Josias Old

Quantum low-density parity check (qLDPC) codes are among the leading candidates to realize error-corrected quantum memories with low qubit overhead. Potentially high encoding rates and large distance relative to their block size make them appealing for practical suppression of noise in near-term quantum computers. In addition to increased qubit-connectivity requirements compared to more conventional topological quantum error correcting codes, qLDPC codes remain notoriously hard to compute with. In this work, we introduce a construction to implement all Clifford quantum gate operations on the recently introduced lift-connected surface (LCS) codes. These codes can be implemented in a 3D-local architecture and achieve asymptotic scaling $[[n, O(n^{1/3}), O(n^{1/3})]]$. In particular, LCS codes realize favorable instances with small numbers of qubits: For the [[15,3,3]] LCS code, we provide deterministic fault-tolerant (FT) circuits of the logical gate set {H, S, CNOT} based on flag qubits. By adding a procedure for FT magic state preparation, we show quantitatively how to realize an FT universal gate set in d=3 LCS codes. Numerical simulations indicate that our gate constructions can attain pseudothresholds in the range $p_th = 4.8 x 10^{-3} - 1.2 x 10^{-2}$ for circuit-level noise. The schemes use a moderate number of qubits and are therefore feasible for near-term experiments, facilitating progress for fault-tolerant error corrected logic in high-rate qLPDC codes.