Anti-concentration is (almost) all you need

Ingo Roth; Jonas Haferkamp; Jonas Helsen; Markus Heinrich

A local random quantum circuit that anti-concentrates is automatically a relative-error state 2-design, with error only about 4ε, making the two properties equivalent for these ensembles and placing log-depth brickwork circuits on par with

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2026-08-04 07:53 UTC pith:4VAFU2AR

load-bearing objection A clean, genuinely new equivalence: for LU-invariant local RQCs, anti-concentration implies relative-error state 2-designs; the proof is short and correct, with restrictions the authors explicitly own.

arxiv 2510.23719 v3 pith:4VAFU2AR submitted 2025-10-27 quant-ph

Anti-concentration is (almost) all you need

Markus Heinrich , Jonas Haferkamp , Ingo Roth , Jonas Helsen This is my paper

classification quant-ph MSC 81P4581P68 PACS 03.67.-a

keywords random quantum circuitsanti-concentrationstate 2-designsrelative-error designscollision probabilitylocal unitary invarianceSchur-Weyl dualitysecond moment

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.

The paper claims that for local random quantum circuits—circuits of Haar-random two-qudit gates arranged in any architecture and made invariant under single-qudit unitaries—anti-concentration is equivalent to having a relative-error approximate state 2-design. The new direction shows that a bound on the collision probability of the form Z_ν ≤ (1+ε) Z_H forces all two-copy expectation values to be within a relative error ε′ ≈ 4ε of their Haar values. The converse was already known, so the two properties coincide for this class. The result matters because the collision probability is far easier to bound than the full second-moment twirling channel; known log-depth anti-concentration results now immediately certify that standard brickwork and all-to-all circuits behave Haar-randomly at the level of two copies.

Core claim

The central claim is Theorem 1: if a local random quantum circuit ν on n qudits is invariant under single-qudit unitaries and its collision probability satisfies Z_ν ≤ (1+ε) Z_H, then the state ensemble {U|0⟩} generated by ν is a relative-error ε′-approximate state 2-design with ε′ = 2(q^n+1)/(q^n−q) · ε/(1−q^{−1}) ≈ 4ε. The inverse implication—that relative-error state 2-designs anti-concentrate—is already known, so the paper establishes that, for this class of circuits, anti-concentration and the state 2-design property are equivalent. The proof expands the two-copy reduced moment m_ν in the local permutation basis, uses Schur–Weyl duality and Hölder's inequality, and shows that a bound on

What carries the argument

The local permutation basis {F_x} on two copies of n qudits, where F_x is the tensor product of flip operators: LU invariance of ν forces the two-copy moment m_ν to be diagonal in this basis, with coefficients m_x = tr(F̂_x m_ν) ≥ 0. Schur–Weyl duality pins the Haar moment to the symmetric subspace; a Hölder/triangle-inequality step then bounds the mass of all 'non-trivial' coefficients (x ∉ {0,1}) by (ε′/2) Z_H. That aggregate bound is exactly what converts the scalar collision-probability bound into an operator-norm guarantee on every two-copy expectation.

Load-bearing premise

The proof requires the coefficients m_x of the two-copy moment in the local permutation basis to be non-negative; the authors import this from prior results for Haar-random two-local gates, and without it, bounding the collision probability does not control the full design error.

What would settle it

Construct a local random circuit with Haar-random two-qudit gates, invariant under single-qudit unitaries, whose collision probability satisfies Z_ν ≤ (1+ε) Z_H at some depth but whose two-copy relative error exceeds ε′ ≈ 4ε—the theorem predicts no such circuit exists. A numerical search over small systems (e.g., n = 8–16 qubits, brickwork architecture, varying depth) checking the operator bound (1−ε′)M_H ≤ M_ν ≤ (1+ε′)M_H would settle it. Alternatively, directly computing the coefficients m_x would reveal whether any are negative, which would show the theorem's premise fails for that ensemble

