Putting fermions onto a digital quantum computer

Quantum computers are expected to become a powerful tool for studying physical quantum systems. Consequently, a number of quantum algorithms for studying the physical properties of such systems have been developed. While qubit-based quantum computers are naturally suited to the study of spin-1/2 systems, systems containing other degrees of freedom must first be encoded into qubits. Transformations to and from fermionic degrees of freedom have long been an important tool in physics and, now the simulation of fermionic systems on quantum computers based on qubits provides yet another application. In this perspective, we review methods for encoding fermionic degrees of freedom into qubits and attempt to dispel the persistent notion that fermionic systems beyond one dimension are fundamentally more difficult to deal with.

💡 Research Summary

This perspective reviews the state‑of‑the‑art methods for encoding fermionic degrees of freedom onto qubit‑based digital quantum computers. After motivating the need for quantum simulation in three major domains—quantum chemistry, condensed‑matter physics, and high‑energy physics—the authors outline a generic four‑step quantum algorithm workflow: representation choice, initialization, state preparation, and observable estimation. The core of the review contrasts two fundamentally different frameworks: first‑quantization, which directly encodes antisymmetric many‑particle wavefunctions in a fixed‑particle‑number subspace, and second‑quantization, which uses mode‑based creation and annihilation operators and is applicable to variable particle numbers. Within second‑quantization, a suite of fermion‑to‑qubit mappings is examined. The Jordan–Wigner transformation provides a conceptually simple but O(N) non‑local mapping; the Bravyi–Kitaev scheme reduces operator length to O(log N). Recent ancilla‑free encodings avoid auxiliary qubits while preserving antisymmetry, and symmetry‑based qubit reduction techniques exploit conserved quantities (electron number, spin, point‑group symmetries) to eliminate unnecessary qubits. Local encodings are highlighted for their compatibility with hardware connectivity constraints. The paper then surveys state‑preparation strategies—including variational quantum eigensolvers, quantum phase estimation, and quantum Metropolis algorithms—and discusses measurement protocols for static observables, dynamical correlators, and multi‑time quantities, emphasizing techniques such as operator grouping and classical post‑processing to reduce sampling overhead. A comparative resource analysis shows that while Jordan–Wigner remains convenient for small molecules, Bravyi–Kitaev, local, or symmetry‑reduced encodings become essential for large lattice models like the Hubbard or gauge‑theory Hamiltonians. The authors argue that the perceived difficulty of simulating fermions beyond one dimension is largely a matter of encoding choice, not a fundamental barrier, and they outline future directions including error‑corrected implementations, automated mapping selection, and integration with emerging quantum hardware.