Randomized search adds new quantum error-correcting codes

What the researcher asked

Can randomly generated quantum error-correcting codes be improved enough to beat entries in Grassl’s tables of qubit stabilizer codes? These codes encode logical qubits across physical qubits and use stabilizer checks to detect errors. The distance, d, is the smallest weight of a logical operator that commutes with the checks but acts nontrivially on encoded information. In the usual code-error model, distance d guarantees correction of up to floor((d−1)/2) errors.

The preprint reports 192 binary stabilizer codes, written as [[n, k, d]], with lengths from 53 to 254 qubits and distances from 5 to 13. At the same (n, k), each has a distance above the lower bound recorded in the table: 161 by one and 31 by two. Standard rules for extending codes to related parameters carry these improvements to 1,873 table entries.

How the search worked

For 190 codes, Girard generated a parent code using a random Clifford encoder: a circuit of random two-qubit Clifford gates. He then enumerated its logical operators below a chosen target distance. A logical operator is a Pauli pattern that can affect stored information. The search selected some low-weight operators and added them as stabilizer checks, reducing the number of logical qubits while arranging for each low-weight operator to anticommute with at least one added check. This targeted reduction is called expurgation.

The remaining two codes began with codes from the tables. The researcher punctured each by removing a qubit, then applied a similar repair procedure. AI agents performed much of the search and computational work under Girard’s direction, with substantial calculations run on the Aurora supercomputer. The preprint says the codes, certificates and an independent checker are publicly available.

Findings and evidence

For each reported distance, the preprint gives two kinds of evidence. An explicit logical operator of weight d shows the distance is no greater than d. To establish that no lighter logical operator exists, Girard used exhaustive meet-in-the-middle enumeration: candidate Pauli operators are split into parts and compared using their error syndromes. The preprint reports complete lower-bound certificates for all 192 codes, with runs enumerating as many as about 4 × 10¹³ Pauli operators. This is a computational result, not a claim that the codes were tested on quantum hardware.

At fixed length and distance, the reported increase in k—the number of encoded logical qubits—ranges from one to nine compared with the value in Grassl’s table. The paper reports larger average gains in cases where the table has no explicit construction than where it does; this does not mean every largest individual gain occurs in the former group. Its explanation is that expurgation can outperform a typical random code by selecting which low-weight logical operators to remove. A first-moment counting argument estimates when suitable choices of stabilizers might exist. It helps explain the results, but does not guarantee that a search will find a code.

The same counting method predicts better codes at 619 additional (n, d) pairs. These are forecasts, not verified constructions. The preprint also reports that the search did not beat the table where known structured codes perform substantially better than random codes. The new codes have dense generators; Girard does not present them as low-density parity-check codes. Their contribution is in their code parameters, not a demonstrated implementation advantage.

Limits and uncertainty

Certification costs grow rapidly with code length and distance. The preprint says cases around n ≈ 250 and d ≈ 13 require a supercomputer, and estimates that some predicted improvements would require enumerations larger than those completed. The reported codes have certificates; many suggested next improvements remain computational targets.

This is an arXiv preprint, not a peer-reviewed publication in the supplied evidence. The comparison uses Grassl’s table as read on 4 November 2026, so later updates could change it. The work provides explicit codes and reproducibility materials, but does not establish how the codes would perform in a particular hardware system or error-correction workflow. Those practical questions require further study.