bibkey: belkinallenclark2025secondmoments authors: Daniel Belkin, James Allen, Bryan K. Clark year: 2025 title: “Apparent Universal Behavior in 2nd Moments of Random Quantum Circuits” doi: null url: https://arxiv.org/abs/2510.23726v2 claim: “Like the brickwork, this architecture can be shown to have a PSD vectorization if the depth is odd. It is unclear if the vectorization is PSD at even depths.” strata_touched:

• D5/S3/Quantum/RandomCircuits/HaarTwoCopyTwirl
• D5/S3/Quantum/RandomCircuits/PermutedBrickworkEvenRefutation license: citation-only triage: anchor

Second moments of random quantum circuits

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Source: https://arxiv.org/abs/2510.23726v2 The locators below use the v2 PDF: Section 2.1 on page 3, Section 5.2 on page 12, and the PSD-circuits appendix (Appendix A.8) on pages 22–23. This arXiv-only citation has no DOI locator.

Section 2.1 defines

$${\Phi }_{\epsilon }(\rho )={\mathbb{E}}_{U\sim \epsilon }[(U^{†}\otimes U^{†})\rho (U\otimes U)],\mathrm{vec}({\Phi }_{\epsilon })={\mathbb{E}}_{U\sim \epsilon }[U^{\ast }\otimes U^{\ast }\otimes U\otimes U].$$

Here the star in the four-factor expectation is entrywise conjugation. The column-first vectorization of a matrix X has entry X(row, column) at index (column, row).

Section 2.3 (page 3) introduces the normalized permutation basis:

$$|\sigma ⟩=\frac{1}{{\sqrt{q}}^t}\sum _{\vec{i}\in \{1,\dots ,q{\}}^t}|\vec{i}⟩\otimes |\sigma (\vec{i})⟩.$$

For $t=2$ the only permutations are identity and swap, so the dimension of the local commutant is always $2$.

For two replicas this normalization is $q^{-1}$. The Haar moment is the orthogonal projection onto the identity/swap span; this is the single-gate projection used in Appendix A.8. The Gram overlaps of the one-site vectors are one for equal permutations and $q^{-1}$ for distinct permutations. The two-site local rule is the corresponding projection, with mixed-vector coefficient $q/(q^2+1)$.

Section 5.2 (page 12) defines complete layers:

Suppose we draw a random two-sided matching of the sites, then apply a layer of Haar-random 2-site gates to those pairs in parallel. In other words, we sample a random complete layer, i.e. a random set of $\frac{N}{2}$ gates such that each site is acted on by exactly one gate. This is the parallel complete-graph (PCG) architecture.

It then defines PB and poses the question:

Suppose we draw layers as in the parallel complete-graph architecture, except that we require each adjacent pair of layers to form a connected block. This guarantees, for example, that we never “waste” a gate by repeating a gate from the previous layer. Avoiding this kind of waste increases the speed of scrambling. This architecture is similar to applying a single period of 1D brickwork to a random permutation of the sites, so I’ll call it the permuted brickwork (PB). Like the brickwork, this architecture can be shown to have a PSD vectorization if the depth is odd. It is unclear if the vectorization is PSD at even depths.

For four sites the complete matchings are {(0,1),(2,3)}, {(0,2),(1,3)} and {(0,3),(1,2)}. The union of a repeated matching has two components; the union of any distinct two is a four-cycle. Thus the connected-block condition is exactly that neighbouring matchings differ. The layer words are uniform among those satisfying this condition; gates on each pair are independent and have normalized Haar law on U(q²).

Appendix A.8 (page 23) conditions an odd-depth PB circuit on its middle matching. Its two remaining halves are conditionally independent and related by inversion, so the moment is an average of A† P A with P the middle-layer projection. An even number of layers has a middle interface between two distinct matching choices; this proof supplies no such single middle projection. Section 2.2 (page 3) states the PSD hypothesis of the optimal-experiment theorem and the remedy:

Furthermore, given a locally invariant ensemble $\mathcal{E}$, one may define an ensemble ${\mathcal{E}}^{\prime }$ with a PSD vectorization by sampling $U{V}^{†}$, with $U,V$ drawn i.i.d. from $\mathcal{E}$.