Optimized Clifford Noise Reduction: Theory, Simulations and Experiments

We propose several optimizations of the CliNR partial error correction scheme which implements Clifford circuits by consuming a resource state. Errors are corrected by measuring a sequence of Pauli operators that we refer to as the verification sequence. We first propose a global optimization algorithm searching for a verification sequence resulting in a low logical error rate using tabu search. Then, we introduce a proxy for the logical error rate which is easier to evaluate and we design a two-step optimization algorithm. First, a verification sequence minimizing the proxy is computed, then this sequence is refined by reintroducing the logical error rate. Finally, we identify a large group of automorphisms of the search space which preserve the proxy and we use this symmetry to reduce the size of the search space. This results in a 168 $\times$ (respectively 20,160 $\times$) reduction of the size of the search space for the optimization of verification sequences with three (respectively four) Pauli operators. Our numerical simulations for 20-qubit Clifford circuits with size 400 under the ion chain model show that our optimization algorithms improve the performance of CliNR by 25% and that the two-step optimization achieves the same results as the global optimization with 64% fewer evaluations of the logical error rate. Finally, we perform experiments on a 36-qubit trapped ion quantum computer, without mid-circuit measurements, showing that the CZNR variant of CliNR is at breakeven.