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

TL;DR

The paper tackles the bottleneck of simulating fermionic systems on quantum devices by minimizing the nonlocal Pauli weight that results from fermion-to-qubit mappings. It reframes mapping design as a Clifford-circuit optimization problem and uses simulated annealing to minimize the average Pauli weight of the transformed Hamiltonian, starting from standard mappings like JWT or BKT. The authors prove that Clifford transformations can reproduce all ternary-tree mappings and also enable non-ternary mappings, showing substantial improvements (up to about 40%) over conventional mappings for intermediate-complexity Hamiltonians, including 2D Hubbard models and hydrogen chains; some cases even outperform all ternary-tree mappings. The work demonstrates a flexible, problem-tailored approach to mappings that can significantly reduce circuit depth and resource costs for fermionic simulations, with potential integration with hardware-aware transpilation and further analytical understanding of optimized structures. Overall, the framework provides a practical route to customized, efficient fermion-to-qubit mappings across a broad class of lattice and chemistry Hamiltonians.

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.

Figures (12)

• Figure 1: Schematic for representing a physical fermionic system on a quantum device consisting of qubits. The physical system is first modelled using a fermionic Hamiltonian expressed using fermionic second-quantized operators. Conventional fermion-to-qubit mappings allows one to represent the fermionic Hamiltonian using operators acting on qubit degrees of freedom, albeit without taking into account the structure of problem Hamiltonian. Our work focuses on using numerical optimization to design a Clifford circuit, whose adjoint action on the qubit Hamiltonian minimizes a cost function (dotted lines).
• Figure 2: The conjugation action of CNOT reduces the weight of some Pauli strings
• Figure 3: Jordan-Wigner tree on $3$ qubits
• Figure 4: Bravyi-Kitaev tree on $5$ qubits
• Figure 5: An optimal ternary-tree on $5$ qubits

Theorems & Definitions (15)

• Lemma 1
• proof
• Theorem 1
• Theorem 2
• Theorem 3
• Lemma 2
• Lemma 3
• proof : Proofs of Lemma \ref{['lem:tree-rotation']} and Lemma \ref{['lem:tree-middle']}
• proof : Proof of Theorem \ref{['thm:jw-bk']}
• Remark 1