Adaptive Loss-tolerant Syndrome Measurements

TL;DR

This paper develops adaptive syndrome measurement protocols for fault-tolerant quantum error correction in mixed error models involving both Pauli errors and erasures, optimizing measurement sequences to reduce overhead.

Contribution

It extends adaptive syndrome measurement techniques to mixed error models, providing new protocols and minimal overhead quantification for loss-tolerant quantum error correction.

Findings

Reduced measurement overhead to a subgroup dimension problem.

Construction of canonical generating sets for stabilizer groups.

Generalization of FTEC conditions to mixed error models.

Abstract

In the presence of qubit losses, the building blocks of fault-tolerant error correction (FTEC) must be revisited. Existing loss-tolerant approaches are mainly architecture-specific, and little attention has been given to optimizing the syndrome measurement sequences under loss. Schemes designed for the standard Pauli error model are not directly applicable because the syndrome patterns differ when both Pauli errors and erasures can occur. Based on recent advances in loss detection units and loss-tolerant syndrome extraction gadgets, we extend the study of adaptive Shor-style measurement sequences to the mixed error model. We begin by discussing how to adaptively convert correctable erasures into located errors. The minimal overhead is quantified by the number of stabilizer measurements, which can be reduced to a subgroup dimension problem for erasures arising in any FTEC circuit for qubits and prime-dimensional qudits. As a byproduct, we provide the construction of the canonical generating set with respect to a given bipartite partition for a stabilizer group on qudits of composite dimension. We then generalize both the weak and strong FTEC conditions. Finally, we present adaptive syndrome-measurement protocols for the mixed error model, generalizing the adaptive protocols for the standard Pauli error model.