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

If this is right

Brickwork and all-to-all local random circuits, already known to anti-concentrate in logarithmic depth, now directly inherit the status of relative-error state 2-designs at that depth—no architectural modifications are needed.
For LU-invariant local random circuits, proving the collision-probability bound Z_ν ≤ (1+ε) Z_H is sufficient to certify the full second-moment convergence of the state ensemble, with a design error of at most ≈4ε.
Past anti-concentration results for short-depth circuits can be promoted to design statements without re-analyzing the spectral gap of the twirling channel.
Because designs imply anti-concentration as well, the two properties are the same for these ensembles, unifying two research directions that were previously thought distinct.
The equivalence does not extend to restricted gate sets: orthogonal-gate circuits anti-concentrate in log depth but need linear depth for relative-error state 2-designs, marking the boundary of the shortcut.

Where Pith is reading between the lines

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

Practical certification shortcut: because the collision probability is far easier to measure or bound than the full two-copy channel, anti-concentration checks can serve as design-certification checks up to the factor-4 error.
The factor ≈4 in the error is probably not tight; the paper itself notes the upper bound is easy and only the lower bound is delicate, so the true design error may be closer to ε than to 4ε.
The same logical structure would promote anti-concentration to a relative-error approximate k-design for any k, provided one had a non-negative basis for k-copy moments; the paper's approach is limited to k=2 exactly because such a representation is available only there.
The break-down for orthogonal-gate circuits suggests the equivalence is tied to the Haar randomness of the local gates supplying non-negative moment coefficients, not to the architecture or depth alone.

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

A clean, genuinely new equivalence: for LU-invariant local RQCs, anti-concentration implies relative-error state 2-designs; the proof is short and correct, with restrictions the authors explicitly own.

read the letter

The paper proves something real: for LU-invariant local random quantum circuits, anti-concentration (collision probability within 1+ε of the Haar value) implies a relative-error state 2-design with error roughly 4ε. Previous log-depth relative-error designs needed coarse-grained circuits [17,18], and [26] only gave the converse, so this closes a genuine gap. I checked the core algebra: Eq. (5) is correct, the constant ε′ follows from the stated identity, and the triangle inequality step is valid. The proof is short and transparent.

The main load-bearing assumption, non-negativity of the m_x coefficients in the local permutation basis, is not hidden: the authors state it explicitly and cite [19]. For Haar 2-local gates it is a known and reasonable input. They also clearly flag that the result needs LU invariance, that it does not extend to unitary 2-designs, that orthogonal-gate circuits are a genuine counterexample, and that the method is specific to second moments. The title is a bit stronger than the theorem; anti-concentration is all you need only within these ensembles. But they do not overclaim in the body.

Minor issues are cosmetic: the operator \hat F_0 + \hat F_1 appears before it is defined in the derivation of Eq. (5), and the step labelled Hölder is really the triangle inequality. Neither affects correctness. The implicit n≥2 boundary case (α_1=1 at n=1) is trivial and not stated. The acknowledgements note that a missing argument in v1 was restored in v2; the version I read appears complete.

Who gets value: anyone studying random quantum circuit convergence, anticoncentration, or design theory. The result is a useful tool because it certifies full second-moment convergence from the much easier collision probability, and it upgrades existing anticoncentration results for brickwork circuits to state 2-designs. The citation practice looks fair: prior related work is cited and clearly distinguished.

Recommendation: send this to peer review. It is a modest but solid result with a correct proof and honest limitations. I would accept after minor revision, mainly to clean up the notational hiccup and state the m_x≥0 assumption as a named condition rather than only in a remark.

Referee Report

0 major / 6 minor

Summary. The paper proves that for a local random quantum circuit (RQC) on n qudits that is invariant under local single-qudit unitaries, anti-concentration in the collision-probability sense (Z_ν ≤ (1+ε) Z_H) implies that the generated state ensemble is a relative-error ε′-approximate state 2-design, with ε′ = 2(q^n+1)/(q^n−q) · ε/(1−q^{−1}) ≈ 4ε. The proof expands the second moment m_ν in the local permutation basis, uses the non-negativity of the coefficients m_x (which holds for Haar 2-local circuits), and converts the collision-probability bound into a control on all off-{0,1} components, thereby bounding every two-copy moment. Together with the known converse (relative-error designs anti-concentrate), the result establishes that anti-concentration and relative-error state 2-designity are equivalent for this class, and implies that standard brickwork and all-to-all local RQCs form state 2-designs in logarithmic depth.

