Quantum Walks, Quantum Gates, and Quantum Computers

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Quantum Walks, Quantum Gates, and Quantum
Computers

• Andrew Hines
• P.C.E. Stamp

Palm Beach, Gold Coast, Australia
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Motivation

• Algorithms
• Implementations
• Decoherence and error-correction

Bells Beach, Torquay, Australia
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Overview

• Quantum Walks simple composite
• Universality Quantum Circuits

1. Background
2. Mappings
3. Decoherence

• Quantum walks, qubit representations
  implementations
• Quantum Walks qubit Hamiltonians quantum
  circuits

• Decoherence models implementation dependent
• Example quantum walk on hypercube

Duranbah, Gold Coast, Australia
Spin, Charge and Topology, Banff, August 2005
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Background
Quantum Walks
Great Barrier Reef, Cairns
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Quantum Walks
Discrete-time or coined
Aharanov, PRA 1993
On the line
Spin, Charge and Topology, Banff, August 2005
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Quantum Walks
Continuous-time
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.
Spin, Charge and Topology, Banff, August 2005
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Quantum Walks
Generalised
1. Simple quantum walk
2. Composite quantum walk
Spin, Charge and Topology, Banff, August 2005
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Background
Quantum Circuits
The 12 Apostles, Great Ocean Road, Victoria
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Quantum Circuits
Basics

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

Spin, Charge and Topology, Banff, August 2005
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Quantum Circuits
Basics

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

For any single-qubit unitary
Spin, Charge and Topology, Banff, August 2005
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Quantum Circuits
Basics

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

Input Input Output Output
Control Target Control Target
0 0 0 0
0 1 0 1
1 0 1 1
1 1 1 0
Spin, Charge and Topology, Banff, August 2005
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Mappings
Quantum Walks to Quantum circuits
Broadbeach, Queensland
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Quantum Walk
Encoding QW in multi-qubit states
1) Single-excitation encoding
jth 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

Spin, Charge and Topology, Banff, August 2005
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Quantum Walk
Single excitation
Example XY-spin chain (1 spin up) QW on a line
Example Implementation pulse sequence, ion trap
,
Approximate Hamiltonian evolution (Trotter
formula)
Spin, Charge and Topology, Banff, August 2005
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Quantum Walk
Multi-excitations excitation
Example XY-spin chain multiple excitations
more complex graph for walk in information space
N 6, M 3
Nodes -
Spin, Charge and Topology, Banff, August 2005
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Quantum Walk
Binary expansion Hypercube
Encoding
Hamiltonian
Dynamics
Spin, Charge and Topology, Banff, August 2005
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QW to gates
Examples The line
Encoding
Hamiltonian
Simulation of evolution
Quantum circuit
Spin, Charge and Topology, Banff, August 2005
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QW to gates
Examples The line
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
Spin, Charge and Topology, Banff, August 2005
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Mappings
Quantum circuits to Quantum Walks
Banff
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Qubit Systems to QW
Generic QC Hamiltonian
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Dynamic Qubit Systems to QW
Generic QC Hamiltonian
(Assume complete, time-varying control over
Hamiltonian parameters)
Single-qubit unitary / gate
Two-qubit entangling operation
Spin, Charge and Topology, Banff, August 2005
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Dynamic Qubit Systems to QW
Basic Gates as Quantum Walks
Spin, Charge and Topology, Banff, August 2005
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Dynamic Qubit Systems to QW
Controlled-NOT
Spin, Charge and Topology, Banff, August 2005
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Dynamic Qubit Systems to QW
Circuits as Quantum Walks
quantum Fourier transform
If all pairs of qubits interact, these gates are
implemented using a single pulse. If only nearest
neighbour interactions more complicated pulse
sequence required
Restrictions on control lead to different basic
gate sets and circuit complexity
Spin, Charge and Topology, Banff, August 2005
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Decoherence
Models a simple example
Wreck Beach, Vancouver
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Decoherence
Error Models
Local, independent error model (Pauli errors),
dissipation dephasing (master equation)
Environments
Spin bath
Oscillator bath
Specific form of errors/environmental couplings
must depend upon what physical system the walk
Hamiltonian is implemented with or describing.
Spin, Charge and Topology, Banff, August 2005
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Decoherence
Quantum Walk on Hypercube
Alagic Russell, PRA 2006
Discrete-time model
(Kendon Tregenna, PRA 2004)
POVM
Spin, Charge and Topology, Banff, August 2005
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Decoherence
Quantum Walk on Hypercube
Continuous-time limit
Time-step ? ! 0
Rate p/? ! ? (constant)
probability p ! 0
Spin, Charge and Topology, Banff, August 2005
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Decoherence
Quantum Walk on Hypercube
Site-Based
Qubit-based
Spin, Charge and Topology, Banff, August 2005
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Decoherence
Quantum Walk on Hypercube
Qubit-based
Site-Based
Spin, Charge and Topology, Banff, August 2005
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Thank you
(Australian wildlife, being eaten by Dusty the
cattle dog)