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The Optimal Distance of Fourteen-Qubit Stabilizer Codes Encoding Three Qubits

Mansehej Singh

Published Aug 10, 2026• 0 citations
Publisher

Abstract

We prove, with computer assistance, that the optimal minimum distance of abinary stabilizer code on $14$ qubits encoding $3$ logical qubits is $4$.Equivalently, no binary quantum stabilizer code with parameters$\code{14,3,5}$ exists, including degenerate codes. In the binary symplecticmodel, this is equivalent to the nonexistence of an $11$-dimensional isotropicsubspace $C\subseteq\F_2^{28}$ satisfying$\min\{\wt(v):v\in C^{\perps}\setminus C\}\ge 5$.The argument separates the two cases determined by the parity of thestabilizer weights. MacWilliams identities and integrality eliminate theall-even case. In the odd case, the shadow enumerator of Rains is combined withtwo averages over the $135$ self-dual extensions of the code: one for theirweight enumerators and one for their shadows. An incidence computation inan eight-dimensional orthogonal geometry of plus type then forces threelow-weight stabilizer profiles. Integrality and a collision count amongweight-three shadow words eliminate these profiles. All parameter-specificcalculations are finite and exact, and reproducible source code accompanies the paper.The nonexistence theorem is additionally formalized and machine-checked inLean~4 over mathlib, with an axiom audit reporting only standard foundationalprinciples.

