manipulating causal order of unitary operations using adiabatic … · 2013-06-10 · manipulating...

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.