Apparent Universal Behavior in Second Moments of Random Quantum Circuits

Bryan K. Clark; Daniel Belkin; James Allen

REVIEW 4 major objections 6 minor 1 cited by

Most random quantum circuits form approximate 2-designs in logarithmic depth, while the bridge graph provably requires quadratically more gates; the paper gives an exact recipe for the optimal experiment that distinguishes any such circuit

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2026-08-04 07:50 UTC pith:BLRDCL4G

load-bearing objection Worth taking seriously, but the two headline quantitative claims—the brickwork constant and the fastest-architecture result—rest on assumptions the paper itself leaves unresolved; needs a focused revision, not a desk reject. the 4 major comments →

arxiv 2510.23726 v2 pith:BLRDCL4G submitted 2025-10-27 quant-ph math-phmath.MP

Apparent Universal Behavior in Second Moments of Random Quantum Circuits

Daniel Belkin , James Allen , Bryan K. Clark This is my paper

classification quant-ph math-phmath.MP

keywords random quantum circuitsapproximate unitary designs2-designHaar measuremultiplicative erroranticoncentrationbrickwork architecturegraph-sampled circuits

verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Random quantum circuits are used to generate approximate unitary designs—ensembles that mimic fully random (Haar) unitaries—and this paper asks how deep such circuits must be before they become good 2-designs. The authors prove that the optimal experiment distinguishing any locally invariant circuit ensemble from Haar is always a tensor product of |00> and singlet states, turning an infinite optimization into a finite search over 2^n choices. With this reduction and exact tensor-network contractions, they compute the multiplicative error for circuits on up to 50 qubits. They find that most graph geometries scramble in depth proportional to log n, give a closed-form estimate for the 1D brickwork, and prove that the bridge graph needs at least ~n² gates. They also show that anticoncentration and 2-design-ness can diverge, and that roughly ten to twenty layers suffice for practical approximate designs.

Core claim

The central claim is Theorem 1: for any circuit ensemble that is invariant under single-site unitaries and whose vectorized second-moment operator has positive-semidefinite difference from Haar, the multiplicative error is exactly the maximum over product states built from |00> and the singlet state. This is a substantial simplification because the multiplicative error is defined as the best possible experiment—a maximization over arbitrary inputs and measurements—and previous approaches only bounded it. The same theorem shows that the collision probability (anticoncentration) is just one particular choice of the optimal experiment, which is why the two notions sometimes coincide. Using this

What carries the argument

The load-bearing object is the vectorized second-moment operator vec(Φ_ε) expressed in the permutation basis, where Schur-Weyl duality gives each site a two-dimensional commutant spanned by the identity and swap states. Theorem 1 shows that the multiplicative error is the maximum over a∈{0,1}^n of the ratio ⟨Ψ(a)|vec(Φ_ε−Φ_Haar)|Ψ(a)⟩ / normalization, where |Ψ(a)> is a product of the corresponding cobasis states; physically this is the experiment that prepares |00> or the singlet on each site. The proof uses the PSD-vectorization assumption to establish diagonal dominance, so the maximum occurs at a diagonal element. The computational engine is exact tensor-network contraction of the transfe

Load-bearing premise

Everything hinges on the assumption that vec(Φ_ε−Φ_Haar) is positive semidefinite for the ensembles studied: if that fails, the optimal experiment need not be a product of |00> and singlet states, and the exact depths and the Ω(n²) bridge lower bound would not follow.

What would settle it

Brute-force search over all input states and measurements for a 4-qubit even-depth permuted brickwork (where PSD-ness is explicitly unclear): if some non-product/singlet experiment beats the best product/singlet experiment, Theorem 1's assumption is violated for that ensemble. Alternatively, computing exact depths for bridge graphs at n=6,8,10 and checking whether they grow like n(n−2)/4·log(1/ε) would test the lower bound.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

