Most logical qubits with check weight at most 6

Task

Find a CSS code in which every check acts on at most 6 qubits and each qubit belongs to at most 6 checks. Maximize the number of logical qubits k within the instance's limits:

• Instance r72: n ≤ 72 and d ≥ 6.
• Instance r108: n ≤ 108 and d ≥ 10.
• Instance r144: n ≤ 144 and d ≥ 12.
• Instance r288: n ≤ 288 and d ≥ 18.

A qubit's degree is the total number of X and Z checks containing it.

The limits come from how a code runs on hardware. Every check is measured in every round by a circuit with one two-qubit gate per qubit in the check, and on hardware with fixed couplers each qubit needs a connection to every check that contains it. Low weight and low degree keep that circuit short and the wiring sparse. Weight 6 and degree 6 is the setting of the bivariate bicycle codes, including IBM's [[144,12,12]] gross code, whose connections fit on two planar layers (Bravyi et al., arXiv:2308.07915). Without a weight limit the task becomes the general table of quantum codes (codetables.de), where codes with dense checks reach much larger k but need long measurement circuits.

Submission file

JSON structure:

{"n": 72, "HX": [[0, 3, 6, 36, 40, 41], ...], "HZ": [[1, 2, 5, 36, 38, 39], ...]}

This shows the file structure. The verifier bundle contains a valid example.

• Each row is a nonempty list of distinct qubit indices in increasing order. Indices run from 0 to n − 1.
• Repeated rows are not allowed within HX or within HZ. HX and HZ may contain at most 2n rows combined.

Validation

• Every row has weight at most 6, and every qubit has degree at most 6.
• HX·HZ^T = 0 (mod 2).
• k = n − rank HX − rank HZ ≥ 1, computed exactly over GF(2).
• The X and Z distances are at least the instance's d. The verifier proves this by an exhaustive search for logical operators below the required distance. If the search finds such an operator, the submission is rejected and the report includes the operator. The search has a budget of six CPU hours on one core: if it does not finish, the result is error and the code is not ranked, even if its distance is in fact large enough. The Verifier tab explains the search.

Scoring

More logical qubits rank first. Ties are ranked by fewer physical qubits, then lower maximum qubit degree, then submission number (lower first). The leaderboard also reports k·d²/n, using the instance's certified distance lower bound for d.

Baseline and reference scores

Codes from Bravyi et al. (arXiv:2308.07915). Lee's census of two-block group-algebra codes (arXiv:2610.13880) finds the gross code on the frontier among abelian weight-6 two-block codes up to n = 300. Twisted-torus codes: Liang et al. (arXiv:2503.03827). Related challenge: the Unitary Foundation QEC Challenge (qecchallenge.com); at r72, r108 and r144 its best entries are the same Bravyi et al. codes.

Application

Increasing k can reduce the number of physical qubits needed per logical qubit.

Current records

Reference scores come from cited papers or derived constructions. Only accepted submissions can hold a QArena record.