preparing topological states on a quantum computer
DESCRIPTION
Preparing Topological States on a Quantum Computer. Martin Schwarz (1) , Kristan Temme (1) , Frank Verstraete (1) Toby Cubitt (2) , David Perez-Garcia (2). (1) University of Vienna (2) Complutense University, Madrid. STV, Phys. Rev. Lett. 108, 110502 (2012) - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: Preparing Topological States on a Quantum Computer](https://reader035.vdocuments.us/reader035/viewer/2022062301/5681514c550346895dbf6e63/html5/thumbnails/1.jpg)
Preparing Topological States on a Quantum Computer
Martin Schwarz(1), Kristan Temme(1),Frank Verstraete(1)
Toby Cubitt(2), David Perez-Garcia(2)
(1)University of Vienna(2)Complutense University, Madrid
STV, Phys. Rev. Lett. 108, 110502 (2012)STVCP-G, (QIP 2012; paper in preparation)
![Page 2: Preparing Topological States on a Quantum Computer](https://reader035.vdocuments.us/reader035/viewer/2022062301/5681514c550346895dbf6e63/html5/thumbnails/2.jpg)
Talk Outline
• Crash course on PEPS
• Growing PEPS in your Back Garden
• The Trouble with Tribbles Topological States
• Crash course on G-injective PEPS
• Growing Topological Quantum States
![Page 3: Preparing Topological States on a Quantum Computer](https://reader035.vdocuments.us/reader035/viewer/2022062301/5681514c550346895dbf6e63/html5/thumbnails/3.jpg)
Crash Course on PEPS!• Projected Entangled Pair State
![Page 4: Preparing Topological States on a Quantum Computer](https://reader035.vdocuments.us/reader035/viewer/2022062301/5681514c550346895dbf6e63/html5/thumbnails/4.jpg)
Crash Course on PEPS!• Projected Entangled Pair State
Obtain PEPS by applying maps to maximally entangled pairs
![Page 5: Preparing Topological States on a Quantum Computer](https://reader035.vdocuments.us/reader035/viewer/2022062301/5681514c550346895dbf6e63/html5/thumbnails/5.jpg)
Crash Course on PEPS!
• Parent Hamiltonian2-local Hamiltonian with PEPS as ground state.
• InjectivityPEPS is “injective” if are left-invertible
(perhaps only after blocking together sites)
• UniquenessAn injective PEPS is the unique ground state of its parent Hamiltonian
![Page 6: Preparing Topological States on a Quantum Computer](https://reader035.vdocuments.us/reader035/viewer/2022062301/5681514c550346895dbf6e63/html5/thumbnails/6.jpg)
Are PEPS Physical?• PEPS accurately approximate ground states of gapped local
Hamiltonians.– Proven in 1D (= MPS) [Hastings 2007]– Conjectured for higher dim (analytic & numerical evidence)
• PEPS preparation would be an extremely powerful computational resource:– as powerful as contracting tensor networks– PP-complete (for general PEPS as classical input)
Cannot efficiently prepare all PEPS, even using a universal quantum computer (unless BQP = PP!)
![Page 7: Preparing Topological States on a Quantum Computer](https://reader035.vdocuments.us/reader035/viewer/2022062301/5681514c550346895dbf6e63/html5/thumbnails/7.jpg)
Are PEPS Physical?
• Is it possible to prepare PEPS on a quantum computer (under mild conditions on PEPS)?
• Which subclass of PEPS are physical?
[V, Wolf, P-G, Cirac 2006]
![Page 8: Preparing Topological States on a Quantum Computer](https://reader035.vdocuments.us/reader035/viewer/2022062301/5681514c550346895dbf6e63/html5/thumbnails/8.jpg)
Talk Outline
• Crash course on PEPS
• Growing PEPS in your Back Garden
• The Trouble with Tribbles Topological States
• Crash course on G-injective PEPS
• Growing Topological Quantum States
![Page 9: Preparing Topological States on a Quantum Computer](https://reader035.vdocuments.us/reader035/viewer/2022062301/5681514c550346895dbf6e63/html5/thumbnails/9.jpg)
Growing PEPS in your Back Garden
• Start with maximally entangled pairs at every edge, and convert this into target PEPS.
