Universal adapters between quantum LDPC codes

Abstract

We propose the repetition code adapter as a way to perform joint logical Pauli measurements within a quantum low-density parity check (LDPC) codeblock or between separate such codeblocks, thus providing a flexible tool for fault-tolerant computation with quantum LDPC codes. This adapter is universal in the sense that it works regardless of the LDPC codes involved and the logical Paulis being measured. The construction achieves joint logical Pauli measurement of $t$ weight $O(d)$ operators using $O(d)$ time and $\tilde O(td)$ additional qubits and checks, up to a factor polylogarithmic in $d$. As a special case, for some geometrically-local codes in fixed $D\ge2$ dimensions, only $O(td)$ additional qubits and checks are required instead. By extending the adapter in the case $t=2$, we also construct a toric code adapter that uses $O(d^2)$ additional qubits and checks to perform addressable logical CNOT gates on arbitrary LDPC codes via Dehn twists. To obtain these results, we develop a novel weaker form of graph edge expansion and the $\mathsf{SkipTree}$ algorithm, which ensures a sparse transformation between different weight-2 check bases for the classical repetition code.