A theory of single-shot error correction for adversarial noise

Single-shot error correction is a technique for correcting physical errors using only a single round of noisy check measurements, such that any residual noise affects a small number of qubits. We propose a general theory of single-shot error correction and establish a sufficient condition called good soundness of the code's measurement checks. Good code soundness in topological (or LDPC) codes is shown to entail a macroscopic energy barrier for the associated Hamiltonian. Consequently, 2D topological codes with local checks can not have good soundness. In tension with this, we also show that for any code a specific choice of measurement checks does exist that provides good soundness. In other words, every code can perform single-shot error correction but the required checks may be nonlocal and act on many qubits. If we desire codes with both good soundness and simple measurement checks (the LDPC property) then careful constructions are needed. Finally, we use a double application of the homological product to construct quantum LDPC codes with single-shot error correcting capabilities. Our double homological product codes exploit redundancy in measurements checks through a process we call metachecking.