman]
![Page 236: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/236.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200759 / 69
� Factorizability lost after projection
R.6/34
D P2
�I2 ˝
�1 0
0 i
�˝ I2
�P2 D
26641 0 0 0
0 i 0 0
0 0 i 0
0 0 0 1
3775(N.B.: delete all null rows and columns)
� Artin’s relation follow from Yang–Baxter eqs only iff theprojection commutes with I ˝ � � � ˝ I ˝ R ˝ I ˝ � � � ˝ I
� Use universal R matrices for TQC [Kauffman]
� Oportunity for topological quantum entanglement
L. Georgiev [INRNE-Sofia] Theory of Elementary Particles Support: EUCLID, NCSR-BG
![Page 237: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/237.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200760 / 69
Embed 1-qubit and 2-qubit gates into 3-qubits
� Phase gates:
S1 � S ˝ I4 D R.8/12; S2 � I2 ˝ S ˝ I2 D R
.8/56;
S3 � I4 ˝ S D R.8/78
L. Georgiev [INRNE-Sofia] Theory of Elementary Particles Support: EUCLID, NCSR-BG
![Page 238: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/238.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200761 / 69
� Hadamard gates:
H1 D H ˝ I4 ' R�112 R�1
23 R�112
De�i �
4
p2
2666666666664
1 0 0 0 1 0 0 0
0 1 0 0 0 1 0 0
0 0 1 0 0 0 1 0
0 0 0 1 0 0 0 1
1 0 0 0 �1 0 0 0
0 1 0 0 0 �1 0 0
0 0 1 0 0 0 �1 0
0 0 0 1 0 0 0 �1
3777777777775L. Georgiev [INRNE-Sofia] Theory of Elementary Particles Support: EUCLID, NCSR-BG
![Page 239: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/239.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200762 / 69
H2 ' I2 ˝ H ˝ I2 ' R�156 R�1
45 R�156
De�i �
4
p2
2666666666664
1 0 1 0 0 0 0 0
0 1 0 1 0 0 0 0
1 0 �1 0 0 0 0 0
0 1 0 �1 0 0 0 0
0 0 0 0 1 0 1 0
0 0 0 0 0 1 0 1
0 0 0 0 1 0 �1 0
0 0 0 0 0 1 0 �1
3777777777775:
L. Georgiev [INRNE-Sofia] Theory of Elementary Particles Support: EUCLID, NCSR-BG
![Page 240: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/240.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200763 / 69
Third Hadamard gate reproduced up to swapping
H3 D I4 ˝ H ' R�178 R�1
45 R�156 R�1
67 R�156 R�1
45 R�178
D�ei �
4
p2
2666666666664
1 1 0 0 0 0 0 0
1 �1 0 0 0 0 0 0
0 0 �1 1 0 0 0 0
0 0 1 1 0 0 0 0
0 0 0 0 1 1 0 0
0 0 0 0 1 �1 0 0
0 0 0 0 0 0 �1 1
0 0 0 0 0 0 1 1
3777777777775:
L. Georgiev [INRNE-Sofia] Theory of Elementary Particles Support: EUCLID, NCSR-BG
![Page 241: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/241.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200764 / 69
Toffoli gate by Controlled-S and CNOT
Toffoli D CCNOT D .I2 ˝ I2 ˝ H / CCZ .I2 ˝ I2 ˝ H /;
![Page 242: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/242.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200764 / 69
Toffoli gate by Controlled-S and CNOT
Toffoli D CCNOT D .I2 ˝ I2 ˝ H / CCZ .I2 ˝ I2 ˝ H /;
where CCZ D diag.1; 1; 1; 1; 1; 1; 1;�1/.
![Page 243: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/243.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200764 / 69
Toffoli gate by Controlled-S and CNOT
Toffoli D CCNOT D .I2 ˝ I2 ˝ H / CCZ .I2 ˝ I2 ˝ H /;
where CCZ D diag.1; 1; 1; 1; 1; 1; 1;�1/.
