Gauge Symmetry in Quantum Simulation

TL;DR

The paper develops universal principles for handling gauge redundancy in quantum simulations of non-Abelian gauge theories, clarifying that physical states correspond to gauge orbits and need not be restricted to gauge singlets. It presents an explicit orbifold-lattice quantum-simulation framework that supports both singlet-projection and non-singlet formulations, including a Haar-averaging projection realized via LCU/QSVT and complete resource estimates. Classical benchmarking on simplified models validates convergence criteria for Hilbert-space truncation and Trotterization, and demonstrates scalable circuit recipes for SU($N$) Yang–Mills dynamics expressed as Pauli-string Hamiltonians. Together, these contributions provide conceptual clarity and practical tools toward quantum advantage in simulating non-Abelian gauge theories.

Abstract

Quantum simulation of non-Abelian gauge theories requires careful handling of gauge redundancy. We address this challenge by presenting universal principles for treating gauge symmetry that apply to any quantum simulation approach, clarifying that physical states need not be represented solely by gauge singlets. Both singlet and non-singlet representations are valid, with distinct practical trade-offs, which we elucidate using analogies to BRST quantization. We demonstrate these principles within a complete quantum simulation framework based on the orbifold lattice, which enables explicit and efficient circuit constructions relevant to real-world QCD. For singlet-based approaches, we introduce a Haar-averaging projection implemented via linear combinations of unitaries, and analyze its cost and truncation errors. Beyond the singlet-approach, we show how non-singlet approaches can yield gauge-invariant observables through wave packets and string excitations. This non-singlet approach is proven to be both universal and efficient. Working in temporal gauge, we provide explicit mappings of lattice Yang-Mills dynamics to Pauli-string Hamiltonians suitable for Trotterization. Classical simulations of small systems validate convergence criteria and quantify truncation and Trotter errors, showing concrete resource estimates and scalable circuit recipes for SU($N$) gauge theories. Our framework provides both conceptual clarity and practical tools toward quantum advantage in simulating non-Abelian gauge theories.

Figures (14)

• Figure 1: Joining/splitting interactions of strings.
• Figure 2: Wilson loops with and without whiskers. For Kogut-Susskind, whiskers disappear (in other words, they become $\textbf{1}$) after the singlet projection. The difference between $\hat{W}^{\rm (non\mathchar'-singlet)}$ with and without whiskers is in $\mathrm{Ker}(\hat{\mathcal{P}})$ and thus redundant. With or without whiskers, a cat is a cat!
• Figure 3: The square of the ground-state wave function for the $n=1$ model, $|\psi_{\rm g.s.}(x_1,x_2)|^2$. We used $m=40$, $R=2$, and $\Lambda=32,64$.
• Figure 4: For the $n=1$ model, the difference between $E_0$ and two-fold degenerate states corresponding to momentum $p=\pm 1$, $\pm 2$, and $\pm 3$ along the U(1) direction should be $\frac{p^2}{2}$ in the infinite-mass limit. Specifically, $E_1-E_0-0.5$, $E_3-E_0-2.0$, and $E_5-E_0-4.5$ should converge to zero. We plotted these numbers for several values of $m$ between 20 and 200. The lines are quadratic fit with respect to $1/m$. We used $R=2$, and $\Lambda$ was take sufficiently large so that the error is less than $10^{-5}$.
• Figure 5: For the $n=1$ model at $R=2$ and $m=20, 30$, and $40$, the low-lying spectrum was calculated varying $\Lambda$ (and hence $\delta_x=2R/\Lambda$), and the difference from the values at $\delta_x=0$ was plotted. The energy eigenvalues at $\delta_x=0$ were estimated using sufficiently small $\delta_x$ where so that the error is smaller than $10^{-10}$. By using $\sqrt{m}\delta_x$ as the horizontal axis, results from different mass line up and show the same exponential decay. This means we should take $\delta_x\lesssim\frac{1}{\sqrt{m}}$ to approximate the spectrum well. Note that we plotted the absolute value $|E_\ell(\delta_x)-E_\ell(\delta_x=0)|$ because the sign oscillates.