![Page 10: Preparing Topological States on a Quantum Computer](https://reader035.vdocuments.us/reader035/viewer/2022062301/5681514c550346895dbf6e63/html5/thumbnails/10.jpg)
Growing PEPS in your Back Garden
• Start with maximally entangled pairs at every edge, and convert this into target PEPS.
• Sequence of partial PEPS |ti are ground states of sequence of parent Hamiltonians Ht :
![Page 11: Preparing Topological States on a Quantum Computer](https://reader035.vdocuments.us/reader035/viewer/2022062301/5681514c550346895dbf6e63/html5/thumbnails/11.jpg)
Growing PEPS in your Back Garden
• Start with maximally entangled pairs at every edge, and convert this into target PEPS.
• Sequence of partial PEPS |ti are ground states of sequence of parent Hamiltonians Ht :
![Page 12: Preparing Topological States on a Quantum Computer](https://reader035.vdocuments.us/reader035/viewer/2022062301/5681514c550346895dbf6e63/html5/thumbnails/12.jpg)
Growing PEPS in your Back Garden
• Start with maximally entangled pairs at every edge, and convert this into target PEPS.
• Sequence of partial PEPS |ti are ground states of sequence of parent Hamiltonians Ht :
![Page 13: Preparing Topological States on a Quantum Computer](https://reader035.vdocuments.us/reader035/viewer/2022062301/5681514c550346895dbf6e63/html5/thumbnails/13.jpg)
Growing PEPS in your Back Garden
• Start with maximally entangled pairs at every edge, and convert this into target PEPS.
• Sequence of partial PEPS |ti are ground states of sequence of parent Hamiltonians Ht :
![Page 14: Preparing Topological States on a Quantum Computer](https://reader035.vdocuments.us/reader035/viewer/2022062301/5681514c550346895dbf6e63/html5/thumbnails/14.jpg)
Growing PEPS in your Back Garden
• Start with maximally entangled pairs at every edge, and convert this into target PEPS.
• Sequence of partial PEPS |ti are ground states of sequence of parent Hamiltonians Ht :
![Page 15: Preparing Topological States on a Quantum Computer](https://reader035.vdocuments.us/reader035/viewer/2022062301/5681514c550346895dbf6e63/html5/thumbnails/15.jpg)
Growing PEPS in your Back Garden
• Start with maximally entangled pairs at every edge, and convert this into target PEPS.
• Sequence of partial PEPS |ti are ground states of sequence of parent Hamiltonians Ht :
![Page 16: Preparing Topological States on a Quantum Computer](https://reader035.vdocuments.us/reader035/viewer/2022062301/5681514c550346895dbf6e63/html5/thumbnails/16.jpg)
Growing PEPS in your Back GardenAlgorithm
1. t = 0
2. Prepare max-entangled pairs (= ground state of H0)
3. Grow the PEPS vertex by vertex:
1. Project onto ground state of Ht+1
2. t = t + 1
![Page 17: Preparing Topological States on a Quantum Computer](https://reader035.vdocuments.us/reader035/viewer/2022062301/5681514c550346895dbf6e63/html5/thumbnails/17.jpg)
Growing PEPS in your Back GardenAlgorithm
1. t = 0
2. Prepare max-entangled pairs (= ground state of H0)
3. Grow the PEPS vertex by vertex:
1. Project onto ground state of Ht+1
2. t = t + 1
![Page 18: Preparing Topological States on a Quantum Computer](https://reader035.vdocuments.us/reader035/viewer/2022062301/5681514c550346895dbf6e63/html5/thumbnails/18.jpg)
Growing PEPS in your Back GardenAlgorithm
1. t = 0
2. Prepare max-entangled pairs (= ground state of H0)
3. Grow the PEPS vertex by vertex:
1. Project onto ground state of Ht+1
2. t = t + 1
![Page 19: Preparing Topological States on a Quantum Computer](https://reader035.vdocuments.us/reader035/viewer/2022062301/5681514c550346895dbf6e63/html5/thumbnails/19.jpg)
Growing PEPS in your Back GardenAlgorithm
1. t = 0