� Construct CCZ gate bay Controlled-S (CS) and CNOT:
CS D diag.1; 1; 1; i/
L. Georgiev [INRNE-Sofia] Theory of Elementary Particles Support: EUCLID, NCSR-BG
![Page 244: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/244.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200765 / 69
Z S S1−S
![Page 245: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/245.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200765 / 69
Z S S1−S
� CS cannot be expressed by braiding:
![Page 246: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/246.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200765 / 69
Z S S1−S
� CS cannot be expressed by braiding:because det CS D i , while det
�R.6/a;aC1
�D �1.
![Page 247: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/247.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200765 / 69
Z S S1−S
� CS cannot be expressed by braiding:because det CS D i , while det
�R.6/a;aC1
�D �1.
� Braid precursor:
![Page 248: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/248.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200765 / 69
Z S S1−S
� CS cannot be expressed by braiding:because det CS D i , while det
�R.6/a;aC1
�D �1.
� Braid precursor: R.6/56
D diag.1; i; 1; i/
L. Georgiev [INRNE-Sofia] Theory of Elementary Particles Support: EUCLID, NCSR-BG
![Page 249: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/249.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200766 / 69
� Bravyi–Kitaev construction: split qubit 1 into twocharge-1=4 quasiholes only if it is in the state j1i
L. Georgiev [INRNE-Sofia] Theory of Elementary Particles Support: EUCLID, NCSR-BG
![Page 250: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/250.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200767 / 69
Toffoli gate by braid-group CCZ precursor
C̃CZ � R.8/12
R.8/34
R.8/56
R.8/78
' diag .�1; 1; 1; 1; 1; 1; 1;�1/
![Page 251: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/251.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200767 / 69
Toffoli gate by braid-group CCZ precursor
C̃CZ � R.8/12
R.8/34
R.8/56
R.8/78
' diag .�1; 1; 1; 1; 1; 1; 1;�1/
� Bravyi–Kitaev construction: split qubit 1 into twocharge-1=4 quasiholes only if it is in the state j1i
![Page 252: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/252.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200767 / 69
Toffoli gate by braid-group CCZ precursor
C̃CZ � R.8/12
R.8/34
R.8/56
R.8/78
' diag .�1; 1; 1; 1; 1; 1; 1;�1/
� Bravyi–Kitaev construction: split qubit 1 into twocharge-1=4 quasiholes only if it is in the state j1i
� Dehn twists!
L. Georgiev [INRNE-Sofia] Theory of Elementary Particles Support: EUCLID, NCSR-BG
![Page 253: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/253.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200768 / 69
Open problems: everything is in the details
� Constructq
R.4/12
by braiding + something
![Page 254: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/254.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200768 / 69
Open problems: everything is in the details
� Constructq
R.4/12
by braiding + something
� Embed all single-qubit and two-qubit gates into three-qubit system
![Page 255: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/255.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200768 / 69
Open problems: everything is in the details
� Constructq
R.4/12
by braiding + something
� Embed all single-qubit and two-qubit gates into three-qubit system
� Construct the Toffoli gate
![Page 256: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/256.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200768 / 69
Open problems: everything is in the details
� Constructq
R.4/12
by braiding + something
� Embed all single-qubit and two-qubit gates into three-qubit system
� Construct the Toffoli gate
� Approximate quantum Fourier transform
![Page 257: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/257.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200768 / 69
Open problems: everything is in the details
� Constructq
R.4/12
by braiding + something
� Embed all single-qubit and two-qubit gates into three-qubit system
� Construct the Toffoli gate
� Approximate quantum Fourier transform
� Analyze the error sources
L. Georgiev [INRNE-Sofia] Theory of Elementary Particles Support: EUCLID, NCSR-BG
![Page 258: Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM](https://reader031.vdocuments.us/reader031/viewer/2022020706/61fc87248d33c02b785e3822/html5/thumbnails/258.jpg)
Statphys23 Satellite Meeting, STATISTICAL FIELD THEORY OF QUANTUM DEVICES, Perugia (Italy) July 16-19, 200769 / 69
L. Georgiev [INRNE-Sofia] Theory of Elementary Particles Support: EUCLID, NCSR-BG