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)