زراعة محفّز الدقة سريع

الملخص الأصلي

كما كتبه المؤلفون، بالإنجليزية arXiv:2606.27358

We introduce a surface-code cultivation protocol for reusable logical catalyst states that implement exact fine dyadic phase gates $$Z^{2^{-b}}$$ by phase kickback. The catalyst is an eigenstate of a high-period Clifford circuit $$U$$, with a direct construction supported on $$O(2^b)$$ logical qubits. Once cultivated, each invocation implements the target phase through a controlled-$$U$$ gadget, removing Clifford+$$T$$ synthesis approximation error from the online gate and making the online non-Clifford depth independent of the target logical accuracy. As a concrete demonstration, we construct a catalyst for $$\sqrt{T}=Z^{1/8}$$, where $$U$$ is a nine-qubit brickwork Clifford circuit and controlled-$$U$$ consists of eight controlled-CNOTs. Starting from nine distance-three rotated-surface-code blocks, we cultivate the catalyst through logical-$$U$$ checks, syndrome extraction and postselection, code growth, and complementary-gap decoding. Due to the intrinsic fault tolerance of the phase read-out, a \emph{single} verification round already reaches the leading error-corrected scaling, in contrast to the repeated logical checks required when cultivating single-qubit magic states. A hybrid tensor-network and stabilizer simulation shows that, at physical error rate $$p=10^{-3}$$, the postselected catalyst can be grown to distance-seven rotated-surface-code blocks with logical leakage rate $$\sim 10^{-6}$$ using around seven expected attempts, and can be suppressed further with stronger postselection. Compared with existing protocols, our approach trades offline, phase-specific catalyst cultivation for exactness, reusability, and constant-depth online implementation of fixed fine dyadic phases in codes with restricted transversal gate sets.