Error-correction is key to building a quantum computer. This includes both storage of quantum information as well as computing on it. Quantum low- density parity check (LDPC) codes offer a route to build these devices with low space overhead. The next question is - how do we fault-tolerantly com- pute on these codes?  Existing proposals (Cohen et al. [2110.10794], Cross et al. [2407.18393]) rely on ancilla systems appended to the original LDPC code. Building on these proposals, our work develops a repetition code adapter that concretely enables joint Pauli measurements, using a number of additional qubits scaling quasilinearly, or in some cases linearly, with the weight of the operator to be measured. This adapter is universal it works regardless of the LDPC codes involved and the Paulis being measured. Our constructions allow a standard computational model, Pauli-based computation, to be performed at the logical level on qubits encoded into LDPC codes.

Using a similarly constructed adapter, we also show a way to unitarily im- plement targeted logical CNOT gates on arbitrary LDPC codes, by efficiently mediating with a toric code and implementing Dehn twists. Our main innovations – a sparse change of basis for repetition codes and a congestion-free cycle basis for D-dimensional local graphs – may be of broader interest.  (This talk is based on arXiv:2410.03628, and will also briefly discuss related work on qLDPC surgery: arXiv:2407.18393 and arXiv:2410.02213).

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