### Abstract

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.

Original language | English (US) |
---|---|

Article number | 062313 |

Journal | Physical Review A |

Volume | 93 |

Issue number | 6 |

DOIs | |

State | Published - Jun 13 2016 |

Externally published | Yes |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Atomic and Molecular Physics, and Optics

### Cite this

*Physical Review A*,

*93*(6), [062313]. https://doi.org/10.1103/PhysRevA.93.062313

**Irreconcilable difference between quantum walks and adiabatic quantum computing.** / Wong, Thomas; Meyer, David A.

Research output: Contribution to journal › Article

*Physical Review A*, vol. 93, no. 6, 062313. https://doi.org/10.1103/PhysRevA.93.062313

}

TY - JOUR

T1 - Irreconcilable difference between quantum walks and adiabatic quantum computing

AU - Wong, Thomas

AU - Meyer, David A.

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.

UR - http://www.scopus.com/inward/record.url?scp=84975168246&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84975168246&partnerID=8YFLogxK

U2 - 10.1103/PhysRevA.93.062313

DO - 10.1103/PhysRevA.93.062313

M3 - Article

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 -