Rains, Hardin, Shor, and Sloane's 1997 letter presented the ((5,6,2)) code, the first quantum error-correcting code strictly better than any stabilizer (additive) code, and sketched a coset/translate construction for further nonadditive codes while stating no explicit open questions. The problem generalizes the question that the letter answered in a single instance: for quantum codes of block length n and minimum distance d, determine whether codes outside the stabilizer class outperform the best stabilizer codes, and by how much; construct such codes systematically; and determine the optimal code dimension over all quantum codes, degenerate and nonadditive included. Later literature resolved this favorably in specific regimes - distance-2 codes, single-error-correcting families 50% larger than optimal stabilizer codes, channel-adapted (amplitude damping, asymmetric) codes, and transversal-gate-optimized codes - but the optimal ((n,K,d)) parameters remain undecided for essentially all block lengths 6 to 19 (with 2026 semidefinite-programming upper bounds lacking matching constructions), and the structure of genuinely nonadditive codes is described in 2024-2026 work as largely uncharted.

Background

A qubit is a two-level quantum system, and an n-qubit register is a 2^n-dimensional complex Hilbert space. A quantum error-correcting code protects quantum information by embedding a K-dimensional logical subspace in that Hilbert space so that noise can be detected and reversed; it is denoted ((n, K, d)), where the minimum distance d means the code can correct arbitrary errors on fewer than d/2 qubits, or equivalently correct the erasure of up to d-1 qubits whose positions are known. Qubit errors are generated by the Pauli matrices sigma_x, sigma_y, sigma_z (bit flip, phase flip, and their combination), whose tensor products form the n-qubit Pauli group. Almost every good quantum code known before 1997 was a stabilizer (also called additive) code: a simultaneous eigenspace of an Abelian subgroup of the Pauli group not containing -I. The Calderbank-Shor-Steane (CSS) construction and Gottesman's stabilizer formalism reduced stabilizer-code construction to classical codes over GF(4), the four-element field, and supplied the codes on which quantum-computing architectures still rest, but left open whether the stabilizer restriction costs anything in principle.

Rains, Hardin, Shor, and Sloane (Phys. Rev. Lett. 79, 953 (1997)) broke this barrier. They exhibited a ((5,6,2)) code - a 6-dimensional subspace of the 5-qubit space that corrects the erasure of any single qubit - strictly better than the best stabilizer code of the same length and distance, which has dimension K = 4. Linear-programming bounds, built on quantum weight enumerators (the quantum analogue of Delsarte's classical method of counting how a code responds to each weight class of Pauli errors), show K = 6 is optimal for n = 5, d = 2. The code was found by a numerical algorithm, and the letter sketched a general construction: take a self-dual additive code C0 with stabilizer code Q0 and a union C of K cosets of C0; the span of the translates of Q0 by elements of C is a ((n,K,d)) code that need not be a stabilizer code. The letter states no explicit open question; the implicit program it leaves is the exploration of further nonadditive codes and the generality of their advantage over stabilizer codes.

Later work reshaped this program. The codeword-stabilized (CWS) framework of Cross, Smith, Smolin, and Zeng represents a code as a graph state (a stabilizer state described by a graph with one qubit per vertex) plus a classical binary code that must correct an error pattern induced by that graph; it covers all known good nonadditive codes and reduces quantum code search to a hard combinatorial problem. The exact criterion any code must satisfy, additive or not, is the Knill-Laflamme condition: the projector P onto the code space must satisfy PEi†EjP=λijPP E_i^\dagger E_j P = \lambda_{ij} P for the correctable error operators Ei,EjE_i, E_j with scalars λij\lambda_{ij}; codes outside the stabilizer class are precisely those whose λij\lambda_{ij} structure no commuting Pauli subgroup can realize. A code is called degenerate when two distinct correctable errors act identically (up to a scalar) on the code space; degenerate and nonadditive codes are both allowed in everything that follows.

Recent work approaches the structure of genuinely nonadditive codes through the Knill-Laflamme coefficients λij\lambda_{ij}: Du, Zhang, Poon, and Zeng note that the structure of these coefficients remains challenging, particularly for nonadditive codes, while Gupta, Sun, He, and Zeng show that exact Pauli-detecting codes form connected continuous families in which stabilizer codes occupy only measure-zero subsets - a geometry that Sun, Cao, Poon, and Zeng develop further.

Problem Statement

The problem is to determine whether, where, and by how much quantum error-correcting codes outside the stabilizer (additive) class outperform the best stabilizer codes, and to build the constructions and theory needed to answer that. Concretely, for quantum codes of block length n and minimum distance d (and, where relevant, a specified noise model or gate requirement, such as amplitude damping - the relaxation channel that takes ∣1⟩|1\rangle to ∣0⟩|0\rangle and can only drive errors in the bit-flip direction, asymmetric Pauli noise - Pauli noise with different bit-flip and phase-flip rates, or a transversal-gate requirement, meaning a logical gate must be implemented by applying the same single-qubit gate to every physical qubit):

  1. Construction: give systematic constructions of nonadditive ((n,K,d)) codes - codes not equivalent to any stabilizer code - whose dimension K strictly exceeds the largest dimension achievable by any stabilizer code of the same n and d, generalizing the coset/translate construction sketched by Rains, Hardin, Shor and Sloane (the span of translates of a self-dual stabilizer code under a union of cosets) and its codeword-stabilized descendants, together with workable encoding and decoding circuits.

  2. Optimality: determine the maximal dimension K_opt(n,d) over all n-qubit quantum codes of distance d, via matching upper bounds (linear-programming and semidefinite-programming bounds valid for degenerate and nonadditive codes) and constructions, and decide for which parameters (n,d) the nonadditive optimum strictly exceeds the stabilizer optimum.

  3. Structure: characterize genuinely nonadditive codes.

These three facets serve one objective: a predictive theory of the nonadditive regime of quantum coding that replaces the current case-by-case collection of constructions and one-way bounds, telling a code designer when giving up the stabilizer structure helps and by how much. The question concerns block lengths 6 <= n <= 19, where the optimal K_opt(n,d) is undecided, and it may turn out that the stabilizer optimum is already attained there - the advantage is refuted for those (n,d) whose optimum is attained by a stabilizer code. Facet 1 is therefore the existence side of the decision posed in facet 2, exhibiting constructions for the (n,d) at which the nonadditive optimum exceeds the stabilizer optimum, and the encoding and decoding circuits of facet 1 and the structural characterization of facet 3 are goals toward the same theory. Partial resolutions are known (distance-2 codes, infinite single-error-correcting families beating all stabilizer codes, channel-adapted and transversal-gate advantages), but the optimal K_opt(n,d) is undecided for essentially all block lengths 6 to 19 and no structural characterization of nonadditive codes exists.

References

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