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If this is right

Exact 2-design depths can now be computed for locally invariant circuits, replacing loose spectral-gap bounds with numbers that match the true multiplicative error.
Most graph-sampled circuits form ε-approximate 2-designs in O(log n) depth, and the 1D brickwork's depth is ~log((3/π²)n/ε)/log(5/4).
The bridge graph requires Ω(n²) gates, so there is no universal gate-count asymptotic for all graph architectures; poor connectivity is the culprit.
Anticoncentration does not imply unitary 2-design-ness: the star graph anticoncentrates much faster than it forms a 2-design.
Only ten to twenty layers are needed for approximate 2-designs in realistic parameter ranges, a constant-factor improvement over earlier constructions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

If Theorem 1 extends to t>2, the same product/singlet reduction could make higher-moment design depths exact, potentially closing the gap between known O(log n) upper bounds and Ω(log n) lower bounds.
The geometry-dependence of the optimal experiment suggests a testable principle: the hardest experiment for a circuit to mimic is typically the one that places entangled states near bottlenecks (edges or connections), not in the bulk. One could probe this by varying the graph's cut structure and tracking which a∈{0,1}^n wins.
The near-flat depth-vs-n curve for permuted brickwork hints that some structured ensembles might scramble in sub-logarithmic or even constant depth; verifying that would require a spectral gap argument the authors do not provide.
The bridge-graph lower bound should generalize to any architecture with a sample-probability bottleneck; replacing the single bridge edge with multiple parallel bridges of increasing width would test how the Ω(n²) scaling degrades.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Desk Editor's Note

Worth taking seriously, but the two headline quantitative claims—the brickwork constant and the fastest-architecture result—rest on assumptions the paper itself leaves unresolved; needs a focused revision, not a desk reject.

read the letter

This paper is worth taking seriously, but it needs a revision before its headline numbers become lore.

The genuinely new part is Theorem 1: for locally invariant ensembles whose vectorized second moment operator is PSD, the optimal distinguishing experiment reduces to a finite set of product/singlet boundary states. That, plus tensor-network contraction, gets exact t=2 multiplicative errors out to 50 qubits—an order of magnitude beyond what anyone else has actually computed. The bridge-graph lower bound is a real theorem conditional on one imported lemma. The star-graph anticoncentration/2-design separation is a clean and surprising observation. Credit also to the authors for marking their conjectures as conjectures and flagging where proofs are deferred.

Soft spots, in proportion. The biggest is the PSD assumption. Theorem 1 collapses if vec(Φε − ΦHaar) isn't positive semidefinite. They prove PSD for graphs, odd-depth brickwork, PCG, PBFE, and odd-depth permuted brickwork, but Section 5.2 says even-depth permuted brickwork is unclear. The claim that permuted brickwork is the fastest scrambler (Fig. 11) is evaluated with the same formula, and if the minimizing depth is even those numbers are not covered by Theorem 1. I did not see a PSD check for even depths. This is not a manufactured concern; it's an explicitly acknowledged gap in the paper. Second, Theorem 2 imports a monotonicity lemma from the authors' own ref [14] with no proof. If that lemma fails, the Ω(n²) gate bound fails. It's likely true, but the dependency should be stated and ideally proved in-house. Third, the brickwork formula Eq. 1 is called semi-empirical with a derivation 'intended for future work'; as presented it's an excellent fit, not a theorem. Fourth, the fast-architecture results are averages over sampled circuit realizations with no error bars, and no code or data is released. That makes the 10-20 layers claim hard to verify independently.

The well-supported parts—Theorem 1's reduction, the graph numerics, the star separation, and the bridge bound modulo the lemma—are solid enough to deserve referee time. The paper is honest about its limits, which I count in its favor. I'd send it to referees with a request for major revision: resolve or qualify the even-depth PB numbers, prove or explicitly flag the imported lemma, include the brickwork derivation, and release code/data. I'd also bring it to the reading group; it's a good paper to argue with. I'd cite it, with a caveat about which results are proven vs empirical.

Referee Report

4 major / 6 minor

Summary. The paper develops a method to compute exactly the t=2 multiplicative error between a random circuit ensemble and the Haar measure, by reducing the optimal distinguishing experiment to a maximum over tensor products of |00> and singlet states (Theorem 1, Eq. 8) under local invariance and PSD vectorization. Using tensor-network contraction, it reports exact 2-design depths up to n=50 for graph-sampled circuits, a semi-empirical brickwork depth formula (Eq. 1), an Ω(n²) lower bound for the bridge graph (Theorem 2), and numerical results for fast parallel architectures. It also studies when collision probability is the optimal experiment, finding cases (star graph, open-boundary brickwork) where anticoncentration and 2-design depths differ.

