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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 I.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]

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Background Quantum Walks [Great Barrier Reef, Cairns]

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Quantum Walks Discrete-time or coined Spin, Charge and Topology, Banff, August 2005 Aharanov, PRA 1993 On the line

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Quantum Walks Continuous-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.

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Quantum Walks Generalised Spin, Charge and Topology, Banff, August 2005 1. Simple quantum walk 2. Composite quantum walk

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Background Quantum Circuits [The 12 Apostles, Great Ocean Road, Victoria

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Quantum Circuits Qubit, quantum wire Single-qubit unitary / gate Two-qubit operation – CNOT Basics Spin, Charge and Topology, Banff, August 2005

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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

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Quantum Circuits Qubit, quantum wire Single-qubit unitary / gate Two-qubit operation – CNOT Basics InputOutput ControlTargetControlTarget 0000 0101 1011 1110

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Mappings Quantum Walks to Quantum circuits [Broadbeach, Queensland]

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Quantum Walk Encoding QW in multi-qubit states Spin, Charge and Topology, Banff, August 2005 1) Single-excitation encoding j th 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 = 2 N nodes Walk in information space efficient encoding, but dynamics can be more difficult to implement {

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Quantum Walk Single 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)

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Quantum Walk Multi-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 -

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Quantum Walk Binary expansion: Hypercube Spin, Charge and Topology, Banff, August 2005 |0i |1i |2i |3i |6i |4i |7i |5i Dynamics Encoding: Hamiltonian:

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QW to gates Examples: The line Spin, Charge and Topology, Banff, August 2005 Encoding: Hamiltonian: Simulation of evolution: Quantum circuit:

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QW to gates Examples: 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

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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 Spin, Charge and Topology, Banff, August 2005 Single-qubit unitary / gate Two-qubit entangling operation (Assume complete, time-varying control over Hamiltonian parameters)

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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 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

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Decoherence Models & a simple example [Wreck Beach, Vancouver]

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Decoherence Error 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

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Decoherence Quantum 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

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Decoherence Quantum Walk on Hypercube Spin, Charge and Topology, Banff, August 2005 Continuous-time limit: Time-step ! 0 probability p ! 0 Rate p/ ! (constant) |0i |1i |2i |3i |6i |4i |7i |5i

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Decoherence Quantum Walk on Hypercube Spin, Charge and Topology, Banff, August 2005 Site-BasedQubit-based

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Decoherence Quantum Walk on Hypercube Spin, Charge and Topology, Banff, August 2005 Site-Based Qubit-based

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

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