Verification of Quantum AutoCCZ States Using the ZX-calculus

Abstract

The modeling, analysis, and verification of techniques used in quantum computing require formal languages designed specifically for the quantum setting. One of such techniques is the use of quantum error-correcting codes like the surface code to protect information from errors. The surface code protects information by defining a logical qubit inside a surface of multiple physical qubits. To execute non-Clifford gates (an essential type of computation) in the surface code, we must perform gate teleportation, a probabilistic protocol that might induce errors on the qubits. Correcting these errors can be time-consuming, but using self-correcting states like the AutoCCZ [Gidney, Fowler 2019] can significantly reduce time overheads. In the current literature, there is no formal verification of the correctness of the AutoCCZ. In this work, we verify the correctness of the AutoCCZ using the ZX-calculus, a formal language for quantum computing that consists of diagrams that can be operated on by a set of rewrite rules. We model the delayed-choice CZ (a major component of the AutoCCZ) and the AutoCCZ as diagrams in the ZX-calculus. Using the rewrite rules, we verify that the delayed-choice CZ matches its definition by being equal to the Identity or to a CCZ depending on a control bit. In the same way, we verify that the AutoCCZ corrects every possible error while reducing the time overhead compared to standard methods.