Significance. If the PSD hypothesis holds, Theorem 1 gives a valuable exact algorithm for multiplicative errors, replacing loose spectral-gap bounds. The numerical results support several conjectures (graph scaling, complete graph fastest, connection-count universality) and the bridge bound answers a question from ref. [1] negatively, conditional on an imported lemma. The brickwork formula provides explicit constants. However, the paper ships no code/data, the headline brickwork formula is deferred, and the fastest-architecture claim rests on an unverified PSD assumption for even depths; these weaknesses currently prevent full certification.

major comments (4)

[§5.2, Appendix A.9, Fig. 11] The claim that permuted brickwork is the fastest scrambler rests on Theorem 1, which requires vec(Φε − ΦHaar) to be positive semidefinite. The text states in §5.2 that it is 'unclear if the vectorization is PSD at even depths' for permuted brickwork, and Appendix A.8 only proves PSD for odd-depth permuted brickwork. Since the numerical errors in Fig. 11 are computed with Eq. 8 (via §2.3), and the PSD condition is used in Appendix A.9 to argue that the maximum is on the diagonal, the reported even-depth results are not covered. No numerical check of PSD for even depths is reported. Provide a proof, a PSD verification, or label the even-depth results as not rigorously justified by Theorem 1.
[Appendix A.11, Theorem 2] The lower bound for the bridge graph relies on Lemma 5, which invokes 'the results of ref. [14]' that composing with tensor-product Haar unitaries cannot increase the multiplicative error. Since ref. [14] is an unpublished/parallel work by the same group and the lemma is not stated or proved here, the Ω(n²) lower bound is conditional. If the monotonicity lemma fails, Theorem 2 collapses. Please include a self-contained proof or a precise theorem statement with conditions, or explicitly mark Theorem 2 as conditional.
[§4.2, Eq. (18)–(19), Fig. 10] The brickwork approximate-2-design depth formula is a headline numerical result (Eq. 1 in the abstract), but its derivation is deferred to 'a future work' and the paper states it is semi-empirical. The formula is used to draw conclusions about constants and layer counts. To be assessable, the derivation or at least the assumptions behind the 'rather tedious calculation' must be included, or the formula must be clearly labeled as a conjecture with supporting numerics rather than a result.
[§5.3, §C] The fast-architecture results in Fig. 11 are obtained by sampling over circuit realizations (Appendix C) but no error bars or statistical confidence intervals are given, and no code or data are provided. Since the paper's main contributions are numerical and the flat-depth claim is an extrapolation, this makes the numerical evidence difficult to verify. Provide error bars for the sampled magnitudes, and release the code/data or a detailed reproducibility statement.

minor comments (6)

[Abstract, §3.4.1] The abstract states 'at least Ω(n) gates per site' while Theorem 2 states Ω(n²) total gates; please reconcile the wording for clarity.
[§5.2] Uses first-person 'I'll call it' in a multi-author paper; change to 'we call it'.
[Acknowledgements] Typo: 'niversit' should be 'Université'.
[Fig. 10 caption] The caption says the 'entangled boundaries bound is tight everywhere tested' but the text says it is a lower bound; clarify the distinction between a proven lower bound and a numerically observed tightness.
[Eq. (11)–(12)] The arrow '∼' discarding e^{-O(n)} terms is abrupt; specify the limit in which the approximation is valid.
[§2.2] The PSD vectorization condition is used crucially but never formally defined; consider adding a definition in the text.

Circularity Check

1 steps flagged

Theorem 2 imports its key monotonicity lemma from the authors' own ref. [14]; otherwise the central Theorem 1 and numerical depths are self-contained.

specific steps

self citation load bearing [Appendix A.11 (Lemma 5 proof), used by Theorem 2 (Section 3.4.1)]
"Let us compose Φ /C with a tensor product of Haar-random unitaries acting on all the qudits on each side of the cut to obtain Φ Haarm ⊗ Φ Haar (n−m). By the results of ref. [14], this composition cannot increase the multiplicative error."

