Exact spectral gap of random Pauli rotations
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Problem
Is the exact Haar-mixing spectral gap of uniformly random Pauli rotations equal to \(2^n(2^n-3)/(8(4^n-1))\) for every \(n\geq4\)?
Let \(\mathcal P_n=\{I,X,Y,Z\}^{\otimes n}\setminus\{I^{\otimes n}\}\). One step chooses \(P\in\mathcal P_n\) and \(\theta\in[0,2\pi)\) uniformly and applies \(e^{i\theta P}\). On \(L^2(\mathrm{SU}(2^n),\mathrm{Haar})\), define
where \(L^2_0\) is the zero-Haar-mean subspace. The conjectured value of the gap in Eq. (1) is
Does Eq. (2) hold for every \(n\geq4\)?
Source
The question is explicitly posed or retained as open in the cited primary literature [Baer26]. The statement is rewritten here to make its hypotheses and success criterion self-contained.
Progress
Baer and Haah prove the bounds
\begin{equation} \frac{4^n+16}{16(4^n-1)} \leq\Delta_n \leq\frac{2^n(2^n-3)}{8(4^n-1)}. \tag{3} \end{equation}The upper bound comes from the fourth-moment representation \(U^{\otimes4}\otimes\overline U^{\otimes4}\); see Theorem 3.16 and Proposition 3.17. [Baer26]
The displayed definitions, constraints, and target bounds are recorded in Eqs. (3).
Proposition 3.17 settles \(n\leq3\). In particular, \(\Delta_3=5/63\), whereas the first unresolved instance has
\begin{equation} \frac1{15}\leq\Delta_4\leq\frac{26}{255}. \tag{4} \end{equation}Conjecture 3.48 asserts that the upper bound is always attained. [Baer26]
The displayed definitions, constraints, and target bounds are recorded in Eqs. (4).
The 23 July 2026 preprint proves a constant gap, but explicitly leaves this exact formula conjectural. No subsequent resolution of Conjecture 3.48 was located. [Baer26]
Comment
The unresolved issue is whether any higher representation mixes more slowly than the identified fourth-moment mode. This would determine the exact spectral bottleneck across all moment orders, not merely establish convergence to Haar randomness.