2. Prepare max-entangled pairs (= ground state of H0)
3. Grow the PEPS vertex by vertex:
1. Project onto ground state of Ht+1
2. t = t + 1
![Page 20: Preparing Topological States on a Quantum Computer](https://reader035.vdocuments.us/reader035/viewer/2022062301/5681514c550346895dbf6e63/html5/thumbnails/20.jpg)
Algorithm
1. t = 0
2. Prepare max-entangled pairs (= ground state of H0)
3. Grow the PEPS vertex by vertex:
1. Project onto ground state of Ht+1
2. t = t + 1
Growing PEPS in your Back Garden
![Page 21: Preparing Topological States on a Quantum Computer](https://reader035.vdocuments.us/reader035/viewer/2022062301/5681514c550346895dbf6e63/html5/thumbnails/21.jpg)
Growing PEPS in your Back GardenAlgorithm
1. t = 0
2. Prepare max-entangled pairs (= ground state of H0)
3. Grow the PEPS vertex by vertex:
1. Project onto ground state of Ht+1
2. t = t + 1
![Page 22: Preparing Topological States on a Quantum Computer](https://reader035.vdocuments.us/reader035/viewer/2022062301/5681514c550346895dbf6e63/html5/thumbnails/22.jpg)
Algorithm
1. t = 0
2. Prepare max-entangled pairs (= ground state of H0)
3. Grow the PEPS vertex by vertex:
1. Project onto ground state of Ht+1
2. t = t + 1
Growing PEPS in your Back Garden
![Page 23: Preparing Topological States on a Quantum Computer](https://reader035.vdocuments.us/reader035/viewer/2022062301/5681514c550346895dbf6e63/html5/thumbnails/23.jpg)
Growing PEPS in your Back GardenAlgorithm
1. t = 0
2. Prepare max-entangled pairs (= ground state of H0)
3. Grow the PEPS vertex by vertex:
1. Project onto ground state of Ht+1
2. t = t + 1
![Page 24: Preparing Topological States on a Quantum Computer](https://reader035.vdocuments.us/reader035/viewer/2022062301/5681514c550346895dbf6e63/html5/thumbnails/24.jpg)
Algorithm
1. t = 0
2. Prepare max-entangled pairs (= ground state of H0)
3. Grow the PEPS vertex by vertex:
1. Project onto ground state of Ht+1
2. t = t + 1
Growing PEPS in your Back Garden
![Page 25: Preparing Topological States on a Quantum Computer](https://reader035.vdocuments.us/reader035/viewer/2022062301/5681514c550346895dbf6e63/html5/thumbnails/25.jpg)
Algorithm
1. t = 0
2. Prepare max-entangled pairs (= ground state of H0)
3. Grow the PEPS vertex by vertex:
1. Project onto ground state of Ht+1
2. t = t + 1
Growing PEPS in your Back Garden
![Page 26: Preparing Topological States on a Quantum Computer](https://reader035.vdocuments.us/reader035/viewer/2022062301/5681514c550346895dbf6e63/html5/thumbnails/26.jpg)
Growing PEPS in your Back Garden
• Even if we could implement this measurement, we cannot choose the outcome, so how can we deterministically project onto P0??
• How can we implement the measurement , when the ground state P0 is a complex, many-body state which we don’t know how to prepare?
??
Algorithm
1. t = 0
2. Prepare max-entangled pairs (= ground state of H0)
3. Grow the PEPS vertex by vertex:
1. Project onto ground state of Ht+1
2. t = t + 1
![Page 27: Preparing Topological States on a Quantum Computer](https://reader035.vdocuments.us/reader035/viewer/2022062301/5681514c550346895dbf6e63/html5/thumbnails/27.jpg)
Measuring the Ground State
• How can we implement the measurement ?
local Hamiltonian ) Hamiltonian simulation )
measure if energy is < or not
QPE
! Use quantum phase estimation:
![Page 28: Preparing Topological States on a Quantum Computer](https://reader035.vdocuments.us/reader035/viewer/2022062301/5681514c550346895dbf6e63/html5/thumbnails/28.jpg)
Measuring the Ground State
measure if energy is < or not
• Condition 1: Spectral gap Ht) > 1/poly
• How can we implement the measurement ?
