Evaluating the performance of QEC primitives on quantum processors at large width and depth

Abstract

Quantum error correction (QEC) relies on repeated parity extraction, mid-circuit measurement (MCM), reset, feed-forward, and scheduling, yet these primitives are usually assessed either in isolation or through resource-demanding experiments. We introduce a few-sample benchmark of QEC-relevant primitives based on an MCM implementation of the quantum approximate optimization algorithm (QAOA) with a fixed set of linear parameters (LR-QAOA). For a chosen code, the QAOA Hamiltonian is constructed from its check structure, such that the resulting circuit mimics the syndrome-extraction connectivity and MCM pattern while producing a direct algorithmic signal, the approximation-ratio r. The LR-QAOA depth, defined by the number of QAOA layers, plays a role analogous to the number of repeated syndrome-extraction rounds in a QEC experiment. From QPU executions, the decay of r with depth defines an effective hardware error, which we map to an equivalent two-qubit depolarizing rate λeff. We compare direct and MCM-mediated implementations across 10 QPUs from IBM, IQM, and Quantinuum, with circuits containing up to 2950 MCM operations. On Quantinuum's Helios-1 and H2-1, we run code-structured LR-QAOA benchmarks for surface-code, triangular color-code, and bivariate-bicycle qLDPC Hamiltonians up to 81, 91, and 48 data qubits, respectively, using up to 480 MCM operations. On IBM ibm_phoenix, we implement the surface-code structure and compare the LR-QAOA response with logical-memory experiments, observing a correlation between the benchmark and the logical performance across different regions of the QPU. Its construction and low resource requirements provide a practical benchmark for comparing hardware generations and QEC implementations before full logical-memory experiments are performed.