Fault tolerance of quantum circuits with tensor networks and symplectic geometry

Abstract:We develop an operator-algebraic framework for analyzing fault tolerance in quantum circuits and establish necessary and sufficient conditions for fault tolerance under any given noise model. For prescribed circuit families and noise models, we derive a semidefinite program (SDP) based test whose infeasibility certifies that there does not exist any recovery map restoring its intended operation. For non-adaptive stabilizer circuits, analyzing the stabilizer symmetries of its components with symplectic geometry, we obtain closed-form detection and logical-effect matrices that characterize circuit distance algebraically. Applied to a finite Hastings--Haah honeycomb Floquet-code circuit, these matrices certify the dynamical encoding of two logical qubits and distance four under Pauli noise including measurement errors. Using these matrices, we then derive MacWilliams identities for circuit weight enumerators and obtain linear-programming (LP) upper bounds on circuit distance. Output-code constraints give additional Singleton-like bounds. Under independent and identically distributed (i.i.d.) depolarizing noise, the weight enumerators determine decoder failure probabilities and yield upper bounds on finite-circuit pseudothresholds. Finally, we derive LP distance bounds for families of flag syndrome-extraction circuits specified by their CNOT orderings, by incorporating the $t$-flag criterion as additional linear constraints to the distance-bounding LP. An analytical CSS Hamming-code benchmark yields a tight distance-three bound attained by the Chao--Reichardt one-flag construction.

arXiv:2610.03429 [quant-ph]

(or arXiv:2610.03429v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2610.03429 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.

Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.

Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.