High-Rate Quantum Code’s Weakness Traced to Weight-3 Logical Operators

A quantum error correcting code with an unusually high encoding rate has now had its weakest point mapped out in full. The code lives on the face centered cubic lattice, the same geometry that describes the densest packing of spheres in three dimensions, and it encodes quantum information at a rate of 67.7 percent, far above the sub-one-percent rates of the standard surface code. A new theoretical study classifies every weight-3 logical operator of the code and shows that these lightweight operators, not any flaw in decoding software, are what cap its error protection at distance 3.

The code is a CSS stabilizer code with parameters [[3L³, 2L³+2, 3]], meaning it uses 3L³ physical qubits to encode 2L³+2 logical qubits while offering only distance-3 protection against errors. All of its stabilizer checks have weight 12, and every coupling is nearest neighbor in three dimensions, which makes the code attractive from a hardware standpoint even though its distance is low. The study answers a question left open by the earlier work that introduced the code: whether a modified construction could raise the distance while keeping the rate. The answer turns out to be no, and the reason is now exact rather than conjectural.

The classification has a strikingly geometric form. In the X sector, every weight-3 logical operator is supported on a triangular face of the lattice, and there are 4L³ of these faces. In the Z sector, every weight-3 logical operator is supported on a claw, the set of three edges of a tetrahedron that meet at one of its four vertices, and there are again 4L³ of these. A lemma shows that both families use exactly one edge from each of the three orthogonal sheets of nearest-neighbor vectors that structure the lattice, a property called the triad decomposition.

Completeness of the classification is proved rather than merely observed. Any weight-3 vector in the relevant kernel must be a three-edge subgraph in which every node has even degree, and the only such graph is a 3-cycle, which in this lattice is precisely a triangular face. On the dual side, an edge lies in exactly two octahedral voids, so three edges produce six incidences that must distribute evenly, forcing the three octahedra to surround a single tetrahedron and the edges to form a claw at one apex. A further proposition rules out mixed operators: no Pauli operator of weight 3 that commutes with the stabilizer group can have both X and Z components, because a support cannot be a face and a claw simultaneously.

The counts behind these families were verified exactly over the finite field GF(2) at lattice sizes L = 4, 6, 8 and 10. At L = 4, direct enumeration found exactly 256 weight-3 vectors in the X-type kernel, matching the predicted 4L³ faces, and exhaustive searches at L = 4 and L = 6 confirmed that the Z-type kernel vectors coincide exactly with the claws. The two families are exact duals under the exchange of vertices and octahedral voids, and the cross-sector symplectic pairing between them has rank 2L³ − 4, three short of each sector’s dimension.

Perhaps the most satisfying result is that the number of independent logical classes in each family follows from the homology of the 3-torus, so it holds for every even L and not only at the sizes checked by computer. The cycle space of the FCC graph has dimension 5L³/2 + 1, the boundary space has dimension 5L³/2 − 2, and subtracting the stabilizer rank of L³/2 − 1 leaves exactly 2L³ − 1 independent weight-3 classes in each sector. Since the code encodes 2L³ + 2 qubits, this means k = (2L³ − 1) + 3: all but three of the logical classes carry a weight-3 representative. The high encoding rate and the distance-3 ceiling are, in a precise sense, the same degrees of freedom counted twice.

The analysis also corrects a conjecture from the companion paper that introduced the code. That work had guessed that the logical operators were localized at tetrahedral voids with two per tetrahedron, and attributed two of the encoded qubits to topology. The correct picture is that the claws are indexed by tetrahedra but there are four claws per tetrahedron, the X sector lives on triangular faces rather than voids, and the topological contribution is 3 rather than 2. All code parameters of the earlier paper reproduce exactly.

Why can this vulnerable sector not simply be monitored? The natural detector would be the three-body parity operator on each face, which anticommutes with the corresponding weight-3 X operator. The study proves this parity is not measurable: it anticommutes with exactly four octahedral stabilizers, one containing all three edges of the face and three containing one edge each, so it does not belong to the normalizer of the stabilizer group. The obstruction is geometric and unavoidable, because every triangular face is shared between one tetrahedron and one octahedron, giving odd overlap with an octahedral check. The result goes further: no measurement of any kind can distinguish an arbitrary encoded state from the state obtained by applying the face logical operator, since that operator acts as a unitary with eigenvectors on the code space, and no measurement distinguishes a state from itself.

Any detector that does work must itself be a logical operator. A claw sharing an odd number of edges with a face commutes with every stabilizer and anticommutes with the face logical, so it detects the face error exactly, but measuring it projects a conjugate logical class. The detector for one weight-3 family is the other weight-3 family. Removing the sector entirely requires adjoining checks that span all 2L³ − 1 independent face directions, which drives the encoded qubit count down to k = 3. At L = 4 the gauged code becomes [[192, 3, 4]], and at L = 6 it becomes [[648, 3, 6]], parameters that coincide with the three-dimensional toric code at the same edge count, though the authors report only the coincidence of parameters and do not claim an equivalence of codes.

The practical consequences are concrete. Two errors on the edges of a face produce a syndrome identical to that of a single error on the third edge, so a maximum-likelihood decoder always chooses the single-edge explanation and, on a double fault, completes the face and applies a logical operation. Two errors therefore suffice to cause a block failure, giving a failure rate scaling as p² rather than p³. This matches Monte Carlo results from the companion paper, which reported a block logical error rate of 1.2 × 10⁻² at p = 10⁻³ under minimum-weight perfect matching at L = 4, exceeding a naive p³ estimate by a factor of about 4.7 × 10⁴. No decoder can separate the two error patterns, because the syndrome data are identical; the limit is set by the code itself. Under noise restricted to a single triad sheet the minimum logical weight rises to 4, but weight-4 operators fit within one sheet, so the gain stops there. The study frames the overall budget clearly: raising the distance on this lattice means changing the chain complex so that faces stop being cycles, and the cost comes out of the same 2L³ − 1 classes that give the code its remarkable rate.

Subject of Research: Classification of weight-3 logical operators in a high-rate face centered cubic lattice CSS quantum error correcting code

Article Title: Complete classification of weight-3 logical operators in the Face Centered Cubic lattice CSS code

Article References: Kulkarni, R. (2026). Complete classification of weight-3 logical operators in the Face Centered Cubic lattice CSS code. Discover Quantum Science, 2(1), Article 43. https://doi.org/10.1007/s44464-026-00047-3

Image Credits: AI Generated

DOI: 10.1007/s44464-026-00047-3

Keywords: quantum error correction, CSS code, FCC lattice, logical operators, qLDPC, code distance, homology, 3-torus, stabilizer codes, encoding rate, decoding, quantum computing