# Isolated Vertices in Continuous-Time Quantum Walks on Dynamic Graphs

## TL;DR

This paper improves the design of dynamic graphs for continuous-time quantum walks, enabling simpler and more efficient implementation of universal quantum gates, including the Pauli, Hadamard, T, and CNOT gates, with validated simulations.

## Contribution

It introduces a distinction between loopless and looped isolated vertices, simplifying the construction of universal quantum gates via quantum walks on dynamic graphs.

## Key findings

- Simpler dynamic graphs for quantum gates
- Reduction in vertices and complexity for T gate
- Successful numerical validation of quantum circuit simulation

## 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 CNOT gates, and these gates are easily extended to multi-qubit 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 four. 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- and two-qubit gates, similar to those in recent quantum supremacy experiments, using a quantum walk.