Quantum Error Correction with Gauge Symmetries

Quantum simulations of Lattice Gauge Theories (LGT) are often formulated on an enlarged Hilbert space containing both physical and unphysical sectors in order to retain a local Hamiltonian. While this has the significant advantage of simplifying the implementation of the time-evolution operator, algorithmic approximations or errors on a quantum device can break gauge invariance by allowing unwanted transitions into the unphysical sector. Recent works have proposed to detect errors that break gauge invariance by means of quantum oracles that check Gauss’s Law. These approaches however are not able to correct errors and are not fault-tolerant. We present in this work a simple fault-tolerant procedure that combines Gauss’s Law with bit and phase flip error correction codes to detect and correct phase and bit-flip errors for a Z2 or truncated U(1) LGT in 1+1 dimensions with a link flux cutoff of 1. For a pure gauge theory on 2N links with periodic boundary conditions, we reduce the space requirement for error correction by exploiting gauge invariance and reach a physical-to-logical qubit ratio of 4.5, below what is possible using the perfect 5-qubit code on each logical qubit alone. The construction also applies to the case where static charges are present and can even accommodate dynamical fermions using a simple extension of our results. The constructions outlined can stimulate further developments in fault-tolerant error correction procedures for LGT systems with larger cutoffs, higher space-time dimensions, and possibly different symmetry groups.