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arXiv:2506.06425 · Physical Review A

Scalable simulation of fermionic encoding performance on noisy quantum computers

Emiliia Dyrenkova, Raymond Laflamme, Michael Vasmer

Published Dec 12, 2025• 0 citations
arXivHTMLPDF

Abstract

A compelling application of quantum computers with thousands of qubits is quantum simulation. Simulating fermionic systems is both a problem with clear real-world applications and a computationally challenging task. In order to simulate a system of fermions on a quantum computer, one has to first map the fermionic Hamiltonian to a qubit Hamiltonian. The most popular such mapping is the Jordan-Wigner encoding, which suffers from inefficiencies caused by the high weight of some encoded operators. As a result, alternative local encodings have been proposed that solve this problem at the expense of a constant factor increase in the number of qubits required. Some such encodings possess local stabilizers, i.e., Pauli operators that act as the logical identity on the encoded fermionic modes. A natural error mitigation approach in these cases is to measure the stabilizers and discard any run where a measurement returns a -1 outcome. Using a high-performance stabilizer simulator, we classically simulate the performance of a local encoding known as the Derby-Klassen encoding and compare its performance with the Jordan-Wigner encoding and the ternary tree encoding. Our simulations use more complex error models and significantly larger system sizes (up to $18\times18$) than in previous work. We find that the high sampling requirements of postselection methods with the Derby-Klassen encoding pose a limitation to its applicability in near-term devices and call for more encoding-specific circuit optimizations.