Significance. If correct, the result is a significant conceptual simplification: to certify second-moment convergence of a local RQC it suffices to analyze the much simpler collision probability. The proof is short and self-contained given Schur-Weyl duality and the cited non-negativity theorem [19]; the error relation is explicit and parameter-free. The paper is honest about the load-bearing assumptions: it identifies m_x ≥ 0 as the key condition, states that the proof does not extend to unitary designs, and correctly places the orthogonal-gate counterexample outside the theorem's scope. These strengths make the contribution a valuable addition to the random-circuit literature.

minor comments (6)

[Theorem 1 (State designs)] Theorem 1 as stated does not explicitly include the LU-invariance assumption and the condition n ≥ 2. The proof requires both: the expansion (4) holds only for LU-invariant ν, and α_1 < 1 fails at n=1 (where α_1=1). Please state these hypotheses in the theorem (or handle n=1 separately).
[Abstract] The abstract's claim that the result holds for 'any random circuit which is invariant under local (single-qubit) unitaries, independent of the architecture' is broader than what is proven: the proof also requires the non-negativity of the coefficients m_x in the permutation-basis expansion (Eq. (4)). This is stated in the remark after Theorem 1, but the abstract should be qualified accordingly.
[Preliminaries] The quantity Z_ν in Eq. (1) is called the 'collision probability', but it is the average of the fourth power of a fixed output amplitude, not the usual sum-over-outcomes collision probability (which is q^n Z_ν). This normalization difference should be stated explicitly to avoid confusion with prior literature.
[Unitary designs] The sentence after Eq. (7) contains an incomplete phrase: 'the relative error is given by.' followed directly by 'In fact, ε = ...'. Please fix. Also, the reduction to diagonal elements is attributed to Belkin et al. [27]; it would be helpful to state clearly at the start of this discussion that Eq. (7) is conditional on the psd-ness of M_ν and the result of [27].
[State designs (proof of Theorem 1)] The bound |tr(A F_x) − tr(A)α_|x|| ≤ 2 tr(A) is stated without justification; it follows from |tr(A F_x)| ≤ tr(A) (since A ≥ 0 and F_x is unitary) and α_|x| ≤ 1. Adding a one-line justification would improve readability.
[Corollary (logarithmic-depth designs)] The claim that brickwork and all-to-all circuits form relative-error state 2-designs in logarithmic depth relies on the anti-concentration results of [19,20]. It would be useful to state precisely which result gives Z_ν ≤ (1+ε)Z_H at log depth with a constant overhead, since the cited papers might only give a constant-factor collision probability.

Circularity Check

0 steps flagged

No significant circularity: the state-2-design bound is derived from anticoncentration via an explicitly stated external nonnegativity result, not from the target design property itself.

full rationale

The derivation chain is not circular. The key reduction is in Eqs. (4)-(6) of the proof of Theorem 1: assuming anti-concentration Zν ≤ (1+ε)ZH and non-negative expansion coefficients m_x, the paper bounds Σ_{x∉{0,1}} m_x ≤ ε/(1−α1) ZH, then converts this into |tr(Amν)−tr(AmH)| ≤ ε′ tr(AmH). The object being bounded (the full two-copy state moment) is not used in the definition of anti-concentration, and ε′ is an explicit closed-form function of ε, not a fitted parameter. The load-bearing nonnegativity m_x ≥ 0 is explicitly cited to Ref. [19] (Dalzell, Hunter-Jones, Brandão), which has no author overlap with the present paper; it is also explicitly flagged as the condition under which the proof works, and the discussion of orthogonal-gate circuits concedes the limitation of that assumption rather than hiding it. The converse direction (designs anti-concentrate) is imported from Ref. [26], again external and independent. Self-citations in Refs. [17] and [31] are contextual or limitation-setting, not load-bearing: Ref. [17] motivates the question but is not used in the proof, and Ref. [31] narrows the theorem's scope. The acknowledgments mention a missing argument in an earlier version that has been restored; this is a versioning/correctness matter, not evidence of circularity. Overall, the proof reduces the design claim to exactly its stated inputs, with no circular step identified.

