Weighted flip graphs from cyclic quantum mutations: pentagon inequalities and exponential separations
Abstract
Positive cyclic quantum mutations at a third root of unity induce an explicit weight on polygon flip graphs. We study the resulting discrepancy between minimum flip count and minimum accumulated normalization loss. Every pentagon three-to-two replacement strictly decreases weight, and in connected types A1, A2, and A3 every minimum-weight path is a flip geodesic. In type A4, however, a six-flip path has smaller weight than the unique five-flip geodesic. A compatible positive projective gluing construction amplifies this example to an exponential success-probability separation in every connected type An, n≥4. We also give a quantitative perturbation bound preserving the separation. The low-rank minimality statement has an exhaustive finite certificate; the pentagon inequality and the all-rank construction are analytic. In the associated sequential postselection model, differences of path weights determine success-probability ratios independently of the input state. The comparison holds for separately normalized exact mutation steps.
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BibTeX
@misc{tang2026weighted,
title = {{Weighted flip graphs from cyclic quantum mutations: pentagon inequalities and exponential separations}},
author = {Xiaomin Tang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.40661},
eprint = {2610.40661},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.40661},
note = {CC0},
}
RIS
TY - UNPB
TI - Weighted flip graphs from cyclic quantum mutations: pentagon inequalities and exponential separations
AU - Xiaomin Tang
PY - 2026
DA - 2026/10/03
PB - arXiv
DO - 10.48550/arxiv.2610.40661
UR - https://arxiv.org/abs/2610.40661
AB - Positive cyclic quantum mutations at a third root of unity induce an explicit weight on polygon flip graphs. We study the resulting discrepancy between minimum flip count and minimum accumulated normalization loss. Every pentagon three-to-two replacement strictly decreases weight, and in connected types A_1, A_2, and A_3 every minimum-weight path is a flip geodesic. In type A_4, however, a six-flip path has smaller weight than the unique five-flip geodesic. A compatible positive projective gluing construction amplifies this example to an exponential success-probability separation in every connected type A_n, n≥4. We also give a quantitative perturbation bound preserving the separation. The low-rank minimality statement has an exhaustive finite certificate; the pentagon inequality and the all-rank construction are analytic. In the associated sequential postselection model, differences of path weights determine success-probability ratios independently of the input state. The comparison holds for separately normalized exact mutation steps.
ER -