Simulating Quantum Walk Hamiltonians without Pauli Decomposition

TL;DR

This paper introduces a novel algorithm called matching decomposition for efficiently simulating continuous-time quantum walks on sparse graphs without using Pauli decomposition, resulting in resource-efficient quantum circuits.

Contribution

The paper presents a new matching decomposition algorithm that simplifies quantum walk simulation, reducing circuit depth and gate count compared to traditional Pauli-based methods.

Findings

Matching decomposition reduces controlled gates by up to 43%.

Circuits generated are up to 54% shallower than Pauli-based methods.

The method can exactly simulate quantum walks on certain graphs.

Abstract

In this work, we present a new algorithm for generating quantum circuits that efficiently implement continuous time quantum walks on arbitrary simple sparse graphs. The algorithm, called matching decomposition, works by decomposing a continuous-time quantum walk Hamiltonian into a collection of exactly implementable Hamiltonians corresponding to matchings in the underlying graph followed by a novel graph compression algorithm that merges edges in the graph. Lastly, we convert the walks to a circuit and Trotterize over these components. The dynamics of the walker on each edge in the matching can be implemented in the circuit model as sequences of CX and CRx gates. We do not use Pauli decomposition when implementing walks along each matching. Furthermore, we compare matching decomposition to a standard Pauli-based simulation pipeline and find that matching decomposition consistently yields substantial resource reductions, requiring up to 43% fewer controlled gates and up to 54% shallower circuits than Pauli decomposition across multiple graph families. Finally, we also present examples and theoretical results for when matching decomposition can exactly simulate a continuous-time quantum walk on a graph.