Pauli-ball discrepancy and stability of quantum codes
Abstract
The Pauli-ball discrepancy, a quadratic discrepancy summed over all error radii, measures how uniformly the Pauli error images of a K-dimensional quantum code cover the d-dimensional Hilbert space. At parameters attaining equality in the quantum Singleton bound with minimum code distance ≥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 e below half the target distance and size Ve, this remainder linearly bounds the squared Hilbert--Schmidt distance of the error-image Gram matrix from the identity. When KVe≤ 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.
Related
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- Unitary fault-tolerant encoding of Pauli states in surface codes
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- Majorana-Pauli stabilizer codes and duality webs of fermionic topological phases
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BibTeX
@misc{yohananov2026pauliball,
title = {{Pauli-ball discrepancy and stability of quantum codes}},
author = {Aryeh Lev Zabokritskiy "Yohananov"},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.34399},
eprint = {2610.34399},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.34399},
}
RIS
TY - UNPB
TI - Pauli-ball discrepancy and stability of quantum codes
AU - Aryeh Lev Zabokritskiy "Yohananov"
PY - 2026
DA - 2026/10/25
PB - arXiv
DO - 10.48550/arxiv.2610.34399
UR - https://arxiv.org/abs/2610.34399
AB - The Pauli-ball discrepancy, a quadratic discrepancy summed over all error radii, measures how uniformly the Pauli error images of a K-dimensional quantum code cover the d-dimensional Hilbert space. At parameters attaining equality in the quantum Singleton bound with minimum code distance ≥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 e below half the target distance and size V_e, this remainder linearly bounds the squared Hilbert--Schmidt distance of the error-image Gram matrix from the identity. When KV_e≤ 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.
ER -