The Ω(n^2)-gate lower bound for the bridge graph (Theorem 2) depends on Lemma 5, whose proof imports the monotonicity statement that composing with Haar unitaries cannot increase multiplicative error from ref. [14]. Ref. [14] is the same three authors' prior preprint (Belkin, Allen, Clark, arXiv:2502.15995), cited without proof or independent verification here. Without that imported lemma, Lemma 5 and Theorem 2 collapse; the claimed theorem is thus supported by a self-citation rather than by a self-contained derivation. This is load-bearing but not definitional, since Theorem 1 and the numerical content are otherwise derived in-paper.

full rationale

The central reduction (Theorem 1, Eq. 8) is proved in-paper in Appendices A.2–A.10 under explicit assumptions (local invariance and PSD vectorization), and the reported 2-design depths are direct tensor-network evaluations of that formula; no fitted constant is relabeled as a prediction. The one load-bearing circularity is the bridge lower bound: Appendix A.11's Lemma 5 proof invokes ref. [14] for the essential monotonicity statement, and that reference is by the same authors and is not independently verified. I weigh this as score 4 rather than higher because the rest of the paper has independent content. I also explicitly flag two non-circular limitations required by the reviewing rule: (i) Sec. 5.2 and App. A.8 state that PSD-ness is 'unclear' for even-depth permuted brickwork, yet Fig. 11 applies Eq. 8 to that architecture; if PSD fails, those 'exact' fast-architecture errors are not covered by Theorem 1. (ii) Eq. 18 for the brickwork leading-order depth is said to follow from a 'rather tedious calculation' with details deferred to future work, so the asymptotic formula is semi-empirical rather than fully derived in this manuscript. Neither is a circularity, but both are assumption/derivation gaps in the chain.

Axiom & Free-Parameter Ledger

0 free parameters · 6 axioms · 2 invented entities

No free parameters are fitted to data; the brickwork constants come from an (undisclosed) asymptotic expansion around the known exact spectral gap. The main reliance is on standard Schur-Weyl machinery, the PSD-vectorization assumption, the authors' own ref [14] monotonicity lemma, and the invented 'connection count' metric behind the universal conjectures.

axioms (6)

standard math Schur-Weyl duality: the single-site commutant of U(q)^⊗t is spanned by permutation states
Section 2.3, Eq. 9, used throughout to express moment operators in a 2^N-dimensional basis.
domain assumption For all studied ensembles, vec(Φε − ΦHaar) is positive-semidefinite
Assumption of Theorem 1; argued for graphs, odd-depth brickwork, and fast architectures in Appendix A.8, but even-depth permuted brickwork is explicitly left unclear (Section 5.2).
domain assumption Composing with tensor-product Haar unitaries on each side of a cut cannot increase multiplicative error
Invoked in Appendix A.11 from the authors' own ref [14]; not proven here and load-bearing for Lemma 5/Theorem 2.
domain assumption Exact spectral gap for 1D brickwork from Deneris et al. [18]
Used in Section 4.2 to derive the asymptotic brickwork depth formula (Eqs. 19-21); the derivation is deferred to future work.
ad hoc to paper The greedy 'connection count' (Appendix B) is the right metric for universal scrambling bounds
Conjectures 3-4 are stated in terms of this newly defined metric; no evidence that another graph invariant would yield the same universal behavior.
domain assumption Finite-size numerical trends persist asymptotically
The Θ(log n) claims for graphs and near-flat fast-architecture depths are inferred from n ≤ 50 data (Figs. 2, 11); authors admit results cannot formally say anything about large n (Section 7).

invented entities (2)

connection count (greedy definition, Appendix B) no independent evidence
purpose: Quantify how many 'connected blocks' an architecture needs before scrambling, to explain lollipop/bridge slowness and to formulate universal bounds
It is a definition plus a greedy estimator introduced here; it is not an independently measurable quantity outside the paper.
Permuted brickwork (PB) and permuted-brickwork with fixed evens (PBFE) independent evidence
purpose: Candidate fast-scrambling architectures designed to beat graph-sampled and parallel-complete-graph ensembles
Their predicted 2-design depths are concrete falsifiable numbers that independent exact contractions could verify; the paper itself verifies them numerically.

