Most efficient weight-6 qLDPC codes with exact distance

Task

Find a CSS code in which every check acts on at most 6 qubits and each qubit belongs to at most 6 checks, X and Z checks together. Give its distance d and one logical operator of weight d. Maximize the efficiency k·d²/n, where n is the number of physical qubits and k the number of logical qubits. Each instance limits the size:

• Instance n144: n ≤ 144.
• Instance n288: n ≤ 288.
• Instance n512: n ≤ 512.

In every instance d is at most 32. A code may enter any instance whose limit it meets, so a small code can also compete in a larger instance. Each instance has its own leaderboard; scores are not combined across instances.

The weight and degree 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. Weight 6 and degree 6 is the setting of the bivariate bicycle codes, including IBM's [[144,12,12]] gross code (Bravyi et al., arXiv:2308.07915). The Weight-6 qLDPC board ranks k at a fixed distance; this board ranks k and d together.

Two limits shape this competition: the size of each instance, and the distance the verifier can prove within its budget of 21,600 s of CPU time. The proof is an exhaustive search whose cost grows quickly with d. A code whose distance is not proved within the budget gets status error and is not ranked, even if the claimed distance is right. Efficiency describes the code alone; it is not a logical error rate under circuit noise, which also depends on the syndrome-extraction circuit and the decoder.

Submission file

JSON structure:

{"n": 144, "HX": [[1, 2, 18, 75, 78, 84], ...], "HZ": [[3, 60, 66, 76, 77, 126], ...], "d": 12,
 "logical": {"type": "X", "support": [0, 3, 9, 15, 63, 66, 80, 81, 82, 97, 98, 114]}}

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

• Each row of HX and HZ 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, and either may be empty.
• logical.type is "X" or "Z". logical.support lists the qubits of the operator in increasing order, exactly d of them.
• No other fields are allowed.

Validation

• Every row has weight at most 6, and every qubit lies in at most 6 rows of HX and HZ together.
• HX·HZ^T = 0 (mod 2).
• k = n − rank HX − rank HZ ≥ 1, computed exactly over GF(2).
• The submitted operator is a nontrivial logical operator of weight d. An X operator must have even overlap with every row of HZ and must not be a sum of rows of HX; for a Z operator, exchange X and Z. So the distance is at most d. This is checked before the search.
• No nontrivial X or Z logical operator has weight below d. The verifier proves this by an exhaustive search, first for each weight from 1 to the smaller of d − 1 and 10, then once for weight up to d − 1 when d − 1 is larger than 10 (no search when d = 1). If the search finds such an operator, the submission is rejected and the report includes the operator. The Verifier tab explains the search.
• Together these make d exact: d = min(d_X, d_Z).
• The whole verification has 21,600 s of CPU time, of which the search gets about 21,500 s, and 2³⁸ search nodes. Otherwise the submission gets status error and is not ranked.

Scoring

Higher efficiency k·d²/n ranks first, compared as exact fractions. Ties are ranked by fewer physical qubits, then lower maximum qubit degree, then submission number (lower first). The leaderboard shows the efficiency rounded to 10 significant digits, together with k, d, n and the maximum qubit degree.

Baseline and reference scores

The gross code is accepted in all three instances. The [[288,16,16]] code is Lee's two-block code over a non-abelian group of order 144, SmallGroup(144,64), rebuilt from Lee's public database; it beats the [[288,12,18]] bivariate bicycle code of Bravyi et al. (efficiency 27/2 = 13.5, also accepted here). Lee's [[254,14,16]] cyclic code (efficiency 1792/127 ≈ 14.11) and the [[72,12,6]] bivariate bicycle code (efficiency 6) are accepted as well. The [[340,16,18]] code is the cyclic form of the twisted-torus code of Liang et al.: L = 170, A = 1 + S⁴ + S²⁵, B = 1 + S + S⁷⁰, H_X = [A | B], H_Z = [B^T | A^T], where S is the cyclic shift of size L. Lee reports that no weight-6 two-block code with n ≤ 300 beats [[288,16,16]].

Verification cost of the baselines on one core of a desktop PC (AMD Ryzen 9 9950X): the gross code 1.8 s, [[288,16,16]] 79 s, [[340,16,18]] 842 s (194 million search nodes), [[288,12,18]] 562 s; memory stays below 13 MB.

Bivariate bicycle codes: Bravyi et al. (arXiv:2308.07915). Twisted-torus codes: Liang et al. (arXiv:2503.03827). [[288,16,16]] and [[254,14,16]]: Lee's census of two-block group-algebra codes (arXiv:2610.13880; database github.com/gideonlsx/bicycle-race).

Application

A code with higher efficiency stores more logical qubits at a given distance in a fixed block of physical qubits. Quantum-memory architects can use such a code as a candidate, then evaluate its syndrome-extraction circuit and decoder.

Current records

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