Finding trail covers: near-optimal decompositions of graph states as linear fusion networks

TL;DR

This paper addresses the challenge of optimizing quantum graph state constructions by developing heuristics for near-optimal trail covers, reducing fusion operations in photonic measurement-based quantum computing.

Contribution

It introduces new graph-theoretic problems related to quantum state construction, proves their NP-hardness, and offers heuristic algorithms and rewrite strategies for efficient quantum compilation.

Findings

Heuristic algorithms effectively find trail covers in complex graphs.

Reduction of the problem to the traveling salesman problem enables practical solutions.

Benchmarks demonstrate improved fusion efficiency in quantum circuit compilation.

Abstract

Quantum compilation requires the development of new algorithms that optimise the cost of implementing quantum computations on physical hardware. Often this gives rise to problems which are asymptotically hard to solve classically, and for which heuristics and reductions to known problems are of great practical use. In this paper, we study three graph-theoretic problems which can be seen as generalisations of the Eulerian and Hamiltonian path problems. These arise in photonic implementations of measurement-based quantum computing, where graph states are constructed by fusing bounded-length linear resource states. Since the fusion operation succeeds with probability smaller than one, we wish to minimise the number of fusions required to build a particular graph state and this corresponds to finding a minimal path or trail cover of the graph. We show that these covering problems are NP-hard in most cases and give heuristic algorithms for finding trail covers in graphs including a reduction to the travelling salesman problem. We propose new rewrite strategies for graph states that reduce the number of fusions required to build a given graph. Finally, we apply these algorithms to the compilation of photonic fusion networks and provide a series of benchmarks showing the performance of our algorithms on common error-correcting codes and circuits from the QASMBench set.