A SAT approach to the initial mapping problem in SWAP gate insertion for commuting gates

TL;DR

This paper introduces a SAT-based method for optimizing initial qubit mappings in quantum circuits with commuting gates, significantly reducing swap gate counts and enabling larger problem sizes on noisy quantum hardware.

Contribution

It presents a novel SAT formulation for initial mapping in commuting gate circuits and combines it with clustering heuristics to handle large-scale problems efficiently.

Findings

Achieves 65% reduction in swap gates for 500-node graphs.

Reduces swap layers by 25% in large 1000-node graphs.

Enables study of quantum algorithms on larger noisy hardware.

Abstract

Most quantum circuits require SWAP gate insertion to run on quantum hardware with limited qubit connectivity. A promising SWAP gate insertion method for blocks of commuting two-qubit gates is a predetermined swap strategy which applies layers of SWAP gates simultaneously executable on the coupling map. A good initial mapping for the swap strategy reduces the number of required swap gates. However, even when a circuit consists of commuting gates, e.g., as in the Quantum Approximate Optimization Algorithm (QAOA) or trotterized simulations of Ising Hamiltonians, finding a good initial mapping is a hard problem. We present a SAT-based approach to find good initial mappings for circuits with commuting gates transpiled to the hardware with swap strategies. Our method achieves a 65% reduction in gate count for random three-regular graphs with 500 nodes. In addition, we present a heuristic approach that combines the SAT formulation with a clustering algorithm to reduce large problems to a manageable size. This approach reduces the number of swap layers by 25% compared to both a trivial and random initial mapping for a random three-regular graph with 1000 nodes. Good initial mappings will therefore enable the study of quantum algorithms, such as QAOA and Ising Hamiltonian simulation applied to sparse problems, on noisy quantum hardware with several hundreds of qubits.