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Simulating fermions with a digital quantum computer

Nature Reviews Physics (2026)Cite this article

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Abstract

Quantum computers are expected to become a powerful tool for studying physical quantum systems. Consequently, a number of quantum algorithms to determine the physical properties of such systems have been developed. Although 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 chemistry, which is now finding another application in the simulation of fermionic systems on quantum computers based on qubits. In this Review, we discuss methods for encoding fermionic degrees of freedom into qubits.

Key points

  • Physical systems have complex interactions that can involve fermions, and computing physically relevant quantities is classically challenging; quantum simulation algorithms to digitally prepare and estimate observables are actively researched.

  • The study of physical systems beyond the reach of classical methods has been identified as one of the most important applications of the emerging technology.

  • Simulating fermions on a quantum computer requires encoding the antisymmetric exchange into qubits, the basic memory elements of most quantum computers.

  • First quantization, in which the number of fermions and their antisymmetrization correlation are explicitly encoded in the many-particle states, provides a compact representation of many-electron systems restricted to a fixed particle-number subspace.

  • Second quantization, a generically applicable formalism natural in many applications areas, is amenable to a variety of encoding methods depending on the structure of the problem and available computing resources.

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Fig. 1: Overview of the four typical simulation steps.
Fig. 2: Two Hamiltonian simulation algorithms to approximate the time evolution of fermionic systems.
Fig. 3: The typical form for state antisymmetrization as part of fermionic simulation in first quantization.
Fig. 4: Second-quantized encoding methods.
Fig. 5: Ancilla-free encoding on a linear tree.
Fig. 6: Local encoding of operators.
Fig. 7: Factors that influence mapping choices.

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Acknowledgements

R.W.C. acknowledges support by the US Department of Energy, Office of Science, National Quantum Information Science Research Centers, Quantum Systems Accelerator, and by the US Department of Energy, Office of Science, Office of Advanced Scientific Computing Research, Exploratory Research for Extreme Scale Science programme. Sandia National Laboratories is a multi-mission laboratory managed and operated by National Technology and Engineering Solutions of Sandia, LLC (NTESS), a wholly owned subsidiary of Honeywell International Inc., for the US Department of Energy’s National Nuclear Security Administration (DOE/NNSA) under contract DE-NA0003525. This written work is authored by an employee of NTESS. The employee, not NTESS, owns the right, title and interest in and to the written work and is responsible for its contents. Any subjective views or opinions that might be expressed in the written work do not necessarily represent the views of the US Government. The publisher acknowledges that the US Government retains a non-exclusive, paid-up, irrevocable, worldwide license to publish or reproduce the published form of this written work or allow others to do so, for US Government purposes. The DOE will provide public access to results of federally sponsored research in accordance with the DOE Public Access Plan. S.S. was supported by the Royal Society University Research Fellowship and “Quantum simulation algorithms for quantum chromodynamics” grant (ST/W006251/1). M.M. acknowledges funding by RQS QLCI grant OMA-2120757. M.C. received the support of a Cambridge Australia Allen & DAMTP Scholarship during the preparation of this document. C.M. acknowledges support from the Intelligence Advanced Research Projects Activity (IARPA), under the Entangled Logical Qubits programme through Cooperative Agreement Number W911NF-23-2-0223. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of IARPA, the Army Research Office, or the US Government. The US Government is authorized to reproduce and distribute reprints for government purposes notwithstanding any copyright notation herein. J.D.W., J.N., B.H., W.W., T.M.H. and G.E.S. acknowledge support by the US Department of Energy, Office of Basic Energy Sciences, under award DE-SC0019374. G.E.S. is a Welch Foundation Chair (C-0036). W.W. and J.D.W. were also partially supported by ARO grant W911NF2410043. J.D.W. holds concurrent appointments at Dartmouth College and as an Amazon Visiting Academic. This paper describes work performed at Dartmouth College and is not associated with Amazon.

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Authors and Affiliations

  1. Sandia National Laboratories, Albuquerque, NM, USA

    Riley W. Chien

  2. DAMTP, Centre for Mathematical Sciences, University of Cambridge, Cambridge, UK

    Mitchell Chiew

  3. Department of Physics and Astronomy, Dartmouth College, Hanover, NH, USA

    Brent Harrison, Jason Necaise, Weishi Wang & James D. Whitfield

  4. Joint Center for Quantum Information and Computer Science, University of Maryland, College Park, MD, USA

    Maryam Mudassar

  5. School of Physics, University of Sydney, Sydney, New South Wales, Australia

    Campbell McLauchlan

  6. Department of Chemistry, Rice University, Houston, TX, USA

    Thomas M. Henderson & Gustavo E. Scuseria

  7. Department of Physics and Astronomy, Rice University, Houston, TX, USA

    Thomas M. Henderson & Gustavo E. Scuseria

  8. Department of Computer Science, University of Oxford, Oxford, UK

    Sergii Strelchuk

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Contributions

R.W.C. managed the project and led the drafting of the manuscript. J.D.W. initiated and supervised the project. M.C. and R.W.C. drafted the figures. All authors contributed to writing and editing of the manuscript.

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Glossary

η

The number of electrons in a system.

Ancilla qubits

Additional qubits that serve an assisting role.

Clifford unitaries

Unitaries that map Pauli operators to Pauli operators by conjugation; also known as Clifford gates.

Encoded Hamiltonian

(\(\mathop{H}\limits^{ \sim }\)). A result of the particular encoding chosen; the tilde notation is used for encoded operators in general.

Encodings

Embeddings of the Hilbert space of a system into the Hilbert space of a collection of qubits.

Fermion parity

The operator which has an eigenvalue of +1 on states with an even number of fermionic particles and −1 on states with an odd number of fermionic particles.

Fermionic modes

Degrees of freedom with two othonormal states, unoccupied and occupied by a fermionic particle.

First quantization

A representation of a many-body system whose degrees of freedom are particles.

H

The Hamiltonian of the fermionic system of interest.

M

The number of orbitals.

n

The number of qubits or fermionic modes in a system.

Non-Clifford gate

Any unitary that is not a Clifford gate; generally expected to be the most expensive operations in a fault-tolerant quantum computer.

Pauli operators

Tensor products of \(I=(\begin{array}{cc}1 & 0\\ 0 & 1\end{array})\), \(X=(\begin{array}{cc}0 & 1\\ 1 & 0\end{array})\), \(Y=(\begin{array}{cc}0 & -i\\ i & 0\end{array})\) and \(Z=(\begin{array}{cc}1 & 0\\ 0 & -1\end{array})\).

Pauli weight

The number of non-identity factors in a Pauli operator.

Qubits

The fundamental units of information storage on a digital quantum computer; each qubit is associated to the Hilbert space \({\mathscr{H}}={{\rm{{\mathbb{C}}}}}^{2}\).

Second quantization

A representation of a many-body system whose degrees of freedom are modes.

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Chien, R.W., Chiew, M., Harrison, B. et al. Simulating fermions with a digital quantum computer. Nat Rev Phys (2026). https://doi.org/10.1038/s42254-025-00914-5

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