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

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.