pith-pipeline@v1.3.0-alltime-deepseek · 21318 in / 18229 out tokens · 163660 ms · 2026-08-04T07:50:19.505268+00:00 · methodology

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Cite this review

Pith. "Pith review of Apparent Universal Behavior in Second Moments of Random Quantum Circuits." pith.science (2026). https://pith.science/paper/BLRDCL4G

@misc{pith2026251023726,
  author       = {Pith},
  title        = {Pith review of: Apparent Universal Behavior in Second Moments of Random Quantum Circuits},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BLRDCL4G}},
  note         = {Machine review of arXiv:2510.23726}
}

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read the original abstract

Just how fast does the brickwork circuit form an approximate 2-design? Is there any difference between anticoncentration and being a 2-design? Does geometry matter? How deep a circuit will I need in practice? We tell you everything you always wanted to know about second moments of random quantum circuits, but were too afraid to compute. Our answers generally take the form of numerical results for up to 50 qubits. Our first contribution is a strategy to determine explicitly the optimal experiment which distinguishes any given ensemble from the Haar measure. With this formula and some computational tricks, we are able to compute $t = 2$ multiplicative errors exactly out to modest system sizes. As expected, we see that most families of circuits form $\epsilon$-approximate $2$-designs in depth proportional to $\log n$. For the 1D brickwork, we work out the leading-order constants explicitly. For graphs, we find some exceptions which are much slower, proving that they require at least $\Omega(n^2)$ gates. This answers a question asked by ref. 1 in the negative. We explain these exceptional architectures in terms of connectedness. Based on this intuition we conjecture universal upper and lower bounds for graph-sampled circuit ensembles. For many architectures, the optimal experiment which determines the multiplicative error corresponds exactly to the collision probability (i.e. anticoncentration). However, we find that the star graph anticoncentrates much faster than it forms an $\epsilon$-approximate $2$-design. Finally, we show that one needs only ten to twenty layers to construct an approximate $2$-design for realistic parameter ranges. This is a large constant-factor improvement over previous constructions. The parallel complete-graph architecture is not quite the fastest scrambler, partially resolving a question raised by ref. 2.

Figures

Figures reproduced from arXiv: 2510.23726 by Bryan K. Clark, Daniel Belkin, James Allen.

**Figure 1.** Figure 1: Multiplicative error M at t = 2 vs. gate count s for four graphs on 12 qubits. These curves share a common structure: A very rapid initial drop in ϵ, followed by a uniform exponential decay. To understand this, let λi be the (unique, ordered) eigenvalues of the vectorized single-step moment operator vec (Φε) and let Pi project into the corresponding eigenspaces. Then at circuit size s, we may write M = max… view at source ↗

**Figure 2.** Figure 2: Circuit size needed to reach an 0.01-approximate 2-design for linear, circle, complete, and lollipop graphs. We see that the linear and circle graphs both give roughly straight lines on this plot, which is to say they form approximate 2-designs in depth1 O(log n). On the other hand, the complete graph appears nearly flat in comparison. There is a lower bound of depth Ω(log n) for the complete graph via ant… view at source ↗

**Figure 3.** Figure 3: gives analogous curves for some other families of graphs. Generally we see more dense graphs tend to form approximate 2-designs faster, with both trees and Ramanujan graphs appearing to interpolate between the linear and complete cases as the degree of the nodes increases. The lollipop is our only exception to this trend. a) b) [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

**Figure 4.** Figure 4: (a) 0.01-approximate 2-design depths for each of three families of graphs. Although the hourglass and bridge look very similar, their scrambling rates are very different. (b) Illustrations of the hourglass, bridge, and lollipop graph families. In each case we assign ⌈ n 2 ⌉ nodes to the upper clique, such that the two regions are of roughly equal size. “stick”. It follows that the vast majority of random g… view at source ↗

**Figure 5.** Figure 5: Mean gates per connection per qubit for each of three architectures, as estimated by the greedy algorithm [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗

