Fermihedral: On the Optimal Compilation for Fermion-to-Qubit Encoding

TL;DR

Fermihedral reframes Fermion-to-qubit encoding as a Boolean SAT problem by encoding Majorana operators into Pauli strings and incorporating constraints (anticommutativity, independence, vacuum preservation) and cost objectives (Hamiltonian-independent and -dependent Pauli weights). It introduces scalable strategies to overcome clause explosion, notably ignoring algebraic independence with a quantifiable low failure probability and employing simulated annealing to optimize Hamiltonian-dependent weight. Across electronic-structure, Fermi-Hubbard, and SYK benchmarks, the SAT-based encodings yield substantial reductions in Pauli weight, gate counts, and circuit depth, improving simulation accuracy in noisy-classical and real-device contexts (IonQ Aria-1). This approach demonstrates a practical pathway to near-optimal Fermion-to-qubit encodings for diverse Hamiltonians, with broad implications for quantum simulation performance and hardware-aware compilation.

Abstract

This paper introduces Fermihedral, a compiler framework focusing on discovering the optimal Fermion-to-qubit encoding for targeted Fermionic Hamiltonians. Fermion-to-qubit encoding is a crucial step in harnessing quantum computing for efficient simulation of Fermionic quantum systems. Utilizing Pauli algebra, Fermihedral redefines complex constraints and objectives of Fermion-to-qubit encoding into a Boolean Satisfiability problem which can then be solved with high-performance solvers. To accommodate larger-scale scenarios, this paper proposed two new strategies that yield approximate optimal solutions mitigating the overhead from the exponentially large number of clauses. Evaluation across diverse Fermionic systems highlights the superiority of Fermihedral, showcasing substantial reductions in implementation costs, gate counts, and circuit depth in the compiled circuits. Real-system experiments on IonQ's device affirm its effectiveness, notably enhancing simulation accuracy.

Figures (11)

• Figure 1: Simulating Fermionic systems with qubit systems
• Figure 2: Overview of Fermihedral framework
• Figure 3: From Pauli string evolution operator $e^{i\lambda P}$ to corresponding circuit
• Figure 4: Probability of $n$$A_k$'s holds simultaneously
• Figure 5: The three types of benchmark Hamiltonians used in this paper. $a^\dagger_*$ and $a_*$ are the creation and annihilation operators. $M_*$ is the Majorana operator.