TY - JOUR

T1 - Irreconcilable difference between quantum walks and adiabatic quantum computing

AU - Wong, Thomas G.

AU - Meyer, David A.

N1 - Funding Information:
Our great appreciation goes to Eleanor Rieffel for her valuable feedback and advice in developing this work. T.W. was supported by the European Union Seventh Framework Programme (FP7/2007-2013) under the QALGO (Grant Agreement No. 600700) project, and the European Research Council (ERC) Advanced Grant MQC. D.M. was partially supported by the Air Force Office of Scientific Research as part of the Transformational Computing in Aerospace Science and Engineering Initiative under grant FA9550-12-1-0046.

PY - 2016/6/13

Y1 - 2016/6/13

N2 - Continuous-time quantum walks and adiabatic quantum evolution are two general techniques for quantum computing, both of which are described by Hamiltonians that govern their evolutions by Schrödinger's equation. In the former, the Hamiltonian is fixed, while in the latter, the Hamiltonian varies with time. As a result, their formulations of Grover's algorithm evolve differently through Hilbert space. We show that this difference is fundamental; they cannot be made to evolve along each other's path without introducing structure more powerful than the standard oracle for unstructured search. For an adiabatic quantum evolution to evolve like the quantum walk search algorithm, it must interpolate between three fixed Hamiltonians, one of which is complex and introduces structure that is stronger than the oracle for unstructured search. Conversely, for a quantum walk to evolve along the path of the adiabatic search algorithm, it must be a chiral quantum walk on a weighted, directed star graph with structure that is also stronger than the oracle for unstructured search. Thus, the two techniques, although similar in being described by Hamiltonians that govern their evolution, compute by fundamentally irreconcilable means.

AB - Continuous-time quantum walks and adiabatic quantum evolution are two general techniques for quantum computing, both of which are described by Hamiltonians that govern their evolutions by Schrödinger's equation. In the former, the Hamiltonian is fixed, while in the latter, the Hamiltonian varies with time. As a result, their formulations of Grover's algorithm evolve differently through Hilbert space. We show that this difference is fundamental; they cannot be made to evolve along each other's path without introducing structure more powerful than the standard oracle for unstructured search. For an adiabatic quantum evolution to evolve like the quantum walk search algorithm, it must interpolate between three fixed Hamiltonians, one of which is complex and introduces structure that is stronger than the oracle for unstructured search. Conversely, for a quantum walk to evolve along the path of the adiabatic search algorithm, it must be a chiral quantum walk on a weighted, directed star graph with structure that is also stronger than the oracle for unstructured search. Thus, the two techniques, although similar in being described by Hamiltonians that govern their evolution, compute by fundamentally irreconcilable means.

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U2 - 10.1103/PhysRevA.93.062313

DO - 10.1103/PhysRevA.93.062313

M3 - Article

AN - SCOPUS:84975168246

VL - 93

JO - Physical Review A - Atomic, Molecular, and Optical Physics

JF - Physical Review A - Atomic, Molecular, and Optical Physics

SN - 1050-2947

IS - 6

M1 - 062313

ER -