A recursive route to higher distance
A preprint describes a code with 13,320 data qubits and 64 logical qubits, with code distance 64, a certified circuit-distance lower bound of at least 48 and check weights no greater than 14. It reaches those figures by applying a flagged [[4, 2, 2]] amplification four times to an [[18, 4, 4]] seed. The guarantee depends on that seed, the certificates used in the analysis and the stated extraction schedule.
That result is the endpoint of a recursive design. The flagged amplifier maps a base code with parameters . Here n is the number of base-code data qubits, k is the number of logical qubits, d is the base-code distance, and m_X and m_Z are the numbers of X and Z checks in the chosen presentation. The rule doubles the logical-qubit count and code distance while adding the selected check terms to the data-qubit count.
The distance guarantee is stated sector by sector. In each X or Z sector, the amplified code distance has a lower bound set by the rounded certified gain times the base-sector distance and an upper bound given by the product of the base and amplifier sector distances. That gives a bounded result for each sector rather than one universal distance figure.
At circuit level, the paper builds a second certificate into the argument. Under the specified extraction conditions, contractions through the amplifier account for the base fault-response certificate, while lifting a complete undetected base history along an amplifier logical supplies the matching upper-bound construction. This allows the certificate to transfer multiplicatively and, at compatible recursive steps, gives exact circuit-distance multiplication under the stated conditions.
Why the circuit certificate matters
That equality is not assumed. The analysis computes fault-response certificates with Dijkstra's algorithm on response states reduced modulo stabilizers. For the two [[18, 4, 4]] examples, both check matrices have rank 7, and there are 2,048 response states per sector after the reduction.
The comparison makes the role of the schedule visible. The unflagged [[4, 2, 2]] schedule has circuit distance 2 and maximum certificates of 1 in both sectors. The flagged schedule also has circuit distance 2, but its maximum certificates are 2 in both sectors. In the reported comparison, the flagged schedule reaches equality between its certificate and circuit distance, unlike the unflagged schedule.
Resource use grows alongside the construction. At amplification levels t = 0, 1, 2 and 3, the reported data-qubit and logical-qubit pairs are (18, 4), (90, 8), (468, 16) and (2,484, 32). For a fixed amplifier, check weights and qubit degrees grow at most linearly with amplification depth, and applying a fixed number of steps preserves the low-density parity-check, or LDPC, property.
Different trade-offs, model-specific simulations
A separate construction uses a rotated surface amplifier. Applied to the same [[18, 4, 4]] seed, a distance-19 amplifier produces a [[9738, 4, 76]] code with circuit distance at least 57 and check weights at most 8. It retains four logical qubits while reporting a much higher code distance.
The simulations cover a specific setup. Runs on the seed and its first three flagged amplifications show increasingly steep logical-error suppression across the sampled physical-error range of 0.1% to 0.6%. The setup uses eight noisy syndrome-extraction rounds followed by two noiseless rounds, independent faults at circuit locations and separate BP+LSD decoding of the CSS sectors.
The recursive construction also carries logical labels. Under the stated full-rank and basis conditions, it provides inherited labels and preserves X-Z pairing, allowing individual logical qubits to be tracked through amplification.
Taken together, the result is a modular route to larger code distance and circuit-distance guarantees, but the strongest claims are conditional. Exact multiplication depends on compatible extraction interfaces and certificate assumptions, and the four-step figures depend on the specified seed, certificates and schedule. The simulation evidence is likewise tied to the reported code, rounds, circuit-location faults and decoder.
The work is identified as arXiv:2609.37231v1, dated 29 September 2026. The paper says code and data for its circuit-level simulations and fault-response certificate computations are available at a listed GitHub repository.
Paper data and sources
Original title: Ultra-high-distance quantum memories from amplified qLDPC codes
Authors: Zijian Liang, Boren Gu, Yu-An Chen et al.
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-09-29
DOI: Not available
Original paper · Full text