manipulating causal order of unitary operations using adiabatic … · 2013-06-10 · manipulating...
TRANSCRIPT
Manipulating Causal Order of
Unitary Operations
using Adiabatic Quantum Computation
Kosuke Nakago, Quantum Information group,
The University of Tokyo, Tokyo,Japan.
Joint work with,
Mio murao, Michal Hajdusek, Shojun Nakayama
Outline
• Motivation
• Background (Review)
▫ Quantum Switch
▫ Adiabatic Gate Teleportation
• Result
▫ Formalism(Assumptions)
▫ Parallelization
▫ Manipulating order of operations
▫ Superposing order of operations
• Conclusion
Outline
• Motivation
• Background (Review)
▫ Quantum Switch
▫ Adiabatic Gate Teleportation
• Result
▫ Formalism(Assumptions)
▫ Parallelization
▫ Manipulating order of operations
▫ Superposing order of operations
• Conclusion
Motivation
• Analyze the Potential of Quantum Computation
Rule Input/output relations proceed from left to right and there are no loops in the circuit.
There’s no restriction about causal structure in Quantum mechanics axiom!
Quantum Circuit model is one standard model
Indeed, we can consider the operation which does not follow this rule.
Time Input
However
qubit
Output
Motivation
• Analyze the Potential of Quantum Computation
• Rule Input/output relations proceed from left to right and there are no loops in the circuit.
Quantum Circuit model is one standard model
?
CTC (Closed Timelike Curve)
These are examples which does not have definite causal order.
?
Let’s consider “non ordered operation” and its implementation.
Outline
• Motivation - Theme:Causal order
• Background (Review)
▫ Quantum Switch
▫ Adiabatic Gate Teleportation
• Result
▫ Formalism(Assumptions)
▫ Parallelization
▫ Manipulating order of operations
▫ Superposing order of operations
• Conclusion
Quantum switch[1]
Switches the order of operation
・・・Control qubit determines the order of unitary operation.
Quantum Switch
• It implements superposition of order
2 qubit space
Input
Output
Control qubit and Target qubit
Order superposed operation
[1] Chiribella G., D’Ariano G.M., Perinotti P., and Valiron B. Beyond causally ordered quantum computers. ArXiv e-prints, dec 2009.
• Implementation?
Control
Target
Ancilla
Input Output
Quantum switch[1]
[1] Chiribella G., D’Ariano G.M., Perinotti P., and Valiron B. Beyond causally ordered quantum computers. ArXiv e-prints, dec 2009.
Pauli X operation
We must use the same unitary gate twice.
1. Quantum circuit
Switches the order of operation
2. Quantum circuit with superposition of wire
Superposed wire
• Implementation?
How to construct superposed wire?
Adiabatic Quantum Computation can simulate!!
Quantum switch[1]
[1] Chiribella G., D’Ariano G.M., Perinotti P., and Valiron B. Beyond causally ordered quantum computers. ArXiv e-prints, dec 2009.
Result
Switches the order of operation
2. Quantum circuit with superposition of wire
Superposed wire
• Implementation?
How to construct superposed wire?
Adiabatic Quantum Computation can simulate!!
Quantum switch[1]
[1] Chiribella G., D’Ariano G.M., Perinotti P., and Valiron B. Beyond causally ordered quantum computers. ArXiv e-prints, dec 2009.
Result
Outline
• Motivation - Theme:Causal order • Background (Review)
▫ Quantum Switch ▫ Adiabatic Gate Teleportation
1.Teleportation 2.Gate teleportation 3.Adiabatic gate teleportation
• Result ▫ Formalism(Assumptions) ▫ Parallelization ▫ Manipulating order of operations ▫ Superposing order of operations
Review1: Teleportation
Teleportation
0
1
2
Telepotation
: maximally entangled state
Probabilistic measurement
Success: 25%
Review1: Teleportation
Teleportation
0
1
2
This probabilistic measurement virtually sends the state back to
the past.
BSS type CTC (Closed Timelike Curves)
: maximally entangled state
Review2: Gate Teleportation
Gate teleportation
0
1
2
Can we do it deterministically?
It allows preparing input state
after acting on desired operation .
Review2: Gate Teleportation
Gate teleportation
0
1
2
Can we do it deterministically?
Use Adiabatic method!!
It allows preparing input state
after acting on desired operation .
Shifting the state as the ground state of Hamiltonian.
