Abstract: We introduce a surface-code cultivation protocol for reusable logical catalyst states that implement exact fine dyadic phase gates Z2−b by phase kickback. The catalyst is an eigenstate of a high-period Clifford circuit U, with a direct construction supported on O(2b) 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 T=Z1/8, 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 \emphsingle 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 ∼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.
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