QPE
! Use quantum phase estimation:
![Page 29: Preparing Topological States on a Quantum Computer](https://reader035.vdocuments.us/reader035/viewer/2022062301/5681514c550346895dbf6e63/html5/thumbnails/29.jpg)
Projecting onto the Ground State
• How can we deterministically project from P0(t) to P0
(t+1)?
! Use Marriot-Watrous measurement rewinding trick:
P0(t+1) =
00
-s c
c s
00
P0(t) =
00
01
00
“Jordan’s lemma” (or “CS decomposition”)• Start in Jordan block of P0
(t) containing |ti
• Measure {P0(t+1),P0
(t+1)?} ! stay in same Jordan block
Condition 2: Unique ground state (= injective PEPS)
![Page 30: Preparing Topological States on a Quantum Computer](https://reader035.vdocuments.us/reader035/viewer/2022062301/5681514c550346895dbf6e63/html5/thumbnails/30.jpg)
Projecting onto the Ground State
! Use Marriot-Watrous measurement rewinding trick:
• How can we deterministically project from P0(t) to P0
(t+1)?
![Page 31: Preparing Topological States on a Quantum Computer](https://reader035.vdocuments.us/reader035/viewer/2022062301/5681514c550346895dbf6e63/html5/thumbnails/31.jpg)
Projecting onto the Ground State
! Use Marriot-Watrous measurement rewinding trick:
• Measure {P0(t+1),P0
(t+1)?}
• How can we deterministically project from P0(t) to P0
(t+1)?
![Page 32: Preparing Topological States on a Quantum Computer](https://reader035.vdocuments.us/reader035/viewer/2022062301/5681514c550346895dbf6e63/html5/thumbnails/32.jpg)
Projecting onto the Ground State
! Use Marriot-Watrous measurement rewinding trick:
c
• Measure {P0(t+1),P0
(t+1)?}
• Outcome P0(t+1) ) done
• How can we deterministically project from P0(t) to P0
(t+1)?
![Page 33: Preparing Topological States on a Quantum Computer](https://reader035.vdocuments.us/reader035/viewer/2022062301/5681514c550346895dbf6e63/html5/thumbnails/33.jpg)
Projecting onto the Ground State
! Use Marriot-Watrous measurement rewinding trick:
c
s
• Measure {P0(t+1),P0
(t+1)?}
• Outcome P0(t+1) ) done
• Outcome P0(t+1) ? …
• How can we deterministically project from P0(t) to P0
(t+1)?
![Page 34: Preparing Topological States on a Quantum Computer](https://reader035.vdocuments.us/reader035/viewer/2022062301/5681514c550346895dbf6e63/html5/thumbnails/34.jpg)
Projecting onto the Ground State
! Use Marriot-Watrous measurement rewinding trick:
c
s
• Measure {P0(t+1),P0
(t+1)?}
• Outcome P0(t+1) ) done
• Outcome P0(t+1) ?
) rewind by measuring {P0(t),P0
(t)?}
• How can we deterministically project from P0(t) to P0
(t+1)?
![Page 35: Preparing Topological States on a Quantum Computer](https://reader035.vdocuments.us/reader035/viewer/2022062301/5681514c550346895dbf6e63/html5/thumbnails/35.jpg)
Projecting onto the Ground State
! Use Marriot-Watrous measurement rewinding trick:
c
s
• Measure {P0(t+1),P0
(t+1)?}
• Outcome P0(t+1) ) done
• Outcome P0(t+1) ?
) go back by measuring {P0(t),P0
(t)?}
• How can we deterministically project from P0(t) to P0
(t+1)?
![Page 36: Preparing Topological States on a Quantum Computer](https://reader035.vdocuments.us/reader035/viewer/2022062301/5681514c550346895dbf6e63/html5/thumbnails/36.jpg)
Projecting onto the Ground State
! Use Marriot-Watrous measurement rewinding trick:
c
s
• Measure {P0(t+1),P0
(t+1)?}
• Outcome P0(t+1) ) done
• Outcome P0(t+1) ?
) go back by measuring {P0(t),P0
(t)?}
• How can we deterministically project from P0(t) to P0
(t+1)?
