Magic Tricycles: Efficient Magic-State Generation with Finite Block-Length Quantum LDPC Codes

The preparation of high-fidelity non-Clifford (magic) states is an essential subroutine for universal quantum computation but imposes substantial space-time overhead. Magic-state factories based on high-rate and -distance quantum low-density parity-check (LDPC) codes equipped with transversal non-Clifford gates can potentially reduce these overheads significantly, by circumventing the need for multiple rounds of distillation and by producing a large number of magic states in a single code block. As a step toward realizing efficient, fault-tolerant magic-state production, we introduce a class of finite block-length quantum LDPC codes which we name tricycle codes, generalizing the well-known bicycle codes to three homological dimensions. These codes can support constant-depth physical circuits that implement logical C C Z gates between three code blocks. To construct these constant-depth C C Z circuits, we develop analytical and numerical techniques that apply to a broad class of three-dimensional homological and balanced-product codes. We further show that tricycle codes enable single-shot state preparation and error correction, leading to a highly efficient magic-state generation protocol. Numerical simulations of specific codes confirm robust performance under circuit-level noise, demonstrating a high circuit-noise threshold of > 0.5 % . With modest postselection, certain tricycle codes of block lengths of only approximately 50–100 qubits are shown to produce | C C Z ¯ ⟩ magic states with logical error rates ranging from 2 × 10 − 8 to below 3 × 10 − 11 . Finally, we construct optimal depth syndrome extraction circuits for tricycle codes and present a protocol for implementing them efficiently on a reconfigurable neutral-atom platform.