Universal adapters between quantum LDPC codes: Fault-tolerant Logical gates and measurements

We propose the repetition code adapter as a novel method to perform joint logical Pauli measurements within a quantum low-density parity check (LDPC) codeblock or between separate such codeblocks. This adapter is universal in the sense that it works regardless of the LDPC codes involved and the Paulis being measured. Since logical Pauli measurements offer

a route to universal quantum computation provided magic states are available, our repetition code adapter is a flexible tool to compute fault-tolerantly with arbitrary LDPC codes. Our construction achieves joint logical Pauli measurement of t weight O(d) operators using O(t d log^2 (d)) additional qubits and checks and O(d) time. To obtain these results, we develop a novel weaker form of graph edge expansion. As a special case, for some geometrically-local codes in fixed D>=2 dimensions, only O(t d) 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 targeted logical CNOT gates on arbitrary LDPC codes via Dehn twists. The toric code adapter is a space-efficient alternative to teleportation in order to mediate between an LDPC code and a toric code. Further, this is an explicit example of how our adapters can be used as a tool during code deformation to connect between codes with different properties, such as those with large rate for space-efficient memory and those with additional symmetries enabling fault-tolerant logical computation.