Putting fermions onto a digital quantum computer

TL;DR

This review analyzes how to encode fermionic degrees of freedom onto qubit-based quantum computers, focusing on first and second quantization frameworks and a variety of fermion-to-qubit mappings. It surveys applications across quantum chemistry, condensed matter, and high-energy physics, and presents core algorithms for state preparation and observable estimation, alongside Hamiltonian-simulation primitives such as Trotterization and quantum signal processing. The article outlines encoding strategies (Jordan–Wigner, Bravyi–Kitaev, ancilla-free, symmetry-based tapering, local encodings) and discusses their resource trade-offs, including qubit counts, operator weights, and locality considerations. It concludingly emphasizes that artifact-level encodings and symmetry exploitation can significantly shape scalability, with fault-tolerant quantum computing as a long-term pathway and near-term locality-preserving methods offering practical progress in model systems and lattice gauge theories.

Abstract

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.

Figures (6)

• Figure 1: Simulation overview through four steps: 0. Choose a representation 1. Initialize the easy-to-prepare physical starting state $|\psi_0\rangle$ 2. Prepare the desired state of interest $\rho$. 3. Perform measurements $\rho\to\mathcal{M}[\rho]$ and estimate observables.
• Figure 2: Some Hamiltonian simulation algorithms applied to fermionic systems. The fermion-to-qubit mapping determines the qubit Hamiltonian, which in turn determines the resource cost of simulation algorithms.
• Figure 3: The typical form for state anti-symmetrization as part of fermionic simulation in first quantization. To represent a fermionic state, the circuit input is a product of qubit states $|\varphi_{m_i}\rangle$ which indicate that the $i$th fermion is in the spin-orbital that has label $m_i$. The output of the circuit is an antisymmetrized qubit representation of the desired fermionic state.
• Figure 4: Second-quantized encoding methods.
• Figure 5: The ternary tree transformation $\gamma_i, \overline{\gamma}_i \mapsto \pm P_i, \overline{P}_i$ is an ancilla-free encoding that maps from Majorana to Pauli operators using the structure of a tree graph.