DOI
10.5281/zenodo.21874337

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We prove, with computer assistance, that the optimal minimum distance of abinary stabilizer code on 141414 qubits encoding 333 logical qubits is 444.Equivalently, no binary quantum stabilizer code with parameters\code14,3,5\code{14,3,5}\code14,3,5 exists, including degenerate codes. In the binary symplecticmodel, this is equivalent to the nonexistence of an 111111-dimensional isotropicsubspace C⊆\F228C\subseteq\F_2^{28}C⊆\F228​ satisfyingmin⁡{\wt(v):v∈C\perps∖C}≥5\min\{\wt(v):v\in C^{\perps}\setminus C\}\ge 5min{\wt(v):v∈C\perps∖C}≥5.The argument separates the two cases determined by the parity of thestabilizer weights. MacWilliams identities and integrality eliminate theall-even case. In the odd case, the shadow enumerator of Rains is combined withtwo averages over the 135135135 self-dual extensions of the code: one for theirweight enumerators and one for their shadows. An incidence computation inan eight-dimensional orthogonal geometry of plus type then forces threelow-weight stabilizer profiles. Integrality and a collision count amongweight-three shadow words eliminate these profiles. All parameter-specificcalculations are finite and exact, and reproducible source code accompanies the paper.The nonexistence theorem is additionally formalized and machine-checked inLean~4 over mathlib, with an axiom audit reporting only standard foundationalprinciples.
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We prove that no binary qubit stabilizer code with parameters [[14,3,d>=5]] exists, settling the remaining distance gap for stabilizer codes encoding three qubits into fourteen. The proof first applies a signed affine-shadow inequality and an integrality congruence to restrict low-weight stabilizer words. Exact split-shadow infeasibility certificates then exclude weight-two words and all configurations with three weight-four checks, leaving a unique weight-four check. That check yields a ten-coordinate binary-additive code of size 1,024, minimum distance at least five, and symplectic hull dimension four. We reconstruct all 37 relevant equivalence classes by exhaustive lengthening: 25 have the wrong hull dimension, and the other twelve violate a necessary 64-coset capacity bound. Together with the known [[14,3,4]] construction, this proves d_max(14,3)=4 for binary stabilizer codes.
The MacWilliams extension theorem fails for module alphabets with non-cyclic socle, and the label alphabet of qudit stabilizer codes, $\F_{q^2}$ over $\F_q$, is such an alphabet. Quantum error correction, however, only ever sees \emph{self-orthogonal} additive codes, and whether that rigidity rescues the theorem---equivalently, whether every weight-preserving isomorphism of stabilizer groups is implemented by local Cliffords and a qudit permutation---was asked by Gluesing-Luerssen and Pllaha and answered negatively by Pllaha for particular qubit codes. We develop the negative answer systematically and at the smallest possible scales. For every prime power qqq we construct a pair of [[q+1,q−1]]q[[q+1,q-1]]_q[[q+1,q−1]]q​ stabilizer codes and a weight-preserving isomorphism between them extending to no monomial transformation; the codespaces are inequivalent even under arbitrary local unitaries combined with permutations, though they share Shor--Laflamme enumerators. Self-orthogonality is automatic here, by two elementary lemmas which also show that Dyshko's threshold-length counterexamples were already self-orthogonal, unremarked. For qubits we prove by exhaustive search that length 333 is minimal and the counterexample essentially unique. Dropping the ``idle qudit'' invariant that detects these, we find the minimal full-support lengths: 444 for a non-extendable isometry, 555 for a weight-isometric pair that is not monomially equivalent, realized by explicit [[5,2]][[5,2]][[5,2]] codes; at length 666 all nontrivial stabilizer elements can have weight ≥4\ge 4≥4. Whether these codespaces are locally unitarily equivalent is posed as an open problem, connecting the extension problem to the LU--LC circle of questions.
Stabilizer codes are often constructed within the Calderbank--Shor--Steane (CSS) framework, where two mutually orthogonal binary classical codes define XXX and ZZZ-type stabilizer generators. While this structure is algebraically convenient, additional non-CSS constraints may help suppress low-weight logical operators and improve decoding performance in the finite-length regime. We thus introduce quantum XYZ stabilizer codes, whose parity-check matrix (PCM) is built from three pairwise orthogonal binary PCMs associated with XXX-, YYY-, and ZZZ-type stabilizer generators. A nontrivial point is that an XYZ code instance is not automatically genuinely non-CSS: the same stabilizer group may admit a CSS generating set. We characterize this collapse, obtaining algebraic and rank conditions for deciding when the YYY-type checks are redundant and when they define genuinely non-CSS stabilizer constraints. We also derive upper and lower bounds on the quantum minimum distance, including bounds for mixed Pauli logical operators. The novel framework includes a known non-CSS topological code, namely the XYZ2^22 hexagonal code, and yields also sparse finite-length quantum low-density parity-check (qLDPC) constructions from intersecting-subset and quasi-dyadic code families. Simulations under depolarizing code-capacity noise and quaternary belief propagation decoding show that the proposed XYZ qLDPC instances can outperform representative CSS qLDPC instances with similar finite-length parameters.
Linear programming approaches have been applied to derive upper bounds on the size of classical and quantum codes. In this paper, we derive similar results for general quantum codes with entanglement assistance by considering a type of split weight enumerator. After deriving the MacWilliams identities for these enumerators, we are able to prove algebraic linear programming bounds, such as the Singleton bound, the Hamming bound, and the first linear programming bound. Our Singleton bound and Hamming bound are more general than the previous bounds for entanglement-assisted quantum stabilizer codes. In addition, we show that the first linear programming bound improves the Hamming bound when the relative distance is sufficiently large. On the other hand, we obtain additional constraints on the size of Pauli subgroups for quantum codes, which allow us to improve the linear programming bounds on the minimum distance of quantum codes of small length. In particular, we show that there is no [[27, 15, 5]] or [[28, 14, 6]] stabilizer code. We also discuss the existence of some entanglement-assisted quantum stabilizer codes with maximal entanglement. As a result, the upper and lower bounds on the minimum distance of maximal-entanglement quantum stabilizer codes with length up to 20 are significantly improved.
An [[n,k,d]][[n,k,d]][[n,k,d]] stabilizer code is phantom if every in-block logical CNOT gate can be realized by a permutation of its physical qubits. This eliminates the large and complicated physical overhead normally required for logical entangling gates. Yet this symmetry is highly restrictive: phantom codes are rare, the number of logical qubits is limited to k=O(log⁡n)k=O(\log n)k=O(logn), and no phantom qLDPC family with growing logical dimension was previously known. We make three contributions. 1. We prove that recognizing phantomness of a given stabilizer code is at least as hard as Graph Isomorphism, even for CSS codes encoding only k=2k=2k=2 logical qubits. 2. We propose the first rate optimal phantom qLDPC families: for every fixed DDD, our CSS families achieve the maximal logical scaling k=Θ(log⁡n)k=Θ(\log n)k=Θ(logn) and distance d≥Dd\geq Dd≥D. 3. We prove a distance no-go theorem: every phantom code family with k=ω(log⁡n)k=ω(\sqrt{\log n})k=ω(logn​) and check weight w=O(1)w=O(1)w=O(1) satisfies d≤wd\leq wd≤w for sufficiently large nnn. Thus our fixed-distance qLDPC families are distance-scaling optimal at the maximal logical scaling k=Θ(log⁡n)k=Θ(\log n)k=Θ(logn).