Clifford Circuit Optimization with Templates and Symbolic Pauli Gates

TL;DR

This paper introduces novel Clifford circuit optimization techniques using template matching and symbolic peephole methods, significantly reducing two-qubit gate counts for large quantum circuits.

Contribution

It presents Clifford-specific templates and a symbolic peephole optimization approach, improving circuit efficiency beyond existing methods.

Findings

Circuits are within 0.2% of optimal for 6 qubits.

Two-qubit gate count reduced by 64.7% on average for circuits with up to 64 qubits.

Methods outperform Aaronson-Gottesman canonical form in gate reduction.

Abstract

The Clifford group is a finite subgroup of the unitary group generated by the Hadamard, the CNOT, and the Phase gates. This group plays a prominent role in quantum error correction, randomized benchmarking protocols, and the study of entanglement. Here we consider the problem of finding a short quantum circuit implementing a given Clifford group element. Our methods aim to minimize the entangling gate count assuming all-to-all qubit connectivity. First, we consider circuit optimization based on template matching and design Clifford-specific templates that leverage the ability to factor out Pauli and SWAP gates. Second, we introduce a symbolic peephole optimization method. It works by projecting the full circuit onto a small subset of qubits and optimally recompiling the projected subcircuit via dynamic programming. CNOT gates coupling the chosen subset of qubits with the remaining qubits are expressed using symbolic Pauli gates. Software implementation of these methods finds circuits that are only 0.2% away from optimal for 6 qubits and reduces the two-qubit gate count in circuits with up to 64 qubits by 64.7% on average, compared with the Aaronson-Gottesman canonical form.