Axiom & Free-Parameter Ledger

0 free parameters · 5 axioms · 0 invented entities

No free parameters: ε is an input variable of the theorem (the anticoncentration error) and ε′ = 2(q^n+1)/(q^n−q)·ε/(1−q^{−1}) is a derived exact function; the constants α_1 and 1−α_1 are computed from q and n. No entities are invented. The only nontrivial imported structure is the non-negativity of m_x from [19], which is external, cited, and explicitly flagged by the authors as the condition on which the proof rests.

axioms (5)

standard math Schur-Weyl duality: the commutant of local (single-qudit) unitaries on the two-copy space is spanned by local permutations {1, F}, justifying the expansion m_ν = Σ_x m_x F_x.
Invoked in Eq. (4) of the main text and in the supplement; standard representation theory, no ad hoc content.
domain assumption Non-negativity of the expansion coefficients m_x ≥ 0 for local random quantum circuits.
Load-bearing: the triangle-inequality step that turns the collision-probability bound into a design bound requires m_x ≥ 0. Cited to Ref. [19] (Dalzell, Hunter-Jones, Brandão), explicitly acknowledged in the remark after Theorem 1. Fails for orthogonal-gate circuits (discussion citing [30,31]), which is the exact separation that shows this assumption is doing real work.
domain assumption LU invariance of the measure ν (invariance under local single-qudit unitaries), imposable by a layer of single-qudit gates at the start of the circuit.
Stated in the preliminaries; used to place m_ν in the local permutation basis. The authors argue it is satisfied by many ensembles and can be imposed by construction, but it is an explicit restriction on the theorem's scope.
standard math Haar invariance M_H = M_H M_ν, i.e., the Haar measure is a fixed point of any unitary channel.
Used in Eq. (5) to express Z_H in terms of the coefficients m_x of m_ν; this is the step that lets the collision probability of the actual circuit determine the Haar reference value.
standard math Reduction to psd operators A and to the globally symmetric subspace when evaluating relative error of second moments.
Definition (2)-(3) quantifies over psd A; the proof argues tr(A m_ν) and tr(A m_H) depend only on the symmetric part of A because m_ν and m_H are swap-invariant. Standard, and the argument is present in the last paragraph of the proof.

pith-pipeline@v1.3.0-alltime-deepseek · 8225 in / 37208 out tokens · 346649 ms · 2026-08-04T07:53:33.752777+00:00 · methodology

0 comments

read the original abstract

Until very recently, it was generally believed that the (approximate) 2-design property is strictly stronger than anticoncentration of random quantum circuits, mainly because it was shown that the latter anticoncentrate in logarithmic depth, while the former generally need linear depth circuits. This belief was disproven by recent results which show that so-called relative-error approximate unitary designs can, in fact, be generated in logarithmic depth, implying anticoncentration. Their result does however not apply to ordinary local random circuits, a gap which we close in this letter, at least for 2-designs. More precisely, we show that anticoncentration of local random quantum circuits already implies that they form relative-error approximate state 2-designs, making them equivalent properties for these ensembles. Our result holds more generally for any random circuit which is invariant under local (single-qubit) unitaries, independent of the architecture.

discussion (0)

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Reference graph

Works this paper leans on

32 extracted references · 11 linked inside Pith · cited by 3 Pith papers

[1]

Emerson, R

J. Emerson, R. Alicki, and K. Zyczkowski, Scalable noise estimation with random unitary operators, Journal of Optics B: Quantum and Semiclassical Optics7, S347 (2005), publisher: IOP Publishing

