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arXiv:2606.15558

Degeneracy Cannot Violate the Quantum Hamming Bound

Yu-Xuan Zhang, Jing-Ling Chen

quant-phSubmitted Jun 14, 2026• Announced Jun 16, 2026• 0 citations
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Abstract

The quantum Hamming bound is the standard finite-length sphere-packing bound for exact correction of arbitrary qubit errors. Whether degeneracy can evade this bound has remained unresolved in full generality for nearly three decades: distinct correctable errors may act identically on the code space, so the usual disjoint-sphere argument breaks down. We prove that every exact binary quantum subspace code with $K>1$ obeys the bound, without assuming either nondegeneracy or additivity. Our proof turns the Li--Xing linear-programming polynomial into an exact intersection count for quaternary Hamming balls. Monotonicity in block length and in ball-center separation then reduces the problem to a local node--edge charging inequality at the shortest admissible length. Thus degeneracy can merge correctable error sectors, but cannot enlarge the finite-length binary Hamming bound.

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The quantum Hamming bound is the standard finite-length sphere-packing bound for exact correction of arbitrary qubit errors. Whether degeneracy can evade this bound has remained unresolved in full generality for nearly three decades: distinct correctable errors may act identically on the code space, so the usual disjoint-sphere argument breaks down. We prove that every exact binary quantum subspace code with K>1K>1K>1 obeys the bound, without assuming either nondegeneracy or additivity. Our proof turns the Li--Xing linear-programming polynomial into an exact intersection count for quaternary Hamming balls. Monotonicity in block length and in ball-center separation then reduces the problem to a local node--edge charging inequality at the shortest admissible length. Thus degeneracy can merge correctable error sectors, but cannot enlarge the finite-length binary Hamming bound.
The quantum Hamming bound arises from a direct dimension estimate for exact quantum error-correcting codes. For nondegenerate codes, distinct correctable errors map the code space into orthogonal subspaces, so dimension counting directly gives the bound; degeneracy allows these subspaces to overlap and invalidates the argument. Although linear-programming, asymptotic, and structural approaches have advanced the problem from several directions, the general local-dimension case has remained open. Here we prove the quantum Hamming bound for every local dimension q≥3q\ge3q≥3. Building on the exact two-center Fourier reduction, we develop new overlap estimates for nonbinary Hamming spaces. For q≥4q\ge4q≥4, Jacobi estimates and monotonicity give a uniform gap; qutrits form the boundary case where the gap vanishes and are handled by a filtered correction. This completes the finite-length nonbinary case; together with the existing qubit result, it establishes the quantum Hamming bound for arbitrary local dimension.
We prove a conjecture of Yu, Lai, and Oh: their strengthened quantum Hamming bounds, a family that interpolates between Hamming-type bounds and the Singleton bound, hold for all quantum codes of dimension K>1K>1K>1, including impure and nonadditive codes. The result covers every integer local dimension p≥2p\ge2p≥2, not necessarily a prime power, every integer t≥1t\ge1t≥1, both distances 2t+12t+12t+1 and 2t+22t+22t+2, and every interpolation parameter e∈{0,…,t}e\in\{0,\ldots,t\}e∈{0,…,t}. The case e=0e=0e=0 implies the ordinary quantum Hamming bound for general codes, recently proved by Zhang and Chen. Yu, Lai, and Oh proved their bounds for pure codes and for general codes of distance three or four. More generally, for every integer p≥2p\ge2p≥2 and every quantum code of length nnn, dimension K>1K>1K>1 and distance d≥3d\ge3d≥3, we prove Kf(0)≤pnEf(X)Kf(0)\le p^n\mathbb{E} f(X)Kf(0)≤pnEf(X), where X∼Bin⁡(n,1−p−2)X\sim\operatorname{Bin}(n,1-p^{-2})X∼Bin(n,1−p−2), for every real polynomial fff of degree less than ddd that is nonnegative on {0,…,n}\{0,\ldots,n\}{0,…,n} and positive at zero. For pure codes this inequality is implicit in the argument of Yu, Lai, and Oh. The proof uses the linear-programming method with quantum weight enumerators.
No ((n,K,d<127))((n,K,d< 127))((n,K,d<127)) Code Can Violate the Quantum Hamming Bound
It is well-known that non-degenerate quantum error correcting codes (QECCs) are constrained by a quantum version of the Hamming bound. Whether degenerate codes also obey such a bound, however, remains a long-standing question with practical implications for the efficacy of QECCs. We employ a combination of previously derived bounds on QECCs to demonstrate that a subset of all codes must obey the quantum Hamming bound. Specifically, we combine an analytical bound due to Rains with a numerical bound due to Li and Xing to show that no((n,K,d))((n,K,d))((n,K,d))code withd<127d< 127d<127can violate the quantum Hamming bound.
The quantum Hamming bound was originally put forward as an upper bound on the parameters of nondegenerate quantum codes, but over the past few decades much work has been done to show that many degenerate quantum codes must also obey this bound. In this paper, we show that there is a Hamming-like bound stricter than the quantum Hamming bound that applies to degenerate ttt-error-correcting stabilizer codes of length greater than some positive integer N(t)N(t)N(t). We show that this bound holds for all single-error-correcting degenerate stabilizer codes, forcing all but a handful of optimal distance-3 stabilizer codes to be nondegenerate.
The Pauli-ball discrepancy, a quadratic discrepancy summed over all error radii, measures how uniformly the Pauli error images of a KKK-dimensional quantum code cover the ddd-dimensional Hilbert space. At parameters attaining equality in the quantum Singleton bound with minimum code distance ≥3\geq3≥3, or in the one-error Hamming bound with prime-power local dimension, the discrepancy of every projector of the prescribed rank equals an explicit benchmark plus an exact nonnegative remainder, which vanishes precisely for pure quantum maximum-distance-separable codes and nondegenerate perfect codes, respectively. For a Pauli ball of radius eee below half the target distance and size VeV_eVe​, this remainder linearly bounds the squared Hilbert--Schmidt distance of the error-image Gram matrix from the identity. When KVe≤dKV_e\leq dKVe​≤d, it linearly bounds the diamond error and entanglement infidelity of a noise-independent polar recovery under any noise in that ball's span. For five qubits, we obtain the sharp rank-two Gram constant and exact logical channels of rotated codes attaining this linear order in diamond error. For pure states, the discrepancy equals a pure-state benchmark plus a positive weighted sum of marginal-purity defects, so absolutely maximally entangled states, when they exist, are exactly the minimizers.
We give an infinite family of degenerate entanglement-assisted quantum error-correcting codes (EAQECCs) which violate the EA-quantum Hamming bound for non-degenerate EAQECCs and achieve the EA-quantum Singleton bound, thereby proving that the EA-quantum Hamming bound does not asymptotically hold for degenerate EAQECCs. Unlike the previously known quantum error-correcting codes that violate the quantum Hamming bound by exploiting maximally entangled pairs of qubits, our codes do not require local unitary operations on the entangled auxiliary qubits during encoding. The degenerate EAQECCs we present are constructed from classical error-correcting codes with poor minimum distances, which implies that, unlike the majority of known EAQECCs with large minimum distances, our EAQECCs take more advantage of degeneracy and rely less on the error correction capabilities of classical codes.