Clifford circuit based heuristic optimization of fermion-to-qubit mappings

TL;DR

This paper introduces a heuristic numerical optimization framework using Clifford circuit optimization and simulated annealing to improve fermion-to-qubit mappings, significantly reducing Pauli weight and enhancing quantum simulation efficiency.

Contribution

It presents a novel heuristic optimization method for fermion-to-qubit mappings that outperforms traditional mappings, including ternary-tree-based ones, for various fermionic Hamiltonians.

Findings

Optimized mappings outperform conventional mappings in Pauli weight reduction.

Achieved 15-40% improvements in average Pauli weight for complex Hamiltonians.

Significant improvements (>40%) for 6x6 nearest-neighbor hopping and Hubbard models.

Abstract

Simulation of interacting fermionic Hamiltonians is one of the most promising applications of quantum computers. However, the feasibility of analysing fermionic systems with a quantum computer hinges on the efficiency of fermion-to-qubit mappings that encode non-local fermionic degrees of freedom in local qubit degrees of freedom. While recent works have highlighted the importance of designing fermion-to-qubit mappings that are tailored to specific problem Hamiltonians, the methods proposed so far are either restricted to a narrow class of mappings or they use computationally expensive and unscalable brute-force search algorithms. Here, we address this challenge by designing a $\mathrm{\textbf{heuristic}}$ numerical optimization framework for fermion-to-qubit mappings. To this end, we first translate the fermion-to-qubit mapping problem to a Clifford circuit optimization problem, and then use simulated annealing to optimize the average Pauli weight of the problem Hamiltonian. For all fermionic Hamiltonians we have considered, the numerically optimized mappings outperform their conventional counterparts, including ternary-tree-based mappings that are known to be optimal for single creation and annihilation operators. We find that our optimized mappings yield between $15\%$ to $40\%$ improvements on the average Pauli weight when the simulation Hamiltonian has an intermediate level of complexity. Most remarkably, the optimized mappings improve the average Pauli weight for $6 \times 6$ nearest-neighbor hopping and Hubbard models by more than $40\%$ and $20\%$, respectively. Surprisingly, we also find specific interaction Hamiltonians for which the optimized mapping outperform $\mathrm{\textbf{any}}$ ternary-tree-based mapping. Our results establish heuristic numerical optimization as an effective method for obtaining mappings tailored for specific fermionic Hamiltonian.