Fig. 1: Error correction with Gauss’s law.
a Schematic illustration of the differences between two error-correcting schemes for a simple one-dimensional LGT: a traditional bit-flip encoding scheme and the two schemes proposed here exploiting the Gauss’ law gauge symmetry. b Number of qubits required (excluding ancillas) for performing fault-tolerant error-correction with different encodings on a \({{\mathbb{Z}}}_{2}\) or truncated U(1) 1+1 dimensional LGT system with 2N links, 2N staggered fermions and a 2+1 dimensional LGT with 8NxNy links and 4NxNy sites, both with a flux cutoff of 1. The second row gives the cost for an encoding where Gauss’s law is exploited to give a bit-flip encoding via an extra even-numbered link qubit in 1+1 D. The last row of the 1+1 and 2+1 D cases gives the cost when only the redundancies from Gauss’s law are used at the logical level to perform error correction. The case with non-dynamical fermions requires the same resources as the pure gauge case and is omitted from the table.
