### Abstract

It was recently shown that continuous-time quantum walks on dynamic graphs, i.e., sequences of static graphs whose edges change at specific times, can implement a universal set of quantum gates. This result treated all isolated vertices as having self-loops, so they all evolved by a phase under the quantum walk. In this paper, we permit isolated vertices to be loopless or looped, and loopless isolated vertices do not evolve at all under the quantum walk. Using this distinction, we construct simpler dynamic graphs that implement the Pauli gates and a set of universal quantum gates consisting of the Hadamard, T, and controlled-not gates, and these gates are easily extended to multiqubit systems. For example, the T gate is simplified from a sequence of six graphs to a single graph, and the number of vertices is reduced by a factor of 4. We also construct a generalized phase gate, of which Z, S, and T are specific instances. Finally, we validate our implementations by numerically simulating a quantum circuit consisting of layers of one- A nd two-qubit gates, similar to those in recent quantum supremacy experiments, using a quantum walk.

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

Article number | 062325 |

Journal | Physical Review A |

Volume | 100 |

Issue number | 6 |

DOIs | |

State | Published - Dec 18 2019 |

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### All Science Journal Classification (ASJC) codes

- Atomic and Molecular Physics, and Optics

### Cite this

**Isolated vertices in continuous-time quantum walks on dynamic graphs.** / Wong, Thomas G.

Research output: Contribution to journal › Article

*Physical Review A*, vol. 100, no. 6, 062325. https://doi.org/10.1103/PhysRevA.100.062325

}

TY - JOUR

T1 - Isolated vertices in continuous-time quantum walks on dynamic graphs

AU - Wong, Thomas G.

PY - 2019/12/18

Y1 - 2019/12/18

N2 - It was recently shown that continuous-time quantum walks on dynamic graphs, i.e., sequences of static graphs whose edges change at specific times, can implement a universal set of quantum gates. This result treated all isolated vertices as having self-loops, so they all evolved by a phase under the quantum walk. In this paper, we permit isolated vertices to be loopless or looped, and loopless isolated vertices do not evolve at all under the quantum walk. Using this distinction, we construct simpler dynamic graphs that implement the Pauli gates and a set of universal quantum gates consisting of the Hadamard, T, and controlled-not gates, and these gates are easily extended to multiqubit systems. For example, the T gate is simplified from a sequence of six graphs to a single graph, and the number of vertices is reduced by a factor of 4. We also construct a generalized phase gate, of which Z, S, and T are specific instances. Finally, we validate our implementations by numerically simulating a quantum circuit consisting of layers of one- A nd two-qubit gates, similar to those in recent quantum supremacy experiments, using a quantum walk.

AB - It was recently shown that continuous-time quantum walks on dynamic graphs, i.e., sequences of static graphs whose edges change at specific times, can implement a universal set of quantum gates. This result treated all isolated vertices as having self-loops, so they all evolved by a phase under the quantum walk. In this paper, we permit isolated vertices to be loopless or looped, and loopless isolated vertices do not evolve at all under the quantum walk. Using this distinction, we construct simpler dynamic graphs that implement the Pauli gates and a set of universal quantum gates consisting of the Hadamard, T, and controlled-not gates, and these gates are easily extended to multiqubit systems. For example, the T gate is simplified from a sequence of six graphs to a single graph, and the number of vertices is reduced by a factor of 4. We also construct a generalized phase gate, of which Z, S, and T are specific instances. Finally, we validate our implementations by numerically simulating a quantum circuit consisting of layers of one- A nd two-qubit gates, similar to those in recent quantum supremacy experiments, using a quantum walk.

UR - http://www.scopus.com/inward/record.url?scp=85077237433&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85077237433&partnerID=8YFLogxK

U2 - 10.1103/PhysRevA.100.062325

DO - 10.1103/PhysRevA.100.062325

M3 - Article

AN - SCOPUS:85077237433

VL - 100

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

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

SN - 1050-2947

IS - 6

M1 - 062325

ER -