Matrix phase-space representations for quantum symmetries

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.

• The paper introduces a matrix phase-space representation that embeds symmetry projectors into the kernel, dramatically reducing sampling errors.
• It rigorously develops operator identities and weighting schemes to simulate quantum many-body dynamics efficiently, validated in Gaussian boson sampling.
• Numerical studies show that the matrix-P method achieves orders-of-magnitude improvement in convergence and error reduction compared to conventional approaches.

Matrix Phase-Space Representations for Quantum Symmetries: A Technical Overview

Introduction and Motivation

The exponential growth of Hilbert space with system size poses a core bottleneck for simulating quantum many-body dynamics, especially when quantum symmetries induce conserved quantities that are not easily exploited by conventional phase-space representations. Existing phase-space methods such as positive-P, stochastic gauge-P, and the stochastic Bloch representation can mitigate the curse of dimensionality to some extent but often struggle with sampling errors, particularly when addressing systems with strong global symmetries or high-order correlations. Drummond, Dellios, and Reid address this challenge by introducing the matrix phase-space (matrix-P) representation—a unification and rigorous generalization of earlier approaches built upon explicit incorporation of symmetry projectors within the phase-space kernel.

Formal Development of Matrix Phase-Space Representations

General Framework

The central construct is the expansion of the quantum density matrix $\hat{\rho}(t)$ as a matrix-valued weighted integral over properly projected phase-space operators:

$$\hat{\rho}(t) = \int P_M(\lambda, t) \Lambda^{(M)}(\lambda) \, d\lambda$$

Here, $P_M(\lambda, t)$ is a positive, normalized probability density over phase-space variables $\lambda$ (including coherent amplitudes and the symmetry weights), and $\Lambda^{(M)}(\lambda)$ is a kernel operator that includes projection onto symmetry eigenspaces. The kernel operator is explicitly constructed as

$$\Lambda^{(M)}(\lambda) = \Pi \Lambda^{(u)}(\vec{\alpha}) \Pi^\dagger e^{-w(\lambda)}$$

where symmetry is imposed by the projectors $\Pi$ associated with the global invariant(s). A stochastic density matrix $\Omega$ is also introduced for multiplet control of weights, generalizing gauge amplitudes [18,22].

Choice of Weights and Operator Normalization

Several possible weighting schemes are provided, including the "normal" weight—to recover positive-P and gauge-P as limiting cases—and the "simple" weight used in the stochastic Bloch approach. The normal weight formulation ensures probability normalization and positive semi-definiteness, while other choices such as regularized or "simple" weights can provide improved performance for specific physical regimes (e.g., in cases of broken symmetry).

Rigorous proofs are given for the existence and normalization of such expansions for arbitrary symmetry projections and for both bosonic and fermionic systems, leveraging the completeness and orthogonality properties of the projected coherent basis.

Operator Representations and Dynamical Equations

The authors derive the explicit forms of operator identities required to translate time evolution (via Liouville or master equations) and observable calculation into efficient phase-space sampling procedures. These include:

• Differential identities for projected kernel matrices under both normal and anti-normal ordering.
• Renormalization factors (matrices $T$) linking phase-space trajectories to symmetry sectors.
• Exact functional forms for marginal and grouped photon-counting observables in symmetry-projected subspaces.

For dynamics, it is shown that for symmetry-preserving Hamiltonians the projected distribution functions inherit considerable efficiency from only evolving the physically relevant sector, with sampling errors significantly reduced compared to unprojected methods.

Treatment of Quantum Symmetries

Discrete and Continuous Symmetries

The matrix-P construction generalizes to all quantum symmetries possessing group-theoretical generators, including phase (number) conservation and translation (momentum or crystal symmetry). For discrete groups, symmetry projectors are constructed as finite sums over group elements, while the $M \to \infty$ (continuous) limit recovers number or momentum eigenstate projections. The mapping is formally justified using the group equivalence lemma and direct analogues of Schur's lemma.

Parity Symmetry and GBS Applications

The paradigm application is Gaussian boson sampling (GBS), where the global parity symmetry (photon number mod 2) is preserved in lossless evolution of squeezed vacuum input states. For matrix-P with $$\hat{\rho}(t) = \int P_M(\lambda, t) \Lambda^{(M)}(\lambda) \, d\lambda$$0 (parity sectors), photon-counting probabilities can be computed extremely efficiently—even for 300+ modes—with sampling errors negligible relative to positive-P or unprojected approaches. The behavior generalizes to higher-order correlations and binning/grouping of detectors, as demonstrated by explicit simulation and comparison to analytic results for lossless and lossy GBS.

Crucially, when losses break symmetry (as in experimental realizations), the symmetry-projected phase-space approach remains valid but loses some of its exponential sampling advantage, smoothly interpolating to the usual positive-P framework in the high-loss regime.

Numerical and Analytical Results

In simulation studies of high-mode-number GBS, matrix-P methods achieve convergence to exact probability distributions using orders-of-magnitude fewer samples than positive-P, especially for observables sensitive to symmetry (e.g., even/odd oscillations in total counts arising from squeezed inputs). Figures in the paper illustrate the near-zero difference error achieved by matrix-P, contrasted with persistent systematic errors in positive-P under the same sample complexity. This computational advantage becomes even more pronounced as the number of modes and thus the dimension of Hilbert space increases.

Theoretical analysis backs up the empirical findings, with the origin of sampling improvement traced to the restriction of stochastic sampling to the conserved symmetry sector and the elimination of trajectory weight "tails" dominating error in standard unprojected phase-space approaches.

Implications and Future Directions

The matrix-P framework opens new directions for phase-space methods by providing a fully scalable, symmetry-adapted representation for quantum simulation. Its main practical implications are:

• Quantum verification: Dramatic error reduction in quantum computing validation tasks (e.g., GBS) with global symmetries, setting a new standard for classical simulation and experimental benchmarking.
• Many-body dynamics: Extension to other strongly correlated or symmetry-protected systems (e.g., FQHE, quantum magnets, gauge theories), where exact eigenvalue sectors dominate statistical effort.
• Hybridization with classical approximators: By enabling controlled projection onto relevant sectors, matrix-P representations allow seamless integration with classical sampling or approximate counting algorithms for scalable quantum device simulation.

Theoretical implications are equally significant:

• Unification of prior phase-space techniques via a universal symmetry-projected construction.
• Platform for studying the interplay of symmetry breaking and sampling complexity, e.g., in open quantum systems or non-equilibrium phase transitions.

Future work is anticipated on systematic incorporation of non-Abelian symmetries, explicit optimization of weight functions for sampling efficiency, application to quantum error correction codes relying on symmetry-enforced logical encoding, and extension to fermionic field theories.

Conclusion

Matrix phase-space representation provides a rigorously founded, computationally efficient method for simulating quantum systems endowed with global symmetries. By embedding symmetry projectors into the phase-space kernel, this framework dramatically suppresses sampling errors endemic to conventional positive-P and related approaches, while admitting transparent theoretical analysis and wide extensibility. These advances render matrix-P a vital tool for ongoing and future research in quantum simulation, quantum computational verification, and the exploitation of symmetry in complex quantum dynamics.

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Reference: "Matrix phase-space representations for quantum symmetries" (2606.12769)