Implement and optimize the simulator subroutines

Implement and optimize the implicit subroutines of the third-order-circuit failure-rate simulator, including the linear-algebraic operations and parallelized procedures used to evaluate the decoding and condition-checking steps, in order to improve its practical performance for large and structured circuits.

Background

The paper presents an efficient simulator for estimating logical failure rates of fault-tolerant third-order circuits composed of CSS operations, diagonal third-level Clifford-hierarchy gates, and Pauli errors. Its main procedures rely on evaluating binary linear equations, calculating the matrices that characterize twisted detectors and twisted errors, checking the contractibility condition C, and solving associated decoding problems.

Although these operations are identified as computationally efficient in principle and potentially amenable to optimization and parallelization, the paper does not provide explicit implementations or performance-optimized versions of all subroutines. The authors therefore leave their implementation and optimization as a concrete task for future work, particularly for large circuits with locality, translation invariance, or group-translation symmetry.

References

In the description of the algorithms above we have left many subroutines implicit. These steps all involve evaluating equations over $Z_2$ or solving linear equations over $Z_2$ which can be done efficiently. In practice, these subroutines can also be highly optimized and parallelized, given the locality structure of the circuits. We expect that our methods will work particularly well for large and structured circuits, for example circuits that admit a form of translation invariance, or more general group-translation symmetry. We leave explicit implementations and optimizations to future work.

— Efficiently estimating failure rates of fault-tolerant logical non-Clifford blocks  (2609.19485 - Fuente et al., 16 Sep 2026) in Remark following Algorithm 3 (GenLinking), Section 3, subsection “Overview and subroutines”