![Page 37: Preparing Topological States on a Quantum Computer](https://reader035.vdocuments.us/reader035/viewer/2022062301/5681514c550346895dbf6e63/html5/thumbnails/37.jpg)
Projecting onto the Ground State
! Use Marriot-Watrous measurement rewinding trick:
c
s
c
c
• Measure {P0(t+1),P0
(t+1)?}
• Outcome P0(t+1) ) done
• Outcome P0(t+1) ?
) go back by measuring {P0(t),P0
(t)?}
• How can we deterministically project from P0(t) to P0
(t+1)?
![Page 38: Preparing Topological States on a Quantum Computer](https://reader035.vdocuments.us/reader035/viewer/2022062301/5681514c550346895dbf6e63/html5/thumbnails/38.jpg)
Projecting onto the Ground State
! Use Marriot-Watrous measurement rewinding trick:
c
s
cs
sc
• Measure {P0(t+1),P0
(t+1)?}
• Outcome P0(t+1) ) done
• Outcome P0(t+1) ?
) go back by measuring {P0(t),P0
(t)?}
• How can we deterministically project from P0(t) to P0
(t+1)?
![Page 39: Preparing Topological States on a Quantum Computer](https://reader035.vdocuments.us/reader035/viewer/2022062301/5681514c550346895dbf6e63/html5/thumbnails/39.jpg)
Projecting onto the Ground State
! Use Marriot-Watrous measurement rewinding trick:
c
s
c
s
s
c
• Measure {P0(t+1),P0
(t+1)?}
• Outcome P0(t+1) ) done
• Outcome P0(t+1) ?
) go back by measuring {P0(t),P0
(t)?}
• How can we deterministically project from P0(t) to P0
(t+1)?
![Page 40: Preparing Topological States on a Quantum Computer](https://reader035.vdocuments.us/reader035/viewer/2022062301/5681514c550346895dbf6e63/html5/thumbnails/40.jpg)
Projecting onto the Ground State
! Use Marriot-Watrous measurement rewinding trick:
c
s
c
s
s
c
s
s
c
c
• Measure {P0(t+1),P0
(t+1)?}
• Outcome P0(t+1) ) done
• Outcome P0(t+1) ?
) go back by measuring {P0(t),P0
(t)?}
• How can we deterministically project from P0(t) to P0
(t+1)?
![Page 41: Preparing Topological States on a Quantum Computer](https://reader035.vdocuments.us/reader035/viewer/2022062301/5681514c550346895dbf6e63/html5/thumbnails/41.jpg)
Projecting onto the Ground State
! Use Marriot-Watrous measurement rewinding trick:
c
s
c
s
s
c
s
s
c
c
• Lemma: where
• How can we deterministically project from P0(t) to P0
(t+1)?
• ) exp fast
• Condition 3: Condition number At > 1/poly
![Page 42: Preparing Topological States on a Quantum Computer](https://reader035.vdocuments.us/reader035/viewer/2022062301/5681514c550346895dbf6e63/html5/thumbnails/42.jpg)
Algorithm:1. t = 0
2. Prepare max-entangled pairs (= ground state of H0)3. Grow the PEPS vertex by vertex:
1. Project onto ground state of Ht+1
2. t = t + 1
Growing PEPS in your Back Garden
![Page 43: Preparing Topological States on a Quantum Computer](https://reader035.vdocuments.us/reader035/viewer/2022062301/5681514c550346895dbf6e63/html5/thumbnails/43.jpg)
Growing PEPS in your Back GardenAlgorithm:1. t = 0
2. Prepare max-entangled pairs (= ground state of H0)3. Grow the PEPS vertex by vertex:
1. Measure {P0(t+1),P0
(t+1)?}
2. While outcome P0(t)
1. Measure {P0(t),P0
(t)?}
2. Measure {P0(t+1),P0
(t+1)?}3. t = t + 1
![Page 44: Preparing Topological States on a Quantum Computer](https://reader035.vdocuments.us/reader035/viewer/2022062301/5681514c550346895dbf6e63/html5/thumbnails/44.jpg)
Are PEPS Physical?
• Is it possible to prepare PEPS on a quantum computer (under mild conditions on PEPS)?
• Which subclass of PEPS are physical?
