A new mathematical framework turns coherence between codewords in a class of quantum codes into a certified lower bound on robustness of magic, a measure here of how far a target lies beyond the stabilizer-state set used for comparison. The work gives an exact threshold for equal-weight superpositions and an exact fixed-parameter procedure for weighted ones.
The result applies to codeword-stabilized, or CWS, codes. In standard form, a CWS code is specified by a graph state and a classical set of binary words. The paper's central move is to treat the arrangement of those words as an affine-geometric problem, linking a quantum certification task to intersections among structured sets of binary vectors.
A threshold for coherence
The proposed CWS coherence witness starts with the projector onto a coherent superposition of the codewords and subtracts the diagonal population of each individual codeword. What remains is the off-diagonal coherence between codewords, rather than their individual populations.
To turn that quantity into a certification, the witness is compared with the largest absolute value it can take on any stabilizer state. A calibrated Hermitian witness then gives a lower bound on robustness of magic by dividing the target state's absolute expectation by that stabilizer maximum. The bound becomes nontrivial when the target expectation is larger than the stabilizer threshold.
For weighted CWS targets, the paper reports an analytic numerator of the form 1 - sum of |alpha_c|^4 and uses it in the robustness lower bound.
The classical shape decides the equal-weight case
The key formula concerns equal-weight superpositions. Their exact stabilizer threshold is determined by an affine-intersection profile, written M_r(C). The threshold is the maximum, over r from 0 to n, of M_r(C)(M_r(C)-1)/(K 2^r), where K is the number of CWS words.
That formula identifies a structural dividing line. A linear CWS word set produces a stabilizer state in the equal-weight mode. Nontrivial equal-weight magic therefore requires nonlinear affine geometry in the underlying word set.
For weighted CWS superpositions, the exact calculation enumerates affine intersections and affine-quadratic phases with a fixed-parameter algorithm in the number of CWS words. The equal-weight threshold is reported to be computable exactly by subset enumeration in time O(2^K(n^3 + K n^2)). That dependence on 2^K makes exhaustive calculation expensive as the word set grows.
Exact certificates for selected code families
The paper applies the framework to a binary affine-simplex family with n at least 2. For that family, it reports a stabilizer threshold of 3/[2(n+1)] and a robustness-of-magic lower bound of 2n/3. Because the expressions depend on n, they describe a family of code states rather than one numerical example.
It also reports exact rational robustness lower bounds for several listed nonadditive CWS examples: 10/3 for ((5, 6, 2)), 44/15 for ((9, 12, 3)), 17/7 for ((10, 18, 3)), 76/33 for ((10, 20, 3)), and 14/5 for the ((7, 22, 2)) SSW example. These are lower-bound certificates produced for the listed examples.
The method is not equally tractable for every structured family. For the Rains family, the word-set size grows exponentially with n. That makes exact subset enumeration non-scalable and leaves affine-intersection bounds as the necessary route for larger instances.
A specified noise benchmark
A global-depolarizing benchmark for the ((5, 6, 2)) state gives a noisy robustness lower bound of 10(1-p)/3, which remains nontrivial for p less than 7/10. The result is specific to the stated global-depolarizing benchmark.
The document is an arXiv preprint, version 1, dated 26 August 2026. The numerical code is available from the corresponding author upon reasonable request, and a public GitHub release is planned upon publication.
The broader point is that nonlinear word-set geometry provides a route to certifiable magic, while exact computation remains limited by the size of the word set. For larger structured families, the paper leaves affine-intersection bounds as the necessary tool when exhaustive enumeration no longer scales.
Paper data and sources
Original title: Affine-Profile Stabilizer Thresholds for Magic in Codeword-Stabilized Quantum Codes
Authors: Li-Yi Hsu, En-Jui Kuo
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
Original paper · Full text