Logical CCZ state cultivation

Abstract

Logical CCZ states enable Toffoli gates, a central building block of fault-tolerant quantum arithmetic. We introduce an end-to-end protocol for direct CCZ-state preparation on bivariate tricycle and bicycle (BT/BB) codes. Our factory combines the cup-product CCZ gates native to BT codes with two novel ingredients, both enabled by engineering the logical coupling between BT and BB codes through projection-preserving lifts. Index-2 lifted BB ancillas enable selective $Z$-basis initialization, allowing the native cup-product gates to directly prepare two disjoint logical CCZ states. The prepared CCZ states are faithfully transferred to index-3 lifted BB memories with high code distance. We use decoder-based postselection to obtain high-fidelity output states with low overhead. We perform exact simulations of our factory under a circuit-level noise model motivated by neutral-atom hardware, obtaining joint infidelities of $1.84\times10^{-5}$ at $21.94\%$ acceptance and $4.10\times10^{-7}$ at $8.89\%$ acceptance for single-qubit, CZ, and CCZ error rates of $0.01\%$, $0.1\%$, and $0.5\%$, respectively. At the output-infidelity target of our smallest factory, its estimated spacetime cost per CCZ state is lower than that of $T$-state cultivation followed by CCZ conversion, providing an efficient route to logical CCZ states for fault-tolerant quantum computation.