Condition 1: Spectral gap Ht) > 1/poly
Condition 3: Condition number At > 1/poly
Run-time:
Condition 2: Unique ground state (= injective PEPS)
Rules out all topological quantum states!
![Page 45: Preparing Topological States on a Quantum Computer](https://reader035.vdocuments.us/reader035/viewer/2022062301/5681514c550346895dbf6e63/html5/thumbnails/45.jpg)
Talk Outline
• Crash course on PEPS
• Growing PEPS in your Back Garden
• The Trouble with Tribbles Topological States
• Crash course on G-injective PEPS
• Growing Topological Quantum States
![Page 46: Preparing Topological States on a Quantum Computer](https://reader035.vdocuments.us/reader035/viewer/2022062301/5681514c550346895dbf6e63/html5/thumbnails/46.jpg)
Projecting onto the Ground State
P0(t+1) =
00
-s1 c1
c1 s1
“Jordan’s lemma” (or “CS decomposition”)
• State could be spread over any of the Jordan blocks of P0
(t) containing |t(k)i.
• Probability of measuring P0(t+1) can be 0.
P0(t) =
00
01
01
-s2 c2
c2 s2
![Page 47: Preparing Topological States on a Quantum Computer](https://reader035.vdocuments.us/reader035/viewer/2022062301/5681514c550346895dbf6e63/html5/thumbnails/47.jpg)
Projecting onto the Ground State
• Probability of measuring P0(t+1) could be 0.
![Page 48: Preparing Topological States on a Quantum Computer](https://reader035.vdocuments.us/reader035/viewer/2022062301/5681514c550346895dbf6e63/html5/thumbnails/48.jpg)
Projecting onto the Ground State
• Probability of measuring P0(t+1) could be 0.
s
![Page 49: Preparing Topological States on a Quantum Computer](https://reader035.vdocuments.us/reader035/viewer/2022062301/5681514c550346895dbf6e63/html5/thumbnails/49.jpg)
Projecting onto the Ground State
• Probability of measuring P0(t+1) could be 0.
s
![Page 50: Preparing Topological States on a Quantum Computer](https://reader035.vdocuments.us/reader035/viewer/2022062301/5681514c550346895dbf6e63/html5/thumbnails/50.jpg)
Projecting onto the Ground State
• Probability of measuring P0(t+1) could be 0.
![Page 51: Preparing Topological States on a Quantum Computer](https://reader035.vdocuments.us/reader035/viewer/2022062301/5681514c550346895dbf6e63/html5/thumbnails/51.jpg)
Projecting onto the Ground State
• Probability of measuring P0(t+1) could be 0.
We can get stuck! (never make it to )
![Page 52: Preparing Topological States on a Quantum Computer](https://reader035.vdocuments.us/reader035/viewer/2022062301/5681514c550346895dbf6e63/html5/thumbnails/52.jpg)
Talk Outline
• Crash course on PEPS
• Growing PEPS in your Back Garden
• The Trouble with Tribbles Topological States
• Crash course on G-injective PEPS
• Growing Topological Quantum States
![Page 53: Preparing Topological States on a Quantum Computer](https://reader035.vdocuments.us/reader035/viewer/2022062301/5681514c550346895dbf6e63/html5/thumbnails/53.jpg)
Crash Course on G-injective PEPS! [Schuch, Cirac, P-G 2010]
• G-injective PEPSPEPS maps left-invertible on invariant subspace of symmetry group G.
• G-isometric PEPSG-injective PEPS where = projector onto G-invariant subspace.
• Topological stateDegenerate ground state of Hamiltonian whose ground states cannot be distinguished by local observables.
