Simulating Fully Gauge-Fixed SU(2) Hamiltonian Dynamics on Digital Quantum Computers

TL;DR

This work tackles real-time dynamics of SU(2) lattice gauge theories on quantum devices by adopting a fully gauge-fixed, mixed-basis formulation that maps continuous gauge degrees of freedom to a finite qubit register. It develops and analyzes two time-evolution strategies—the differential-operator approach and Pauli-string truncation—demonstrating that three qubits per plaquette suffice to capture low-energy physics with ~$10^{-3}$ precision for two-plaquette systems, across couplings $g$, and validates the methods with FEM benchmarks and on IBM hardware using extensive error mitigation. Key contributions include a detailed digitization scheme (ω-grids, DST-based Laplacian, finite-difference first derivatives), explicit circuit constructions, and concrete resource estimates guiding scalability to larger lattices. The results establish a practical path toward scalable, gauge-invariant simulations of SU(2) (and potentially SU(3)) gauge theories at all couplings on near-term quantum hardware, offering insight into nonperturbative real-time dynamics and phase structure. The combination of mixed-basis truncation, rigorous benchmarking, and hardware demonstration constitutes a significant step toward quantum simulations of non-Abelian gauge theories.

Abstract

Quantum simulations of many-body systems offer novel methods for probing the dynamics of the Standard Model and its constituent gauge theories. Extracting low-energy predictions from such simulations rely on formulating systematically-improvable representations of lattice gauge theory Hamiltonians that are efficient at all values of the gauge coupling. One such candidate representation for SU(2) is the fully gauge-fixed Hamiltonian defined in the mixed basis. This work focuses on the quantum simulation of the smallest non-trivial system: two plaquettes with open boundary conditions. A mapping of the continuous gauge field degrees of freedom to qubit-based representations is developed. It is found that as few as three qubits per plaquette is sufficient to reach per-mille level precision on predictions for observables. Two distinct algorithms for implementing time evolution in the mixed basis are developed and analyzed in terms of quantum resource estimates. One algorithm has favorable scaling in circuit depth for large numbers of qubits, while the other is more practical when qubit count is limited. The latter algorithm is used in the measurement of a real-time observable on IBM's Heron superconducting quantum processor, ibm_fez. The quantum results match classical predictions at the percent-level. This work lays out a path forward for two- and three-dimensional simulations of larger systems, as well as demonstrating the viability of mixed-basis formulations for studying the properties of SU(2) gauge theories at all values of the gauge coupling.

Figures (8)

