A spectral gap for almost every pair of quantum gates
Let $G = \mathrm{SU}(2)$ with Haar probability measure $\mu$. For $a, b \in G$ let $P_{a,b}$ be the lazy averaging operator on $L^2(G, \mu)$, $$P_{a,b} f(g) = \frac12 f(g) + \frac18 \bigl( f(ag) + f(a^{-1}g) + f(bg) + f(b^{-1}g) \bigr).$$ Gamburd, Jakobson and Sarnak constructed tuples in $\mathrm{SU}(2)$ with a spectral gap from quaternions, and Bourgain and Gamburd proved a spectral gap for every finite set with algebraic entries that generates a dense subgroup, but pairs with algebraic entries form a null set. Pikhurko and Sakamoto proved that the set of pairs with a spectral gap has Haar measure zero or full measure, as Fisher had shown for $n$-tuples with $n \ge 3$.
Problem. Is it true that for $(\mu \otimes \mu)$-almost every pair $(a, b)$ there is $\delta(a, b) > 0$ such that $$\|P_{a,b} f\|_2 \le (1 - \delta(a, b)) \|f\|_2$$ for every $f \in L^2(G, \mu)$ with $\int_G f \, d\mu = 0$? The gap may depend on the pair; no bound uniform over all pairs is asked for.
Source: Gamburd, Jakobson and Sarnak (1999), for almost every n-tuple in SU(2), n ≥ 2; see Pikhurko and Sakamoto (2026). Stated by AIM — Open Applied Problems for pairs, with the lazy averaging operator.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- GR — Group Theory
- Secondary
- DS, SP
- Keywords
- AIM Open Applied Problems, random walks on compact groups, spectral gaps
References
@article{gamburd1999,
author = {Gamburd, Alex and Jakobson, Dmitry and Sarnak, Peter},
title = {Spectra of elements in the group ring of {SU}(2)},
journal = {J. Eur. Math. Soc. (JEMS)},
volume = {1},
number = {1},
year = {1999},
pages = {51--85},
doi = {10.1007/PL00011157},
}
@article{pikhurko2026,
author = {Pikhurko, Oleg and Sakamoto, Kohki},
title = {A zero--one law for the spectral gap property of pairs in {SU}(2)},
journal = {Bull. Lond. Math. Soc.},
volume = {58},
number = {10},
year = {2026},
pages = {e70527},
doi = {10.1112/blms.70527},
eprint = {2603.17869},
archivePrefix = {arXiv},
}
@article{fisher2006,
author = {Fisher, David},
title = {{Out}({$F_n$}) and the spectral gap conjecture},
journal = {Int. Math. Res. Not.},
year = {2006},
pages = {Art. ID 26028},
doi = {10.1155/IMRN/2006/26028},
}
@article{bourgain2008,
author = {Bourgain, Jean and Gamburd, Alex},
title = {On the spectral gap for finitely-generated subgroups of {SU}(2)},
journal = {Invent. Math.},
volume = {171},
number = {1},
year = {2008},
pages = {83--121},
doi = {10.1007/s00222-007-0072-z},
}
@article{bourgain2012,
author = {Bourgain, Jean and Gamburd, Alex},
title = {A spectral gap theorem in {SU}$(d)$},
journal = {J. Eur. Math. Soc. (JEMS)},
volume = {14},
number = {5},
year = {2012},
pages = {1455--1511},
doi = {10.4171/JEMS/337},
eprint = {1108.6264},
archivePrefix = {arXiv},
}
@misc{aim-su-two-random-gate-gap,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{A spectral gap for almost every pair of quantum gates}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/213-su-two-random-gate-gap.md},
note = {Entry 213 at commit a153f46, 2026-10-13},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00480, "A spectral gap for almost every pair of quantum gates", clopen.solutions.