2005
[2]

Knill, D

E. Knill, D. Leibfried, R. Reichle, J. Britton, R. B. Blakestad, J. D. Jost, C. Langer, R. Ozeri, S. Seidelin, and D. J. Wineland, Randomized benchmarking of quan- tum gates, Physical Review A77, 012307 (2008)

2008
[3]

Magesan, J

E. Magesan, J. M. Gambetta, and J. Emerson, Char- acterizing quantum gates via randomized benchmarking, Physical Review A85, 042311 (2012)

2012
[4]

Helsen, I

J. Helsen, I. Roth, E. Onorati, A. Werner, and J. Eis- ert, General Framework for Randomized Benchmarking, PRX Quantum3, 020357 (2022)

2022
[5]

Heinrich, M

M. Heinrich, M. Kliesch, and I. Roth, Randomized benchmarking with random quantum circuits (2023), 2212.06181

Pith/arXiv arXiv 2023
[6]

A. R. Brown and L. Susskind, Second law of quantum complexity, Physical Review D97, 086015 (2018), pub- lisher: American Physical Society

2018
[7]

Akers, A

C. Akers, A. Bouland, L. Chen, T. Kohler, T. Met- ger, and U. Vazirani, Holographic pseudoentanglement and the complexity of the AdS/CFT dictionary (2024), 2411.04978

Pith/arXiv arXiv 2024
[8]

A.W.HarrowandR.A.Low,RandomQuantumCircuits are Approximate 2-designs, Communications in Mathe- matical Physics291, 257 (2009)

2009
[9]

Dankert, R

C. Dankert, R. Cleve, J. Emerson, and E. Livine, Ex- act and approximate unitary 2-designs and their applica- tion to fidelity estimation, Physical Review A80, 012304 (2009)

2009
[10]

W. G. Brown and L. Viola, Convergence rates for ar- bitrary statistical moments of random quantum cir- cuits,PhysicalReviewLetters104,250501(2010),arXiv: 0910.0913

Pith/arXiv arXiv 2010
[11]

Brown and O

W. Brown and O. Fawzi, Scrambling speed of ran- dom quantum circuits, arXiv preprint arXiv:1210.6644 (2012)

Pith/arXiv arXiv 2012
[12]

F. G. Brandao, A. W. Harrow, and M. Horodecki, Local random quantum circuits are approximate polynomial- designs, Communications in Mathematical Physics346, 397 (2016)

2016
[13]

Harrow and S

A. Harrow and S. Mehraban, Approximate unitary $t$-designs by short random quantum circuits us- ing nearest-neighbor and long-range gates, Commu- nications in Mathematical Physics401, 1531 (2023), arXiv:1809.06957

Pith/arXiv arXiv 2023
[14]

Haferkamp and N

J. Haferkamp and N. Hunter-Jones, Improved spectral gaps for random quantum circuits: Large local dimen- sions and all-to-all interactions, Physical Review A104, 022417 (2021)

2021
[15]

Haferkamp, Random quantum circuits are approxi- mate unitaryt-designs in depthO(nt 5+o(1)), Quantum 6, 795 (2022)

J. Haferkamp, Random quantum circuits are approxi- mate unitaryt-designs in depthO(nt 5+o(1)), Quantum 6, 795 (2022)

2022
[16]

C.-F. Chen, J. Haah, J. Haferkamp, Y. Liu, T. Metger, and X. Tan, Incompressibility and spectral gaps of ran- dom circuits, arXiv preprint arXiv:2406.07478 (2024)

Pith/arXiv arXiv 2024
[17]

Schuster, J

T. Schuster, J. Haferkamp, and H.-Y. Huang, Random unitaries in extremely low depth, Science389, 92 (2025)

2025
[18]

LaRacuente and F

N. LaRacuente and F. Leditzky, Approximate Unitary k- Designs from Shallow, Low-Communication Circuits, in 16th Innovations in Theoretical Computer Science Con- ference (ITCS 2025), Leibniz International Proceedings 5 in Informatics (LIPIcs), Vol. 325, edited by R. Meka (Schloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl, Germany, 2025) pp. 69:1–...

