Fault-tolerant quantum computing with a high-rate symplectic double code

High-rate and large-distance quantum codes are expected to make fault-tolerant quantum computing more efficient, but most of them lack efficient fault-tolerant encoded-state preparation methods. We propose such a fault-tolerant encoder for a [30, 6, 5] symplectic double code. The advantage of this code is its compactness, in addition to its high encoding rate, allowing for early experimental realization. Detecting crucial errors during encoding with as few auxiliary qubits as possible, our encoder can reduce resource overheads while keeping low logical error rates, compared to more naive methods. Numerical simulations with a circuit-level noise model demonstrate the reliability and effectiveness of the proposed method. We also develop an arbitrary-state encoder that enables the injection of arbitrary quantum states into the code space. Combined with basic fault-tolerant operations, this supports universal quantum computation. We thus demonstrate that efficient and reliable logical state preparation is achievable even for a compact and high-rate code, offering a potential step toward efficient fault-tolerant quantum computing suitable for near-term experiments.