• G-injective PEPS = Topological stateParent Hamiltonian has topologically degenerate ground states (degeneracy = # “pair conjugacy classes” of G)
![Page 54: Preparing Topological States on a Quantum Computer](https://reader035.vdocuments.us/reader035/viewer/2022062301/5681514c550346895dbf6e63/html5/thumbnails/54.jpg)
Crash Course on G-injective PEPS! [Schuch, Cirac, P-G 2010]
• Many important topological quantum states areG-injective PEPS:
• Kitaev’s toric code
• Quantum double models
• Resonant valence bond states[Schuch, Poilblanc, Cirac, P-G, arXiv:1203.4816]
• …
![Page 55: Preparing Topological States on a Quantum Computer](https://reader035.vdocuments.us/reader035/viewer/2022062301/5681514c550346895dbf6e63/html5/thumbnails/55.jpg)
Talk Outline
• Crash course on PEPS
• Growing PEPS in your Back Garden
• The Trouble with Tribbles Topological States
• Crash course on G-injective PEPS
• Growing Topological Quantum States
![Page 56: Preparing Topological States on a Quantum Computer](https://reader035.vdocuments.us/reader035/viewer/2022062301/5681514c550346895dbf6e63/html5/thumbnails/56.jpg)
Growing Topological Quantum States
• A(t) no longer invertible (only invertible on G-invariant subspace) ) zero eigenvalues ) = 1 ) c = 0 (bad!)
• Recall key Lemma relating probability c of successful measurement to condition number:
where
• However, G-injectivity ) restriction of A(t) to G-invariant subspace is invertible.
• How can we exploit this?
![Page 57: Preparing Topological States on a Quantum Computer](https://reader035.vdocuments.us/reader035/viewer/2022062301/5681514c550346895dbf6e63/html5/thumbnails/57.jpg)
Algorithm
1. t = 0
2. Prepare max-entangled pairs (ground state of H0)
3. Grow the PEPS vertex by vertex:
1. Project onto ground state of Ht+1
2. t = t + 1
Growing Topological Quantum StatesIdea:• Get into the G-invariant subspace.• Stay there!
![Page 58: Preparing Topological States on a Quantum Computer](https://reader035.vdocuments.us/reader035/viewer/2022062301/5681514c550346895dbf6e63/html5/thumbnails/58.jpg)
Growing Topological Quantum States
Algorithm
1. t = 0
2. Prepare G-isometric PEPS (ground state of H0)
3. Deform vertex by vertex to G-injective PEPS:
1. Project onto ground state of Ht+1
2. t = t + 1
Idea:• Get into the G-invariant subspace.• Stay there!
For (suitable representation of) trivial group G = 1,G-isometric PEPS = maximally entangled pairs! recover original algorithm
![Page 59: Preparing Topological States on a Quantum Computer](https://reader035.vdocuments.us/reader035/viewer/2022062301/5681514c550346895dbf6e63/html5/thumbnails/59.jpg)
Growing Topological Quantum StatesAlgorithm
1. t = 0
2. Prepare G-isometric PEPS (ground state of H0)
3. Deform vertex by vertex to G-injective PEPS:
1. Project onto ground state of Ht+1
2. t = t + 1
G-isometric PEPS = quantum double models ! algorithms known for preparing these exactly [e.g. Aguado, Vidal, PRL 100, 070404 (2008)]
![Page 60: Preparing Topological States on a Quantum Computer](https://reader035.vdocuments.us/reader035/viewer/2022062301/5681514c550346895dbf6e63/html5/thumbnails/60.jpg)
Growing Topological Quantum StatesAlgorithm
1. t = 0
2. Prepare G-isometric PEPS (ground state of H0)
3. Deform vertex by vertex to G-injective PEPS:
1. Project onto ground state of Ht+1
2. t = t + 1
Key Lemma: If initial state is already in G-invariant subspace, prob. successful measurement
is condition number restricted to G-invariant subspace
! Marriot-Watrous measurement rewinding trick works!
![Page 61: Preparing Topological States on a Quantum Computer](https://reader035.vdocuments.us/reader035/viewer/2022062301/5681514c550346895dbf6e63/html5/thumbnails/61.jpg)
Conclusions• Injective PEPS can be prepared efficiently on a quantum computer,
under the following conditions:– Sequence of parent Hamiltonians is gapped– PEPS maps A(v) are well-conditioned
• G-injective PEPS can be prepared efficiently under similar conditions includes many important topological states
• Alternatives to Marriot-Watrous trick:– Jagged adiabatic thm? [Aharonov, Ta-Shma, 2007]
(Worse run-time, may not work for G-injective case)– Quantum rejection sampling ! quadratic speed-up
[Ozols, Roetteler, Roland, 2011]