2025
[19]

A. M. Dalzell, N. Hunter-Jones, and F. G. S. L. Brandão, Random quantum circuits anti-concentrate in log depth, PRX Quantum3, 010333, 2011.12277

Pith/arXiv arXiv 2011
[20]

Barak, C.-N

B. Barak, C.-N. Chou, and X. Gao, Spoofing Linear Cross-Entropy Benchmarking in Shallow Quantum Cir- cuits, in12th Innovations in Theoretical Computer Sci- ence Conference (ITCS 2021), Leibniz International Pro- ceedings in Informatics (LIPIcs), Vol. 185, edited by J. R. Lee (Schloss Dagstuhl–Leibniz-Zentrum für Informatik, Dagstuhl, Germany, 2021) pp. 30...

2021
[21]

Hangleiter and J

D. Hangleiter and J. Eisert, Computational advantage of quantum random sampling, Reviews of Modern Physics 95, 035001 (2023)

2023
[22]

Nahum, S

A. Nahum, S. Vijay, and J. Haah, Operator Spreading in Random Unitary Circuits, Physical Review X8, 021014 (2018)

2018
[23]

Hunter-Jones, Unitary designs from statistical me- chanics in random quantum circuits, arXiv:1905.12053 [cond-mat, physics:hep-th, physics:quant-ph] (2019), 1905.12053

N. Hunter-Jones, Unitary designs from statistical me- chanics in random quantum circuits, arXiv:1905.12053 [cond-mat, physics:hep-th, physics:quant-ph] (2019), 1905.12053

Pith/arXiv arXiv 1905
[24]

Zhou and A

T. Zhou and A. Nahum, Emergent statistical mechan- ics of entanglement in random unitary circuits, Physical Review B99, 174205 (2019)

2019
[25]

Belkin, J

D. Belkin, J. Allen, and B. K. Clark, Absence of cen- soring inequalities in random quantum circuits (2025), arXiv:2502.15995

Pith/arXiv arXiv 2025
[26]

Hangleiter, J

D. Hangleiter, J. Bermejo-Vega, M. Schwarz, and J. Eis- ert, Anticoncentration theorems for schemes showing a quantum speedup, Quantum2, 65 (2018)

2018
[27]

Belkin, J

D. Belkin, J. Allen, and B. K. Clark, Apparent universal behavior in 2nd moments of random quantum circuits (2025), to appear on arXiv on 29 October 2025

2025
[28]

Laracuente, Quantum relative entropy decay com- position yields shallow, unstructured k-designs, arXiv preprint arXiv:2510.08537 (2025)

N. Laracuente, Quantum relative entropy decay com- position yields shallow, unstructured k-designs, arXiv preprint arXiv:2510.08537 (2025)

arXiv 2025
[29]

Suzuki, H

R. Suzuki, H. Katsura, Y. Mitsuhashi, T. Soejima, J. Eis- ert, and N. Yoshioka, More global randomness from less random local gates, arXiv preprint arXiv:2410.24127 (2024)

Pith/arXiv arXiv 2024
[30]

Sauliere, B

A. Sauliere, B. Magni, G. Lami, X. Turkeshi, and J. D. Nardis, Universality in the Anticoncentration of Chaotic Quantum Circuits (2025), arXiv:2503.00119

Pith/arXiv arXiv 2025
[31]

Grevink, J

L. Grevink, J. Haferkamp, M. Heinrich, J. Helsen, M. Hinsche, T. Schuster, and Z. Zimborás, Will it glue? On short-depth designs beyond the unitary group (2025), arXiv:2506.23925. 6 Anti-concentration is (almost) all you need – Supplemental material – EXP ANSIONS IN THE LOCAL PERMUT A TION BASIS Using Schur’s lemma and the projectors onto the globally sym...

arXiv 2025
[32]

