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arXiv:2504.19317

Classical simulation of parity-preserving quantum circuits

Carolin Wille, Sergii Strelchuk

Published Apr 27, 2025• 0 citations
arXivHTMLPDF

Abstract

We present a classical simulation method for fermionic quantum systems which, without loss of generality, can be represented by parity-preserving circuits made of two-qubit gates in a brick-wall structure. We map such circuits to a fermionic tensor network and introduce a novel decomposition of non-Matchgate gates into a Gaussian fermionic tensor and a residual quartic term, inspired by interacting fermionic systems. The quartic term is independent of the specific gate, which allows us to precompute intermediate results independently of the exact circuit structure and leads to significant speedups when compared to other methods. Our decomposition suggests a natural perturbative expansion which can be turned into an algorithm to compute measurement outcomes and observables to finite accuracy when truncating at some order of the expansion. For particle number conserving gates, our decomposition features a unique truncation cutoff reducing the computational effort for high precision calculations. Our algorithm significantly lowers resource requirements for simulating parity-preserving circuits while retaining high accuracy, making it suitable for simulations of interacting systems in quantum chemistry and material science. Lastly, we discuss how our algorithm compares to other classical simulation methods for fermionic quantum systems.

DOI
10.48550/arxiv.2504.19317

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We present a classical simulation method for fermionic quantum systems which, without loss of generality, can be represented by parity-preserving circuits made of two-qubit gates in a brick-wall structure. We map such circuits to a fermionic tensor network and introduce a novel decomposition of non-Matchgate gates into a Gaussian fermionic tensor and a residual quartic term, inspired by interacting fermionic systems. The quartic term is independent of the specific gate, which allows us to precompute intermediate results independently of the exact circuit structure and leads to significant speedups when compared to other methods. Our decomposition suggests a natural perturbative expansion which can be turned into an algorithm to compute measurement outcomes and observables to finite accuracy when truncating at some order of the expansion. For particle number conserving gates, our decomposition features a unique truncation cutoff reducing the computational effort for high precision calculations. Our algorithm significantly lowers resource requirements for simulating parity-preserving circuits while retaining high accuracy, making it suitable for simulations of interacting systems in quantum chemistry and material science. Lastly, we discuss how our algorithm compares to other classical simulation methods for fermionic quantum systems.
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The mapping of fermionic states onto qubit states, as well as the mapping of fermionic Hamiltonian into quantum gates enables us to simulate electronic systems with a quantum computer. Benefiting the understanding of many-body systems in chemistry and physics, quantum simulation is one of the great promises of the coming age of quantum computers. One challenge in realizing simulations on near-term quantum devices is the large number of qubits required by such mappings. In this work, we develop methods that allow us to trade-off qubit requirements against the complexity of the resulting quantum circuit. We first show that any classical code used to map the state of a fermionic Fock space to qubits gives rise to a mapping of fermionic models to quantum gates. As an illustrative example, we present a mapping based on a non-linear classical error correcting code, which leads to significant qubit savings albeit at the expense of additional quantum gates. We proceed to use this framework to present a number of simpler mappings that lead to qubit savings with only a very modest increase in gate difficulty. We discuss the role of symmetries such as particle conservation, and savings that could be obtained if an experimental platform could easily realize multi-controlled gates.
Abstract This work explores displaced fermionic Gaussian operators with nonzero linear terms. We first demonstrate equivalence between several characterizations of displaced Gaussian states. We also provide an efficient classical simulation protocol for displaced Gaussian circuits and demonstrate their computational equivalence to circuits composed of nearest-neighbor matchgates augmented by single-qubit gates on the initial line. Finally, we construct a novel Gaussianity-preserving unitary embedding that maps n-qubit displaced Gaussian states to ( n + 1 ) -qubit even Gaussian states. This embedding facilitates the generalization of existing Gaussian testing protocols to displaced Gaussian states and unitaries. Our results provide new tools to analyze fermionic systems beyond the constraints of parity super-selection, extending the theoretical understanding and practical simulation of fermionic quantum computation.
Fermionic Gaussian states (FGSs) and the associated matchgate circuits play a central role in quantum information theory and condensed matter physics. Despite being possibly highly entangled, they can still be efficiently simulated on classical computers. We address the question of how to optimally create such states when using matchgate circuits acting on product states. To this end, we derive lower bounds on the number of gates required to prepare an arbitrary pure FGS: We establish both an asymptotic bound on the minimal gate count over general nearest-neighbor gate sets and an exact bound for circuits composed solely of matchgates. We present explicit algorithms whose constructions saturate these bounds, thereby proving their optimality. We furthermore determine when an FGS can be prepared with a circuit of any given depth, and derive an algorithm that constructs such a circuit whenever this condition is satisfied, either exactly or approximately. Our results have direct applications to (approximate) state preparation and to disentangling procedures. Moreover, we introduce a new classical simulation algorithm for matchgate circuits, based entirely on manipulating the generating circuits of the FGSs. Finally, we briefly study an extension of our framework for ttt-doped Gaussian states and circuits.
The mapping of fermionic states onto qubit states, as well as the mapping of fermionic Hamiltonian into quantum gates enables us to simulate electronic systems with a quantum computer. Benefiting the understanding of many-body systems in chemistry and physics, quantum simulation is one of the great promises of the coming age of quantum computers. Interestingly, the minimal requirement of qubits for simulating Fermions seems to be agnostic of the actual number of particles as well as other symmetries. This leads to qubit requirements that are well above the minimal requirements as suggested by combinatorial considerations. In this work, we develop methods that allow us to trade-off qubit requirements against the complexity of the resulting quantum circuit. We first show that any classical code used to map the state of a fermionic Fock space to qubits gives rise to a mapping of fermionic models to quantum gates. As an illustrative example, we present a mapping based on a nonlinear classical error correcting code, which leads to significant qubit savings albeit at the expense of additional quantum gates. We proceed to use this framework to present a number of simpler mappings that lead to qubit savings with a more modest increase in gate difficulty. We discuss the role of symmetries such as particle conservation, and savings that could be obtained if an experimental platform could easily realize multi-controlled gates.
We develop and analyze a method for simulating quantum circuits on classical computers by representing quantum states as rooted tree tensor networks. Our algorithm first determines a suitable, fixed tree structure adapted to the expected entanglement generated by the quantum circuit. The gates are sequentially applied to the tree by absorbing single-qubit gates into leaf nodes, and splitting two-qubit gates via singular value decomposition and threading the resulting virtual bond through the tree. We theoretically analyze the applicability of the method as well as its computational cost and memory requirements, and identify advantageous scenarios in terms of required bond dimensions as compared to a matrix product state representation. The study is complemented by numerical experiments for different quantum circuit layouts up to 37 qubits.