**Figure 6.** Figure 6: Mean connected blocks per gate per site, as estimated by the greedy algorithm described in Appendix [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗

**Figure 7.** Figure 7: Connection count needed to reach an 0.01-approximate 2-design for each of six graph families. We see that all require roughly comparable connection counts, although some rise slightly with n and others fall. a) b) [PITH_FULL_IMAGE:figures/full_fig_p009_7.png] view at source ↗

**Figure 8.** Figure 8: Connection count needed to reach an 0.01-approximate 2-design for (a) tree and star graphs, (b) Ramanujan graphs. In terms of connection count, the slowest-scrambling architecture tested is the linear graph, while the fastest is the star graph. Note that star graph has O(n log n) gates per connection, where as the linear graph need O(n log n) gates for the first connection but only O(n) for later connecti… view at source ↗

**Figure 9.** Figure 9: Conjectured bounds. (a) Circuit size needed to reach an 0 [PITH_FULL_IMAGE:figures/full_fig_p010_9.png] view at source ↗

**Figure 10.** Figure 10: Depth needed for the brickwork to reach an 0 [PITH_FULL_IMAGE:figures/full_fig_p011_10.png] view at source ↗

**Figure 11.** Figure 11: Approximate 2-design depths for each of the fast architectures. All are faster than the complete graph, [PITH_FULL_IMAGE:figures/full_fig_p013_11.png] view at source ↗

**Figure 12.** Figure 12: Error ratios and optimal experiments for complete-graph architectures. Upper panels show the error [PITH_FULL_IMAGE:figures/full_fig_p014_12.png] view at source ↗

**Figure 13.** Figure 13: Anticoncentration vs. 2-design-ness for the star graph. (a) Depths needed for the multiplicative and [PITH_FULL_IMAGE:figures/full_fig_p015_13.png] view at source ↗

**Figure 14.** Figure 14: (a) 0.01-approximate 2-design depth and 0.01-anticoncentration depths for 1D brickworks, with open and periodic boundary conditions. We see that anticoncentration and being a 2-design are usually equivalent with periodic boundaries, but inequivalent with open boundaries. (b) The gap between the depths needed to reach relative error ϵ for the entangled boundaries experiment and the collision probability, f… view at source ↗

**Figure 15.** Figure 15: Tensor network depiction of a permutation operator [PITH_FULL_IMAGE:figures/full_fig_p019_15.png] view at source ↗

**Figure 16.** Figure 16: On the other hand, we can make some attempt to use the three reduction rules above to reduce this number. We use a greedy algorithm, which proceeds as follows: • Add gates to the current block until it becomes connected. • For each gate in the last layer of the current block, i.e. each gate which commutes with every gate after it, check if it can be removed without disconnecting the block. If it can, remo… view at source ↗

**Figure 16.** Figure 16: Mean gates per connection per qubit for the linear graph, as counted by the naive and greedy methods. [PITH_FULL_IMAGE:figures/full_fig_p027_16.png] view at source ↗

**Figure 17.** Figure 17: Estimated mean connection count vs. circuit size for each of four graphs on 12 qubits. [PITH_FULL_IMAGE:figures/full_fig_p027_17.png] view at source ↗

discussion (0)

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

Unitary Designs from Doped Matchgate Circuits
quant-ph 2026-06 unverdicted novelty 7.0

Doped matchgate circuits achieve approximate parity-preserving 2-designs in polylogarithmic depth using a sparse number of non-Gaussian gates, with the design formation mapped exactly to a birth-death Markov chain.

Reference graph

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We may split any gate into two consecutive copies of itself. For example, given four qubits labeleda, b, c, d, the following two architectures are equivalent: ab, ad, bc ab, ad, bc, ad B.2 Naive and greedy algorithms We do not know of a guaranteed way to compute the connection count, as defined above, since there may be many possible ways to rearrange and...

Apparent Universal Behavior in Second Moments of Random Quantum Circuits

Core claim §

What carries the argument

Load-bearing premise §

What would settle it §

If this is right

Where Pith is reading between the lines

discussion (0)

Core claim

Load-bearing premise

What would settle it