Estimating Low Logical Error Rates from Few Syndrome Extraction Cycles

Quantum error correcting codes allow quantum information to be protected

against physical qubit noise. One (or a few) logical qubits are encoded within

many physical qubits, and then actively monitored using syndrome

measurements, so that as long as the physical qubits’ error rate is below a critical

threshold, the logical qubits error rate is suppressed far below the physical

qubits’ error rate. In repetition codes, protection against a limited set of errors has

been demonstrated experimentally down to the extraordinarily low rate of 1e-

10/cycle. Measuring and verifying such extraordinary low failure rates requires

an extraordinary amount of effort and data. We investigate whether very low

logical error rates can be estimated accurately using far less data, by analyzing

the syndrome data used to perform error correction. We demonstrate that when

physical qubit noise is restricted to weight-1 and weight-2 errors, we can

accurately estimate a repetition code logical error rate of E using data from N <<

O(1/E) cycles of syndrome extraction. The naïve approach, counting logical

errors, requires O(1/E) cycles. We verify that our estimation strategy is robust to

varying the decoder, deploy it on data from real quantum testbeds, and consider

its extension to fully quantum codes like the surface code.