Review3: Adiabatic Gate Teleportation[2]
1. Prepare input state and gate state
→ initial Hamiltonian
2. Final state on should be
→ final Hamiltonian
3. We will shift initial Hamiltonian towards final Hamiltonian slowly enough.
2
1
0
is free
is free
[2] Bacon D. and Flammia S.T. Adiabatic gate teleportation. Physical Review Letters, 103(12):120504, sep 2009. 0905.0901
is free
Review3: Adiabatic Gate Teleportation[2]
1. Prepare input state and gate state
→ initial Hamiltonian
[2] Bacon D. and Flammia S.T. Adiabatic gate teleportation. Physical Review Letters, 103(12):120504, sep 2009. 0905.0901
2. Final state on should be
→ final Hamiltonian
3. We will shift initial Hamiltonian towards final Hamiltonian slowly enough.
2
1
0
is free
Review3: Adiabatic Gate Teleportation[2]
1. Prepare input state and gate state
→ initial Hamiltonian
2. Final state on should be
→ final Hamiltonian
3. We will shift initial Hamiltonian towards final Hamiltonian slowly enough.
2
1
0
[2] Bacon D. and Flammia S.T. Adiabatic gate teleportation. Physical Review Letters, 103(12):120504, sep 2009. 0905.0901
is free
is free
is free
Review3: Adiabatic Gate Teleportation[2]
1. Prepare input state and gate state
→ initial Hamiltonian
2. Final state on should be
→ final Hamiltonian
3. We will shift initial Hamiltonian towards final Hamiltonian slowly enough.
2
1
0
is free
Ground state
1st excited state
2nd excited state
Energy Gap
Energy eigenvalue
Gate teleportation is implemented! [2] Bacon D. and Flammia S.T. Adiabatic gate teleportation. Physical Review Letters, 103(12):120504, sep 2009. 0905.0901
is free
Review3: Adiabatic Gate Teleportation[2]
1. Prepare input state and gate state
→ initial Hamiltonian
2. Final state on should be
→ final Hamiltonian
3. We will shift initial Hamiltonian towards final Hamiltonian slowly enough.
2
1
0
is free
Ground state
1st excited state
2nd excited state
Energy Gap
Energy eigenvalue
Gate teleportation is implemented!
There is 2-degeneracy in the ground state. How can we check that information is preserved?
[2] Bacon D. and Flammia S.T. Adiabatic gate teleportation. Physical Review Letters, 103(12):120504, sep 2009. 0905.0901
why AGT works?
• Because logical space is preserved.
First, let us consider most easiest case U=I (Adiabatic Teleportation).
Ground state is stabilized by
Introduce Logical operator
It commutes with the Hamiltonian! i.e.
Logical operator is preserved.
why AGT works?
• Because logical space is preserved.
First, let’s consider most easiest case U=I (Adiabatic teleportation).
Ground state is stabilized by
Introduce Logical operator
It commutes with the Hamiltonian! i.e.
Logical operator is preserved.
Energy eigenvalue
0
1
s
Ground state
1st exited state
No jump!
Preserved by
Ground state
1st excited state
2nd excited state
Energy Gap
Energy eigenvalue
why AGT works?-(2)
• Unitary conjugation form.
Adiabatic Teleportation Adiabatic Gate Teleportation
conjugation
initial
final
Outline
• Motivation - Theme:Causal order
• Background (Review)
▫ Quantum Switch
▫ Adiabatic Gate Teleportation
• Result
▫ Formalism(Assumptions)
▫ Parallelization
▫ Manipulating order of operations
▫ Superposing order of operations
• Conclusion
Formalism(Assumptions) in AGT
• Gate Hamiltonian corresponding to
Unitary
• Ground state of Oracle Hamiltonian
can be prepared.
• Controlling the strength s of the Hamiltonians.
Parallelization of AGT
• Consider 2 gate Hamiltonians and with 5 qubits sys.
1
0
2
4
3
We can perform consecutive operations in 1 step.
We use
Then,
Ordered
Parallelization of AGT
• Consider 2 oracle Hamiltonians and
1
0
2
4
3
We can perform consecutive operations in 1 step.
We use
Then,
Ordered
Ground state
1st excited state
Energy Gap
Manipulating order of operations
1
0
2
4
3
• If we change final Hamiltonian,,,
Changing final Hamiltonian changes the order of operation!
We use
Then,
Opposite order
Superposing order of operations • We introduce control qubit (1+5 qubits system)
Input state
1
0
2
4
3
1
0
2
4
3
Superposing order of operations • We introduce control qubit (1+5 qubits system)
Input state
1
0
2
4
3
1
0
2
4
3 This is Quantum Switch operation!!
Outline
• Motivation - Theme:Causal order
• Background (Review)
▫ Quantum Switch
▫ Adiabatic Gate Teleportation
• Result
▫ Formalism(Assumptions)
▫ Parallelization
▫ Manipulating order of operations
▫ Superposing order of operations
• Conclusion
Conclusion
• Adiabatic gate teleportation scheme allows us to manipulate order of operations.
We can control the order by changing only final Hamiltonian.
• We can simulate superposition of wire in quantum circuit model, by using this scheme.
Problems and future works
• Compare the difference between Quantum circuit model and Adiabatic quantum computation.
• Analyze computational time scale of our scheme.