Manipulating Causal Order of Unitary Operations using Adiabatic Quantum Computation

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

Motivation qubit Analyze the Potential of Quantum Computation Quantum Circuit model is one standard model Input Time Rule Input/output relations proceed from left to right and there are no loops in the circuit. Output However There s no restriction about causal structure in Quantum mechanics axiom! Indeed, we can consider the operation which does not follow this rule.

Motivation Analyze the Potential of Quantum Computation Quantum Circuit model is one standard model These are examples which does not have definite causal order.?? Rule Input/output relations proceed CTC from (Closed left to Timelike right Curve) and there are no loops in the circuit. 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] It implements superposition of order Control qubit determines the order of unitary operation. 2 qubit space Input Control qubit and Target qubit Quantum Switch Output [1] Chiribella G., D Ariano G.M., Perinotti P., and Valiron B. Beyond causally ordered quantum computers. ArXiv e-prints, dec 2009. Switches the order of operation Order superposed operation

Quantum switch[1] Implementation? 1. Quantum circuit Input Output Control Target Ancilla Pauli X operation We must use the same unitary gate twice. [1] Chiribella G., D Ariano G.M., Perinotti P., and Valiron B. Beyond causally ordered quantum computers. ArXiv e-prints, dec 2009.

Quantum switch[1] Implementation? 2. Quantum circuit with superposition of wire Superposed wire Switches the order of operation How to construct superposed wire? Result Adiabatic Quantum Computation can simulate!! [1] Chiribella G., D Ariano G.M., Perinotti P., and Valiron B. Beyond causally ordered quantum computers. ArXiv e-prints, dec 2009.

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 Probabilistic measurement Success: 25% 1 Telepotation 2 : maximally entangled state

Review1: Teleportation Teleportation 0 1 BSS type CTC (Closed Timelike Curves) This probabilistic measurement virtually sends the state back to the past. 2 : maximally entangled state

Review2: Gate Teleportation Gate teleportation 0 1 2 It allows preparing input state after acting on desired operation. Can we do it deterministically?

Review2: Gate Teleportation Gate teleportation 0 1 2 It allows preparing input state after acting on desired operation. Can we do it deterministically? Use Adiabatic method!! Shifting the state as the ground state of Hamiltonian.

Review3: Adiabatic Gate Teleportation[2] 1. Prepare input state and gate state initial Hamiltonian is free 2. Final state on should be final Hamiltonian is free 0 1 2 3. We will shift initial Hamiltonian towards final Hamiltonian slowly enough. [2] Bacon D. and Flammia S.T. Adiabatic gate teleportation. Physical Review Letters, 103(12):120504, sep 2009. 0905.0901

Review3: Adiabatic Gate Teleportation[2] 1. Prepare input state and gate state initial Hamiltonian is free Energy eigenvalue 2nd excited state 2. Final 1st excited state on state should be final Hamiltonian Energy Gap is free 0 1 2 3. We will shift initial Hamiltonian towards final Hamiltonian Ground slowly state enough. Gate teleportation is implemented! [2] Bacon D. and Flammia S.T. Adiabatic gate teleportation. Physical Review Letters, 103(12):120504, sep 2009. 0905.0901

Review3: Adiabatic Gate Teleportation[2] 1. Prepare input state and gate state initial Hamiltonian is free Energy eigenvalue 2nd excited state 2. Final 1st excited state on state should be final Hamiltonian is free Energy Gap There is 2-degeneracy in the ground state. How can we check that information is preserved? 3. We will shift initial Hamiltonian towards final Hamiltonian Ground slowly state enough. 0 1 2 Gate teleportation is implemented! [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, Energy let s eigenvalue consider most easiest case U=I (Adiabatic teleportation). 2nd excited state Ground state is stabilized by 1st excited state 1st exited state Preserved by Introduce Logical operator No jump! Energy Gap It commutes with the Hamiltonian! i.e. 0 s Ground state Ground state Logical operator is preserved. 1

why AGT works?-(2) Unitary conjugation form. conjugation Adiabatic Teleportation Adiabatic Gate Teleportation initial final

Formalism(Assumptions) in AGT Gate Hamiltonian Unitary corresponding to 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. 0 We use 1 2 3 Then, 4 Ordered We can perform consecutive operations in 1 step.

Parallelization of AGT Consider 2 oracle Hamiltonians and 0 We use 1 2 3 4 1st excited state Then, Energy Gap Ground state Ordered We can perform consecutive operations in 1 step.

Manipulating order of operations If we change final Hamiltonian,,, 0 We use 1 2 3 Then, 4 Opposite order Changing final Hamiltonian changes the order of operation!

Superposing order of operations We introduce control qubit (1+5 qubits system) Input state 0 1 2 3 4 0 1 2 3 4

Superposing order of operations We introduce control qubit (1+5 qubits system) Input state 0 1 0 1 2 2 3 3 This is Quantum Switch operation!! 4 4

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.