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arXiv:2510.07158

Haar random codes attain the quantum Hamming bound, approximately

Fermi Ma, Xinyu Tan, John Wright

Published Oct 8, 2025• 0 citations
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Abstract

We study the error correcting properties of Haar random codes, in which a $K$-dimensional code space $\boldsymbol{C} \subseteq \mathbb{C}^N$ is chosen at random from the Haar distribution. Our main result is that Haar random codes can approximately correct errors up to the quantum Hamming bound, meaning that a set of $m$ Pauli errors can be approximately corrected so long as $mK \ll N$. This is the strongest bound known for any family of quantum error correcting codes (QECs), and continues a line of work showing that approximate QECs can significantly outperform exact QECs [LNCY97, CGS05, BGG24]. Our proof relies on a recent matrix concentration result of Bandeira, Boedihardjo, and van Handel.

DOI
10.48550/arxiv.2510.07158

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We study the error correcting properties of Haar random codes, in which a KKK-dimensional code space C⊆CN\boldsymbol{C} \subseteq \mathbb{C}^NC⊆CN is chosen at random from the Haar distribution. Our main result is that Haar random codes can approximately correct errors up to the quantum Hamming bound, meaning that a set of mmm Pauli errors can be approximately corrected so long as mK≪NmK \ll NmK≪N. This is the strongest bound known for any family of quantum error correcting codes (QECs), and continues a line of work showing that approximate QECs can significantly outperform exact QECs [LNCY97, CGS05, BGG24]. Our proof relies on a recent matrix concentration result of Bandeira, Boedihardjo, and van Handel.
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.
kkk-Uniform states are fundamental to quantum information and computing, with applications in multipartite entanglement and quantum error-correcting codes (QECCs). Prior work has primarily focused on constructing exact kkk-uniform states or proving their nonexistence. However, due to inevitable theoretical approximations and experimental imperfections, generating exact kkk-uniform states is neither feasible nor necessary in practice. In this work, we initiate the study of approximate kkk-uniform states, demonstrating that they are locally indistinguishable from their exact counterparts unless massive measurements are performed. We prove that such states can be constructed with high probability from the Haar-random ensemble and, more efficiently, via shallow random quantum circuits. Furthermore, we establish a connection between approximate kkk-uniform states and approximate QECCs, showing that Haar random constructions yield high-performance codes with linear rates, vanishing proximity, and exponentially small failure probability while random circuits can't construct codes with linear code rate in shallow depth. Finally, we investigate the relationship between approximate QECCs and approximate quantum information masking. Our work lays the foundation for the practical application of kkk-uniform states.
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One of the main problems in quantum information systems is the presence of errors due to noise, and for this reason quantum error-correcting codes (QECCs) play a key role. While most of the known codes are designed for correcting generic errors, i.e., errors represented by arbitrary combinations of Pauli X, Y and Z operators, in this paper we investigate the design of stabilizer QECC able to correct a given number egof generic Pauli errors, plus eZPauli errors of a specified type, e.g., Z errors. These codes can be of interest when the quantum channel is asymmetric in that some types of error occur more frequently than others. We first derive a generalized quantum Hamming bound for such codes, then propose a design methodology based on syndrome assignments. For example, we found a [[9, 1]] quantum error-correcting code able to correct up to one generic qubit error plus one Z error in arbitrary positions. This, according to the generalized quantum Hamming bound, is the shortest code with the specified error correction capability. Finally, we evaluate analytically the performance of the new codes over asymmetric channels.
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.
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.