The charge-singlet measurement toolbox

TL;DR

The paper addresses enforcing non-Abelian color-neutrality constraints in lattice gauge theory simulations, where direct singlet-state preparation can be resource-intensive. It introduces charge-singlet measurements (CSMs), a group-theoretical projection readout that confines observables to the singlet subspace via a diagonal operator $\hat{K}$, allowing post-processing corrections without explicit singlet preparation. The authors provide explicit constructions for SU(2) and SU(3) in $(1+1)$-D, connect to thermodynamics (e.g., entropy) and finite-temperature observables, and demonstrate noise-mitigation benefits for time evolution and ground-state preparation, illustrating broad applicability to both quantum devices and tensor-network simulations. These tools enable flexible classical and quantum simulations using tensor networks or quantum hardware, reducing constraints on circuit architectures and enabling deeper studies of gauge-theory thermodynamics and dynamics.

Abstract

Symmetry is fundamental to physical laws across different scales$\unicode{x2014}$from spacetime structure in general relativity to particle interactions in quantum field theory. Local symmetries, described by gauge theories, are central to phenomena such as superconductivity, topological phases, and the Standard Model of particle physics. Emerging simulation techniques using tensor network states or quantum computers offer exciting new possibilities of exploring the physics of these gauge theories, but require careful implementation of gauge symmetry and charge-neutrality constraints. This is especially challenging for non-Abelian gauge theories such as quantum chromodynamics (QCD), which governs the strong interaction between quarks and gluons. In a recent article (arXiv:2501.00579), we introduced "charge-singlet measurements" for quantum simulations, consisting of a projection based technique from group representation theory that allowed us to probe for the first time the phase diagram of (1+1)-dimensional QCD on a quantum computer. In this article, we show more broadly how to apply charge-singlet measurements as a flexible tool for both classical and quantum simulations of discrete and continuous gauge theories. Our approach extends the use of charge-singlet measurements beyond state preparation in the charge neutral (charge-singlet) sector to include noise mitigation in symmetry-preserving time-evolution circuits. We further demonstrate how this method enables the computation of thermodynamic observables$\unicode{x2014}$such as entropy$\unicode{x2014}$within the charge-singlet subspace, providing a new tool for exploring the connection between quantum thermodynamics and gauge symmetry.

Figures (12)

• Figure 1: Electric field versus chemical potential for a SU(2) unit cell. The VQT protocol as introduced in than2024phase is employed to numerically simulate thermal states for a SU(2) unit cell at a fixed temperature $T=0.5$, with Hamiltonian parameters $m=g^2=0.5$. As the chemical potential $\mu$ is varied, the expectation value of the electric field Hamiltonian $\langle \hat{H}_{el}\rangle_0$ changes accordingly. Our VQT protocol combined with the projection technique described in Eq. (\ref{['eq:exp-val-K']}) yields results that closely agree with those from exact diagonalization. In contrast, the expectation value calculated from the reducible density matrix $\hat{\rho}$ without the projection formula leads to inaccurate results due to $\hat{\rho}$ being a mixture of energy eigenstates from different irreducible representations. For each value of $\mu$, the variational optimization is repeated across five independent trials with random initializations. The error bars shown represent the standard deviation of the optimized electric field expectation values across these trials, although they are too small to be visible in the plot.
• Figure 2: Evaluation of charge-singlet entropy in a classical simulation. (a) Matrix product states can be used to evaluate quantities like $\langle \hat{H}\hat{K}\rangle$. In this case, the reducible density operator $\hat{\rho}$ is expanded using $\hat{\rho} = \sum_n p_n |\psi_n\rangle\langle\psi_n|$, where $|\psi_n\rangle$ is written in the MPS form and $\hat{H} = \sum_k c_k \hat{P}_k$, where $\hat{P}_k$ are Pauli strings. $a_i$ are Monte Carlo samples for the single-qubit operators, and $M$ is the number of Monte Carlo samples used to evaluate the expectation value. (b) Instead of using a linear superposition of MPS, one can also use MPO to represent the reducible density matrix $\hat{\rho}$ for evaluating Eq. (\ref{['eq:MC_integral']}).
• Figure 3: Charge-singlet entropy with Monte-Carlo sampling. The entropy of the charge-singlet density matrix for a SU(2) unit cell at finite temperature ($T=0.5$) is computed using Eq.(\ref{['eq:gi_entropy']}) within the tensor network (TN) framework. The TN is implemented as a matrix product operator (MPO) constructed from the optimized reducible density matrix $\hat{\rho}$ obtained at the conclusion of the VQT protocol described in Sec. \ref{['sec:thermal-states']}. Once the MPO is constructed, the quantities $\langle \hat{K}\rangle$ and $\langle \hat{H}\hat{K}\rangle$ in Eq. (\ref{['eq:gi_entropy']}) are evaluated using Monte Carlo (MC) sampling (2000 samples for each value of $\mu$) combined with tensor contractions, as illustrated in Fig. \ref{['fig:MPS_entropy']}. The charge-singlet entropy, computed via the projection operator in Eq. (\ref{['eq:gi_entropy']}) and MC sampling, shows good quantitative agreement with results from exact diagonalization. The discrepancies observed at lower chemical potentials arise from limitations in the expressiveness of the variational circuit used in the VQT protocol, rather than from sampling error. The error bars represent the standard deviation computed over five independent trials, each using a different random seed for the Monte Carlo sampling.
• Figure 4: Noisy time evolution of a SU(2) unit cell with fermionic matter in (1+1)-D. Starting from the strong-coupling vacuum state, the system is evolved in time till $t=10.0$ using the full Hamiltonian with $m=g^2=0.5$. The trotter step size is chosen to be $\delta t = 0.25$. Noisy time evolution is simulated using a noise model with a two-qubit gate fidelity of $\sim99.9\%$, implemented via a depolarizing channel applied after each two-qubit gate.
• Figure 5: Relation between noise strength and performance of the CSM technique for time evolution of a SU(2) unit cell. The behavior of the quantity $R(t)$ with time is shown here for different depolarizing noise strength $\lambda_d$ for a simulation performed in the same setting as described in Sec \ref{['sec:noisy-time-evolve']}. The quantity $R(t)$ approaches an asymptotic value (dashed line) as the density matrix gradually approaches the maximally mixed state. The data points are fitted with a sigmoid function $f(x)=a/(1+e^{-bx})+c$, where the parameter $b$ determines how fast the density matrix approaches the maximally mixed state. The fitted sigmoid functions are shown as solid curves. The inset shows the linear dependence of the fit parameter $b$ on $\lambda_d$.