• Figure 1: Spectrum of the Two Plaquette Systema) Two plaquette system with open boundary conditions. The system contains seven links, numbered $1$ through $7$, with the physical links denoted as $\kappa=1,2$. The magnetic degrees of freedom $\hat{X}_{1},\hat{X}_2$ are associated with each plaquette. The dashed lines that run around each plaquette denote the loops used in the max tree gauge fixing procedure. b) Spectrum of the Hamiltonian for different bare couplings. The solid lines show the solution obtained by FEM. The black dashed line on the bottom shows the energy bound of Eq. \ref{['eq:bound']}. The grey lines denote the strong coupling limit of the character irrep spectrum. This limit is found by ignoring the contribution of the magnetic Hamiltonian, which is only valid when the gauge coupling is large. c) Low-lying spectrum of the character irrep Hamiltonian in the strong coupling limit. The first and fourth excited state are two-fold degenerate and each state is a linear superposition of the stacked basis states. See Table \ref{['tab:CharIrrepStrong']} for further details.
• Figure 2: Digitization of Mixed Basisa) The continuous range of $\omega$ is digitized on a discrete grid. As such the wave function will only be sampled at a discrete set of points. For the closest approximation to the undigitized theory, the maximum value of $\omega_{\text{max}}\leq 2\pi$ must be chosen. b) A slice of the ground state wave function for different values of the gauge coupling. As the gauge coupling decreases, the wave function becomes more concentrated around $\omega_1 = 0$, although it must go to $0$ at both boundaries due to the Dirichlet boundary conditions. c) Convergence of the ground state energy to the undigitized value as the digitization resolution $N_{\omega}$ is increased. The convergence is plotted for a variety of $\omega_{\text{max}}$. After quick convergence to $\epsilon_0\sim 10^{-3}$ a slower exponential approach occurs, followed by a plateau. d)(top) Convergence to the ground state energy as $\omega_{\text{max}}$ is varied. The dashed lines correspond to the full two plaquette system while the solid lines correspond to the system without any first-derivative interactions. The effect of the first derivative reduces the precision that can be achieved. Even though this term limits precision, digitization errors are still below those due to time-evolution algorithms. (bottom) The absolute value of the derivative of the energy w.r.t the truncation. It was heuristically found that the truncation at which this derivative takes a minimum corresponds to the optimal truncation for the analytically solvable case. This method is still applicable even when the exact solution is not known. e) The solution of the stationary condition for a range of gauge couplings. The agreement is very good with the analytical solution Eq. \ref{['eq:OmegaMax']}.
• Figure 3: Accuracy of the $\nu$ Truncationa) Calculation of the time dependent magnetic energy for the low-energy state $\ket{5}$, defined in Eq. \ref{['eq:low_energy_state']}, as the maximum $\nu$ value is increased for $g=0.5$ and $N_{\omega}=10$. This choice of digitization resolution ensures that any errors due to the $\omega_1,\omega_2$ truncation will be at the percent level. b) The probability $\langle\Pi_{\nu>1}\rangle$ of leaking out of the $\nu=0,1$ sector as a function of time for $g=0.5, N_{\omega}=10$ and $\nu_\text{max} = 5$.
• Figure 4: Circuits for the Digitization Presented in Sec. \ref{['subsec:DifOpCircuits']}, which constructs sub-components for each class of differential operatorsa) Implementation of the exact exponential of the second derivative, making use of Discrete Sine Transforms. This subcircuit only entangles qubits within each $\omega$ register and does not tuch the $\nu$ register. The ancillas $a_1,a_2$ associated with each $\omega$ register are the outermost qubits. b) An important circuit identity used in this work: The exponential of an $N-$body Pauli string $P$ is implemented by the staircase of $CX$s connecting all the qubits participating in the gate, followed by an $R_Z(\theta)$ rotation on one of the qubits and then another $CX$ staircase. The green single qubit gates denote a local change of bases to change $P$ into a string of $Z$s on all qubits e.g. if there is an $X$ on qubit $i$ then this will be a Hadamard $H$ and if it is a $Y$ then it will be a Hadamard followed by a phase gate $HS$. The different shades denote that the local basis change can depend on the particular Pauli string $P$ being implemented. c) Circuits for implementing first order derivatives and mixed derivatives using the finite difference representation. (top) Implements the exponential of a single first derivative.Each element in this circuit implements one of the terms in the sum Eq. \ref{['eq:partial_MPO']}. The bottom qubit of each step is the target qubit for both the $V_k$ and $CRZ$ gates. bottom An example of one of circuit elements $e^{-it\frac{\partial}{\partial\omega_1}\frac{\partial}{\partial\omega_2}}$ that interleaves the top circuit. d) Noiseless simulator results for the expectation value of the magnetic Hamiltonian at a different number of Trotter steps.
• Figure 5: Approximation for Time Evolution Circuits Presented in Sec. \ref{['subsec:coarse']}a) The expectation value of the magnetic part of the Hamiltonian for differing digitization resolutions and percentage of terms dropped. Rapid convergence is observed for $n_q\geq 3$ and for truncation levels $\delta<.6$. b) Different stages of the time evolution operator for a second-order Trotter step. The silver circuit denotes the entire Trotter step. The blue sub-circuits acts only on the $\omega$ registers, typically involving the largest angles, and is identical for each. The green sub-circuits couple each $\omega$ register to the $\nu$ register and the circuit structure is again identical for both $\omega$. The yellow sub-circuit couple everything and generally contain the smallest rotations, therefor being the elements most affected by small-angle truncations. It also involves the highest weight terms, and so focusing on truncating these elements allows the circuit to be compressed at only a small cost in algorithmic error. c) The accuracy of first- and second-order Trotter on magnetic observables. Repeating the blue sub-circuit in the second step allows the circuit to take much larger time steps while only needing a small amount of extra gates to implement.