DOI
10.1103/dg5d-m728

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A compelling application of quantum computers with thousands of qubits is quantum simulation. Simulating fermionic systems is both a problem with clear real-world applications and a computationally challenging task. In order to simulate a system of fermions on a quantum computer, one has to first map the fermionic Hamiltonian to a qubit Hamiltonian. The most popular such mapping is the Jordan-Wigner encoding, which suffers from inefficiencies caused by the high weight of some encoded operators. As a result, alternative local encodings have been proposed that solve this problem at the expense of a constant factor increase in the number of qubits required. Some such encodings possess local stabilizers, i.e., Pauli operators that act as the logical identity on the encoded fermionic modes. A natural error mitigation approach in these cases is to measure the stabilizers and discard any run where a measurement returns a -1 outcome. Using a high-performance stabilizer simulator, we classically simulate the performance of a local encoding known as the Derby-Klassen encoding and compare its performance with the Jordan-Wigner encoding and the ternary tree encoding. Our simulations use more complex error models and significantly larger system sizes (up to 18×1818\times1818×18) than in previous work. We find that the high sampling requirements of postselection methods with the Derby-Klassen encoding pose a limitation to its applicability in near-term devices and call for more encoding-specific circuit optimizations.
The most scalable proposed methods of simulating lattice fermions on noisy quantum computers employ encodings that eliminate nonlocal operators using a constant factor more qubits and a nontrivial stabilizer group. In this work, we investigated the most straightforward error mitigation strategy using the stabilizer group, stabilizer postselection, that is very natural to the setting of fermionic quantum simulation. We numerically investigate the performance of the error mitigation strategy on a range of systems containing up to 42 qubits and on a number of fundamental quantum simulation tasks including non-equilibrium dynamics and variational ground state calculations. We find that at reasonable noise rates and system sizes, the fidelity of computations can be increased significantly beyond what can be achieved with the standard Jordan-Wigner transformation at the cost of increasing the number of shots by less than a factor of 10, potentially providing a meaningful boost to near-term quantum simulations. Our simulations are enabled by new classical simulation algorithms that scale with the logical Hilbert space dimension rather than the physical Hilbert space dimension.
The simulation of quantum many-body systems, relevant for quantum chemistry and condensed matter physics, is one of the most promising applications of near-term quantum computers before fault-tolerance. However, since the vast majority of quantum computing technologies are built around qubits and discrete gate-based operations, the translation of the physical problem into this framework is a crucial step. This translation will often be device specific, and a suboptimal implementation will be punished by the exponential compounding of errors on real devices. The importance of an efficient mapping is already revealed for models of spinful fermions in two or three dimensions, which naturally arise when the relevant physics relates to electrons. Using the most direct and well-known mapping, the Jordan-Wigner transformation, leads to a nonlocal representation of local degrees of freedom, and necessities efficient decompositions of nonlocal unitary gates into a sequence of hardware accessible local gates. In this paper, we provide a step-by-step recipe for simulating the paradigmatic two-dimensional Fermi-Hubbard model on a quantum computer using only local operations. To provide the ingredients for such a recipe, we briefly review the plethora of different approaches that have emerged recently but focus on the Derby-Klassen compact fermion mapping to make our discussion concrete. We provide a detailed recipe for an end-to-end simulation including embedding on a physical device, preparing initial states such as ground states, simulation of unitary time evolution, and measurement of observables and spectral functions. We explicitly compute the resource requirements for simulating a global quantum quench and conclude by discussing the challenges and future directions for simulating strongly correlated fermionic matter on quantum computers.
Simulating molecular systems on quantum computers requires efficient mappings from Fermionic operators to qubit operators. Traditional mappings such as Jordan-Wigner or Bravyi-Kitaev often produce high-weight Pauli terms, increasing circuit depth and measurement complexity. Although several local qubit mappings have been proposed to address this challenge, most are specialized for structured models like the Hubbard Hamiltonian and perform poorly for realistic chemical systems with dense two-body interactions. In this work, we utilize a suite of techniques to construct compact and noise-resilient Fermion-to-qubit mappings suitable for general molecular Hamiltonians. Building on the Generalized Superfast Encoding (GSE) and other similar works, we demonstrate that it outperforms prior encodings in both accuracy and hardware efficiency for molecular simulations. Our improvements include path optimization within the Hamiltonian's interaction graph to minimize operator weight, introduction of multi-edge graph structures for enhanced error detection without added circuit depth, and a stabilizer measurement framework that directly maps logical terms and stabilizers to the Z-basis using Clifford simulation. Applying these methods to simulations of (H2)2(H_2)_2(H2​)2​ and (H2)3(H_2)_3(H2​)3​ systems yields significantly improved absolute and correlation energy estimates under realistic hardware noise, with further accuracy gains achieved by increasing code distance. We also propose a [[2N, N, 2]] variant of GSE compatible with square-lattice and (quasi-)linear hardware topologies, demonstrating a twofold reduction in RMSE for orbital rotations on IBM Kingston hardware. These results establish GSE as a very attractive mapping for molecular quantum simulations.
Quantum simulations of fermionic many-body systems crucially rely on mappings from indistinguishable fermions to distinguishable qubits. The non-local structure of fermionic Fock space necessitates encodings that either map local fermionic operators to non-local qubit operators, or encode the fermionic representation in a long-range entangled code space. In this latter case, there is an unavoidable trade-off between two desirable properties of the encoding: low weight representations of local fermionic operators, and a high distance code space. Here it is argued that despite this fundamental limitation, fermionic encodings with low-weight representations of local fermionic operators can still exhibit error mitigating properties which can serve a similar role to that played by high code distances. In particular when undetectable errors correspond to "natural" fermionic noise. We illustrate this point explicitly for two fermionic encodings: the Verstraete-Cirac encoding, and an encoding appearing in concurrent work by Derby and Klassen. In these encodings many, but not all, single-qubit errors can be detected. However we show that the remaining undetectable single-qubit errors map to local, low-weight fermionic phase noise. We argue that such noise is natural for fermionic lattice models. This suggests that even when employing low-weight fermionic encodings, error rates can be suppressed in a similar fashion to high distance codes, provided one is willing to accept simulated natural fermionic noise in their simulated fermionic system.
Quantum simulations before fault tolerance suffer from the intrinsic noise present in quantum computers. In this regime, extracting meaningful results greatly benefits from stability against that noise. This stability, defined as an error in observables that is independent of the system's size, is expected in local systems under local noise. In fermionic systems, the encoding of the fermionic degrees of freedom into qubits can introduce non-locality, making stability more delicate. Here, we investigate the stability to noise of fermion-to-qubit mappings. We consider noisy quantum circuits in DDD dimensions modeled by alternating layers of local unitaries and general, single-qubit Pauli noise. We show that, when using local fermionic encodings, expectation values of quadratic fermionic observables are stable to noise in states with spatially decaying correlations: a power-law decay with exponent μ>Dμ>Dμ>D is sufficient for stability. By contrast, we show that this stability cannot be achieved by non-local encodings such as Jordan-Wigner in 2D2D2D, or quasi-local ones such as the Bravyi-Kitaev transform. Our findings formalize the intuition that decaying correlations of the physical systems under study provide protection against noise for local fermionic encodings, and help inform design principles in near-term quantum simulations.
Simulation of fermionic systems is one of the most promising applications of quantum computers. It spans problems in quantum chemistry, high-energy physics and condensed matter. Underpinning the core steps of any quantum simulation algorithm, fermion-qubit mappings translate the fermionic interactions to the operators and states of quantum computers. This translation is highly non-trivial: a burgeoning supply of fermion-qubit mappings has arisen over the past twenty years to address the limited resources of early quantum technology. Previous literature has presented a dichotomy between ancillafree fermion-qubit mappings, which minimise qubit count, and local encodings, which minimise gate complexity. We present two computational approaches to the construction of general mappings while working with a limited number of qubits, striking a balance between the low-qubit and low-gate demands of present quantum technology. The first method frames the order of fermionic labels as an instance of the quadratic assignment problem to minimize the total and maximum Pauli weights in a problem Hamiltonian. We compare the orderoptimized performance of several common ancilla-free mappings on systems of size up to 225 fermionic modes. The second method is a computational approach to incrementally add ancilla qubits to Jordan-Wigner transformations and further reduce the Pauli weights. By adding up to 10 ancilla qubits, we were able to reduce the total Pauli weight by as much as 67 % in JordanWigner transformations of fermionic systems with up to 64 modes, outperforming the previous state-of-the-art ancilla-free mappings. Reproducibility: source code and data are available at https://github.com/cameton/QCE_QubitAssignment