Trade-offs in Gauss's law error correction for lattice gauge theory quantum simulations

Abstract

Gauss's law-based quantum error correction (GLQEC) offers a promising approach to reducing qubit overhead in lattice gauge theory simulations by leveraging built-in symmetries. For applications of GLQEC to 1+1D lattice quantum electrodynamics (QED), we identify two significant trade-offs. First, we prove via dimension-counting arguments that GLQEC requires periodic electric fields, thereby constraining the design space for lattice QED simulations. Second, we numerically compare GLQEC with a universal quantum error correction (UQEC) code, specifically the $d=3$ bitflip repetition code, and find that while GLQEC can achieve lower logical error rates in single-round error correction, it exhibits faster decoherence to the steady-state mixed ensemble under multiple rounds. The mixing speed penalty is manifest in observables of interest for both memory experiments and Hamiltonian evolution. We identify a mixing speed threshold, $p_{th}=0.277(2)$, above which using GLQEC exhibits even faster decoherence than without error correction. Our results highlight fundamental limitations of symmetry-based error correction schemes and inform corresponding constraints on formulations of lattice gauge theories compatible with error-robust quantum simulation techniques.

Figures (10)

• Figure 1: A concatenated universal QEC code and the RRW QEC code. $X$-checks are black and $Z$ checks are dark gray. a) A universal error correction scheme using Shor's $[\![9,1,3]\!]$ code for each site and link qubit. b) The corresponding 1+1D lattice of four staggered fermionic sites $S_s$ connected by links $L_{s,s+1}$ with periodic boundary conditions. c) The RRW scheme rajput_quantum_2023 using a $d=3$ phase-flip code concatenated with the 3-body Gauss's law checks ($Z^{\otimes 9}$ checks).
• Figure 2: The flux configuration space graph between two neighboring electron sites for a $U(1)_2^-$ theory. Notably, the graph is almost an all-to-all graph, except for one edge corresponding $(0,-1)$, which is not a physical flux configuration for positron sites.
• Figure 3: Logical error rates as a function of physical error rate for a lattice of $n=2$ physical sites subject to universal (UQEC) or Gauss's law error correction (GLQEC) strategies compared to the bare physical qubit performance (noQEC). The noise channel is bitflip noise with probability $p$, and for simplicity, the codes are not concatenated with the phase-flip code (as opposed to fig:rrw lattice). As expected for two $d=3$ codes, GLQEC achieves the same $p^2$ order of asymptotic scaling with fewer qubits using 3-body stabilizers, compared to the 2-body stabilizers used in UQEC.
• Figure 4: Ratio of universal to Gauss's law logical error rates as a function of physical error rate for lattices of increasing volume from $n = 2$--$50\ 000$ physical sites.
• Figure 5: Multi-round memory and simulation results for the truncated $U(1)$ theories for $n=2$ lattice size (8 application qubits) and $p=0.08$ error rate. Exact, density-matrix-based expectation values for fidelity (a), physicality (b), electric field energy (c), and single pair probability (d) are shown for the cases of no error correction (no QEC), universal error correction (UQEC) with the bitflip code, and Gauss's law error correction (GLQEC). As a baseline, the noiseless cases are displayed in gray. All series are shown for memory and simulation channels for the $U(1)_2^{\circ}$ theory. The insets show the highly similar results for $U(1)_2^{-}$ theories.