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arXiv:2606.12769

Matrix phase-space representations for quantum symmetries

Peter D. Drummond, Alexander S. Dellios, Margaret D. Reid

quant-phSubmitted Jun 11, 2026• Announced Jun 12, 2026• 0 citations
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Abstract

We introduce a general phase-space representation that includes global quantum symmetries in the basis expansion. This method, called matrix phase-space, projects the basis onto a reduced Hilbert space, which can greatly reduce sampling errors of many-body quantum simulations and unifies several previous phase-space methods. The purpose of this paper is to provide detailed proofs of basic theorems and operator identities. We also treat several different types of symmetries. To illustrate the benefits of matrix phase-space methods, we give a detailed derivation of a recent application to the topical problem of verifying the outputs of Gaussian boson sampling (GBS) quantum computers with photon number resolving detectors. This has exponential complexity, and using parity symmetry reduces sampling errors by very large factors relative to earlier methods.

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We introduce a general phase-space representation that includes global quantum symmetries in the basis expansion. This method, called matrix phase-space, projects the basis onto a reduced Hilbert space, which can greatly reduce sampling errors of many-body quantum simulations and unifies several previous phase-space methods. The purpose of this paper is to provide detailed proofs of basic theorems and operator identities. We also treat several different types of symmetries. To illustrate the benefits of matrix phase-space methods, we give a detailed derivation of a recent application to the topical problem of verifying the outputs of Gaussian boson sampling (GBS) quantum computers with photon number resolving detectors. This has exponential complexity, and using parity symmetry reduces sampling errors by very large factors relative to earlier methods.
We introduce coherent matrix phase-space distributions. These use conservation laws and symmetries to improve the accuracy and speed of quantum phase-space representations. As an example, this is applied to the validation of low-loss Gaussian boson sampling (GBS) quantum computational advantage experiments, where classical generation of the random photon-number counts is exponentially hard. Large improvements in sampling errors are demonstrated compared to previous methods. Matrix phase-space representations also provide a large numerical speedup due to their (at worst) quadratic scaling, compared to other methods for validating total count probabilities of large-scale, low-loss GBS networks.
Recent papers have developed a cyclic symmetry projected extension for coherent state representations of bosonic Hilbert space. This extension can greatly reduce sampling errors in phase-space simulations of bosonic quantum systems that have symmetries. Here we prove an eigenvalue theorem for the continuum limit of the projected coherent states. Our intuitively appealing result is that in the continuum limit the projected coherent states are eigenstates of the symmetry generating operator. This result gives a differential identity in phase-space that includes the conserved eigenvalues. Detailed applications of these differential identities to obtaining improved Fokker-Planck and stochastic equations in many-body physics will be given elsewhere.
Phase-space representations are of increasing importance as a viable and successful means to study exponentially complex quantum many-body systems from first principles. This paper traces the background of these methods, starting from the early work of Wigner, Glauber and Sudarshan. We focus on modern phase-space approaches using non-classical phase-space representations. These lead to the Gaussian representation, which unifies bosonic and fermionic phase-space. Examples treated include quantum solitons in optical fibers, colliding Bose-Einstein condensates, and strongly correlated fermions on lattices.
Abstract An important challenge with the current generation of noisy, large-scale quantum computers is the question of validation. Does the hardware generate correct answers? If not, what are the errors? This issue is often combined with questions of computational advantage, but it is a fundamentally distinct issue. In current experiments, complete validation of the output statistics is generally not possible because it is exponentially hard to do so. Here, we apply phase-space simulation methods to partially verify recent experiments on Gaussian boson sampling (GBS) implementing photon-number resolving detectors. The positive-P phase-space distribution is employed, as it uses probabilistic sampling to reduce complexity. It is ∼ 10 18 times faster than direct classical simulation for experiments on 288 modes where quantum computational advantage is claimed. When combined with binning and marginalization to improve statistics, multiple validation tests are efficiently computable, of which some tests can be carried out on experimental data. We show that the data as a whole has discrepancies with theoretical predictions for perfect squeezing. However, a modification of the GBS parameters greatly improves agreement for some tests. We suggest that such validation tests could form the basis of feedback methods to improve GBS experiments.
A phase space representation is defined for quantum mechanical state functions. General requirements of quantum mechanics and some additional, very reasonable restrictions lead to a representation that is uniquely determined apart from a scaling. With one choice of this scaling, the representation is equivalent to that of Bargmann, although different aspects are emphasized here. Equivalence between position and momentum representations leads to a phase space representation that is invariant to, or at most rescaled by, Fourier transform. Phase space equivalents of some common functions are presented including Cartesian Gaussian basis functions and the hydrogen atom ground state function.
We present a detailed discussion of a general theory of phase-space distributions, introduced recently by the authors [J. Phys. A 31, L9 (1998)]. This theory provides a unified phase-space formulation of quantum mechanics for physical systems possessing Lie-group symmetries. The concept of generalized coherent states and the method of harmonic analysis are used to construct explicitly a family of phase-space functions which are postulated to satisfy the Stratonovich-Weyl correspondence with a generalized tracing condition. The symbol calculus for the phase-space functions is given by means of the generalized twisted product. The phase-space formalism is used to study the problem of the reconstruction of quantum states. In particular, we consider the reconstruction method based on measurements of displaced projectors, which comprises a number of recently proposed quantum-optical schemes and is also related to the standard methods of signal processing. A general group-theoretic description of this method is developed using the technique of harmonic expansions on the phase space.