Quantum Walks, Quantum Quantum Walks, Quantum Gates, and Quantum Gates, and Quantum

ComputersComputers

Andrew Hines

P.C.E. Stamp

[Palm Beach, Gold Coast, Australia]

MotivationMotivation

• Algorithms

• Implementations

• Decoherence and error-correction

Bell’s Beach, Torquay, Australia]

OverviewI. Background

II. Mappings

III. Decoherence

Spin, Charge and Topology, Banff, August 2005

• Quantum Walks – simple & composite

• Universality & Quantum Circuits

• Quantum walks, qubit representations & implementations

• Quantum Walks $ qubit Hamiltonians $ quantum circuits

• Decoherence models: implementation dependent

• Example – quantum walk on hypercube

[Duranbah, Gold Coast, Australia]

BackgroundQuantum Walks

[Great Barrier Reef, Cairns]

Quantum WalksDiscrete-time or ‘coined’

Spin, Charge and Topology, Banff, August 2005

Aharanov, PRA 1993

On the line

Quantum WalksContinuous-time

Spin, Charge and Topology, Banff, August 2005

Fahri & Guttman, PRA 1998

Childs et al.

Hamiltonian is essentially the adjacency matrix for the corresponding graph, each node corresponding to an orthonormal basis state.

Quantum WalksGeneralised

Spin, Charge and Topology, Banff, August 2005

1. Simple quantum walk

2. Composite quantum walk

BackgroundQuantum Circuits

[The 12 Apostles, Great Ocean Road, Victoria

Quantum Circuits

• Qubit, quantum wire• Single-qubit unitary / gate• Two-qubit operation – CNOT

Basics

Spin, Charge and Topology, Banff, August 2005

Quantum Circuits

• Qubit, quantum wire• Single-qubit unitary / gate• Two-qubit operation – CNOT

Basics

Bloch sphere rotations

For any single-qubit unitary

Spin, Charge and Topology, Banff, August 2005

Spin, Charge and Topology, Banff, August 2005

Quantum Circuits

• Qubit, quantum wire• Single-qubit unitary / gate• Two-qubit operation – CNOT

Basics

Input OutputControl Target Control Target

0 0 0 0

0 1 0 1

1 0 1 1

1 1 1 0

MappingsQuantum Walks to Quantum circuits

[Broadbeach, Queensland]

Quantum WalkEncoding QW in multi-qubit states

Spin, Charge and Topology, Banff, August 2005

1) Single-excitation encodingjth spin

• N qubits = N nodes• Hamiltonian operators:• Walk in physical space• not an efficient encoding, but may be easier to implement operations

2) Binary-expansion encoding

• N qubits = 2N nodes• Walk in information space• efficient encoding, but dynamics can be more difficult to

implement

{

Quantum WalkSingle excitation

Spin, Charge and Topology, Banff, August 2005

Example: XY-spin chain (1 spin up) = QW on a line

Example: Implementation – pulse sequence, ion trap

,

Approximate Hamiltonian evolution (Trotter formula)

Quantum WalkMulti-excitations excitation

Spin, Charge and Topology, Banff, August 2005

Example: XY-spin chain – multiple excitations = more complex graph for walk in information space

N = 6, M = 3

Nodes -

Quantum WalkBinary expansion: Hypercube

Spin, Charge and Topology, Banff, August 2005

|0i |1i

|2i|3i

|6i

|4i

|7i

|5i

Dynamics

Encoding:

Hamiltonian:

QW to gatesExamples: The line

Spin, Charge and Topology, Banff, August 2005

Encoding:

Hamiltonian:

Simulation of evolution: Quantum circuit:

QW to gatesExamples: The line

Spin, Charge and Topology, Banff, August 2005

Components

Generalise to a hyperlattice, where each line represents a dimension. It turns out that `lines’ do not interact, so can simulate QW on arbitrary dimensional hyperlattice

MappingsQuantum circuits to Quantum Walks

[Banff]

Qubit Systems to QWGeneric QC Hamiltonian

Dynamic Qubit Systems to QWGeneric QC Hamiltonian

Spin, Charge and Topology, Banff, August 2005

Single-qubit unitary / gate

Two-qubit entangling operation

(Assume complete, time-varying control over Hamiltonian parameters)

Dynamic Qubit Systems to QWBasic Gates as Quantum Walks

Spin, Charge and Topology, Banff, August 2005

Dynamic Qubit Systems to QWControlled-NOT

Spin, Charge and Topology, Banff, August 2005

Dynamic Qubit Systems to QWCircuits as Quantum Walks

Spin, Charge and Topology, Banff, August 2005

Restrictions on control lead to different basic gate sets and circuit complexity

If all pairs of qubits interact, these gates are implemented using a single pulse.If only nearest neighbour interactions – more complicated pulse sequence required

quantum Fourier transform

DecoherenceModels & a simple example

[Wreck Beach, Vancouver]

DecoherenceError Models

Spin, Charge and Topology, Banff, August 2005

Local, independent error model (Pauli errors), dissipation & dephasing (master equation)

Specific form of errors/environmental couplings must depend upon what physical system the walk Hamiltonian is implemented with or describing.

Oscillator bath Spin bath

Environments

DecoherenceQuantum Walk on Hypercube

Alagic & Russell, PRA 2006

Spin, Charge and Topology, Banff, August 2005

Discrete-time model (Kendon & Tregenna, PRA 2004)

POVM:

|0i |1i

|2i|3i

|6i

|4i

|7i

|5i

DecoherenceQuantum Walk on Hypercube

Spin, Charge and Topology, Banff, August 2005

Continuous-time limit: Time-step ! 0probability p ! 0 Rate p/ ! (constant)

|0i |1i

|2i|3i

|6i

|4i

|7i

|5i

DecoherenceQuantum Walk on Hypercube

Spin, Charge and Topology, Banff, August 2005

Site-Based Qubit-based

DecoherenceQuantum Walk on Hypercube

Spin, Charge and Topology, Banff, August 2005

Site-Based Qubit-based

Thank you

(Australian wildlife, being eaten by Dusty the cattle dog)