Approximate Error Correction for Quantum Simulations of SU(2) Lattice Gauge Theories

Abstract

We present a protocol for actively suppressing Gauss law violations in quantum simulations of SU(2) lattice gauge theory. The protocol uses mid-circuit measurements to extract a characterization of the gauge-violation sector at each lattice vertex, resolving both the total angular momentum and magnetic quantum numbers of the violation via a group quantum Fourier transform. Syndrome-conditional recovery operations map the state back to the gauge-invariant subspace through an iterative sweep over vertices, a procedure we call gauge cooling. We show that while the Knill-Laflamme conditions are not generically satisfied at vertices with nontrivial singlet multiplicity, every single-qubit error is detected by the gauge syndrome. We demonstrate gauge cooling on a single-plaquette simulation of the Kogut-Susskind Hamiltonian truncated to the spin-$1/2$ representation under depolarizing and amplitude damping noise, showing that the protocol restores gauge invariance and improves fidelity at noise rates representative of current superconducting hardware.

Figures (3)

• Figure 1: Circuit for the Gauss law test at a single vertex $v$. The ancillary register is prepared in the uniform superposition $\ket{+_G} = \frac{1}{\sqrt{|G|}} \sum_g \ket{g}$ by the unitary $F_G$. The controlled gauge action applies $U^{(v)}(g)$ to the data register conditioned on the ancilla state $\ket{g}$. The inverse $F_G^\dagger$ is applied and the ancilla is measured; the outcome corresponding to $\ket{0}$ is the "accept" outcome, with acceptance probability $\langle\psi|\Pi^{(v)}_0|\psi\rangle$.
• Figure 2: Syndrome extraction at vertex $v$. The circuit is identical to the circuit in Figure \ref{['fig:gbose_circuit']}, except that the inverse preparation $F_{\mathrm{SU(2)}}^\dagger$ is replaced by the group quantum Fourier transform $\mathrm{QFT}_{\mathrm{SU(2)}}$. Measuring the ancillary register in the Wigner basis yields a syndrome $(J,M,N)$. The operator applied to the data register selects the component in $\mathcal{W}_N^J$ and maps it into $\mathcal{W}_M^J$, leaving the multiplicity degrees of freedom untouched.
• Figure 3: Fidelity with the ideal noiseless evolution as a function of Trotter step for a single-plaquette SU(2) lattice gauge theory simulation with $j_{\mathrm{max}} = 1/2$, coupling $g^2 = 1$, total evolution time $T = 3.0$, and step size $dt = 0.1$. Dashed lines: no error correction. Solid lines: iterative gauge cooling (up to 10 sweeps) applied after each Trotter step. Left panel: qudit depolarizing noise with error rate $p$ per edge per step. Right panel: amplitude damping with damping rate $\gamma$ per edge per step.