Exact and practical pattern matching for quantum circuit optimization

Quantum computations are typically compiled into a circuit of basic quantum gates. Just like for classical circuits, a quantum compiler should optimize the quantum circuit, e.g. by minimizing the number of required gates. Optimizing quantum circuits is not only relevant for improving the runtime of quantum algorithms in the long term, but is also particularly important for near-term quantum devices that can only implement a small number of quantum gates before noise renders the computation useless. An important building block for many quantum circuit optimization techniques is pattern matching, where given a large and a small quantum circuit, we are interested in finding all maximal matches of the small circuit, called pattern, in the large circuit, considering pairwise commutation of quantum gates. In this work, we present a classical algorithm for pattern matching that provably finds all maximal matches in time polynomial in the circuit size (for a fixed pattern size). Our algorithm works for both quantum and reversible classical circuits. We demonstrate numerically that our algorithm, implemented in the open-source library Qiskit, scales considerably better than suggested by the theoretical worst-case complexity and is practical to use for circuit sizes typical for near-term quantum devices. Using our pattern matching algorithm as the basis for known circuit optimization techniques such as template matching and peephole optimization, we demonstrate a significant (~30%) reduction in gate count for random quantum circuits, and are able to further improve practically relevant quantum circuits that were already optimized with state-of-the-art techniques.