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 $[[O(n w^2 \log w), k, \Omega(d w)]]$ 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 $[[n, \tilde O(n^{1/3}), \tilde \Omega(n^{2/3})]]$. 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.