End-to-End Formalization of Quantum Error Correction

2026-05-15

AI Summary

Verifying the distance of quantum error-correcting codes is an NP-hard problem, and existing approaches lack scalability and formal guarantees. This work presents the first end-to-end formally verified framework for stabilizer codes in Lean 4, reducing distance verification to a certified Boolean satisfiability problem. By introducing BitVec flattening and error-location encoding techniques, the framework substantially mitigates combinatorial explosion. It integrates symplectic geometry over binary fields, Pauli group theory, and constructions from CSS and two-variable bicycle codes, enabling scalable automated distance certification within a theorem prover for the first time. The approach successfully verifies multiple industry-scale codes—including [[90,8,10]] and [[70,6,9]]—and supports end-to-end fault-tolerance verification for systems up to 144 qubits.

Abstract

Quantum error-correcting codes (QECCs) sit between noisy quantum hardware and reliable computation, so the code parameters used in practice must be trustworthy. The single number that summarizes a code's strength is its distance, yet certifying a distance lower bound is NP-hard in general, placing it beyond the reach of pen-and-paper proofs as well as direct proof-assistant scripting. As a result, distance values in the literature come either from non-scaling hand proofs, or from unverified solvers that leave a trust gap exactly where the code is supposed to provide a guarantee. We present Lean-QEC, the first Lean 4 formalization of stabilizer-code theory that delivers end-to-end, machine-checked distance certificates at industrial code sizes. Lean-QEC formalizes the linear algebra of qubit states, the Pauli group, stabilizer codes, the binary symplectic representation, classical coding theory, and the CSS and Bivariate Bicycle families. To break the combinatorial barrier, Lean-QEC translates the distance condition into a Boolean satisfiability formula through a verified reduction. The pipeline scales through a BitVec-flattened encoding that replaces Lean's Matrix representation, and an error-location encoding that reduces the variable count from $n$ to $k\lceil \log_2 n\rceil$. With these, we obtain automatically-generated Lean-checked distance proofs for a large range of industrially viable qLDPC codes within the Bivariate Bicycle and Generalized Bicycle families, including [[90, 8, 10]] and [[70, 6, 9]] BB codes, with the formulation scaling up to 144 qubits when performed outside the Lean kernel. The resulting library is reusable and is designed to plug into broader Lean-based efforts toward end-to-end verification of fault-tolerant quantum computation.

Problem

• Quantum Error Correction
• Code Distance
• Formal Verification
• NP-hard
• Stabilizer Codes

Innovation

• quantum error correction
• formal verification
• Lean 4
• distance certification
• qLDPC codes