Simplified Quantum Weight Reduction with Optimal Bounds

Quantum weight reduction is the task of transforming a quantum code with large check weight into one with small check weight. Low-weight codes are essential for implementing quantum error correction on physical hardware, since high-weight measurements cannot be executed reliably. Weight reduction also serves as a critical theoretical tool, which may be relevant to the quantum PCP conjecture. We introduce a new procedure for quantum weight reduction that combines geometric insights with coning techniques, which simplifies Hastings' previous approach while achieving better parameters. Given an arbitrary [[n,k,d]] quantum code with weight w, our method produces a code with parameters \[[O\(n w² log w\), k, Ω(d w)\]] with check weight 5 and qubit weight 6. When applied to random dense CSS codes, our procedure yields explicit quantum codes that surpass the square-root distance barrier, achieving parameters \[[n, O\(n^1/3\), Ω\(n^2/3\)\]]. Furthermore, these codes admit a three-dimensional embedding that saturates the Bravyi-Poulin-Terhal (BPT) bound. As a further application, our weight reduction technique improves fault-tolerant logical operator measurements by reducing the number of ancilla qubits.