Sample complexity of the non-linear optimization route for learning logical error rates from syndrome data

Establish rigorous sample-complexity bounds for estimating logical error rates in non-adaptive Clifford circuits from syndrome data when using the exact non-linear least-squares optimization approach that solves the full system of equations relating syndrome expectations to local Pauli eigenvalues (i.e., log Λ = A log λ together with linear constraints Bp = 0), without the low-error-rate linearization. Quantify the dependence of the required number of samples on system size, circuit depth, code distance, and noise parameters, and determine whether the exponential sampling advantage over direct logical-level estimation persists in this exact optimization setting.

Background

The paper develops two routes for learning physical and logical noise from syndrome data. One route uses an exact non-linear least-squares optimization to solve the full system of equations connecting syndrome expectations to local Pauli eigenvalues; the other route linearizes these equations in the low-error regime and proves sample-complexity guarantees for learning both physical and logical error rates, including an exponential advantage over direct logical measurements.

While the linearized route is analyzed rigorously, the authors explicitly note that they lack a proof of sample complexity for the exact non-linear optimization approach. Resolving this would substantiate the generality of the exponential sampling advantage they observe numerically and extend the theoretical foundations beyond the low-error approximation.