In particular, we have the non-negativity of the principal minor 0≤ ˜m0,0 ˜ma+c,b+d ˜ma+c,b+d ˜m0,0 = ˜m2 0,0 −˜m2 a+c,b+d , thus˜m0,0 ≥˜ma,b for alla, b

We can thus invoke Bochner’s theorem to conclude that the matrix˜Aa,b c,d := ˜ma+c,b+d is psd. In particular, we have the non-negativity of the principal minor 0≤ ˜m0,0 ˜ma+c,b+d ˜ma+c,b+d ˜m0,0 = ˜m2 0,0 −˜m2 a+c,b+d , thus˜m0,0 ≥˜ma,b for alla, b. We can then write Eq. (7) of the main text as ε= qn 2 max a ˜ma,a qn + (−1)|a| 2Da −1. As we have shown abo...

[1] [1]

Emerson, R

J. Emerson, R. Alicki, and K. Zyczkowski, Scalable noise estimation with random unitary operators, Journal of Optics B: Quantum and Semiclassical Optics7, S347 (2005), publisher: IOP Publishing

2005

[2] [2]

Knill, D

E. Knill, D. Leibfried, R. Reichle, J. Britton, R. B. Blakestad, J. D. Jost, C. Langer, R. Ozeri, S. Seidelin, and D. J. Wineland, Randomized benchmarking of quan- tum gates, Physical Review A77, 012307 (2008)

2008

[3] [3]

Magesan, J

E. Magesan, J. M. Gambetta, and J. Emerson, Char- acterizing quantum gates via randomized benchmarking, Physical Review A85, 042311 (2012)

2012

[4] [4]

Helsen, I

J. Helsen, I. Roth, E. Onorati, A. Werner, and J. Eis- ert, General Framework for Randomized Benchmarking, PRX Quantum3, 020357 (2022)

2022

[5] [5]

Heinrich, M

M. Heinrich, M. Kliesch, and I. Roth, Randomized benchmarking with random quantum circuits (2023), 2212.06181

Pith/arXiv arXiv 2023

[6] [6]

A. R. Brown and L. Susskind, Second law of quantum complexity, Physical Review D97, 086015 (2018), pub- lisher: American Physical Society

2018

[7] [7]

Akers, A

C. Akers, A. Bouland, L. Chen, T. Kohler, T. Met- ger, and U. Vazirani, Holographic pseudoentanglement and the complexity of the AdS/CFT dictionary (2024), 2411.04978

Pith/arXiv arXiv 2024

[8] [8]

A.W.HarrowandR.A.Low,RandomQuantumCircuits are Approximate 2-designs, Communications in Mathe- matical Physics291, 257 (2009)

2009

[9] [9]

Dankert, R

C. Dankert, R. Cleve, J. Emerson, and E. Livine, Ex- act and approximate unitary 2-designs and their applica- tion to fidelity estimation, Physical Review A80, 012304 (2009)

2009

[10] [10]

W. G. Brown and L. Viola, Convergence rates for ar- bitrary statistical moments of random quantum cir- cuits,PhysicalReviewLetters104,250501(2010),arXiv: 0910.0913

Pith/arXiv arXiv 2010

[11] [11]

Brown and O

W. Brown and O. Fawzi, Scrambling speed of ran- dom quantum circuits, arXiv preprint arXiv:1210.6644 (2012)

Pith/arXiv arXiv 2012

[12] [12]

F. G. Brandao, A. W. Harrow, and M. Horodecki, Local random quantum circuits are approximate polynomial- designs, Communications in Mathematical Physics346, 397 (2016)

2016

[13] [13]

Harrow and S

A. Harrow and S. Mehraban, Approximate unitary $t$-designs by short random quantum circuits us- ing nearest-neighbor and long-range gates, Commu- nications in Mathematical Physics401, 1531 (2023), arXiv:1809.06957

Pith/arXiv arXiv 2023

[14] [14]

Haferkamp and N

J. Haferkamp and N. Hunter-Jones, Improved spectral gaps for random quantum circuits: Large local dimen- sions and all-to-all interactions, Physical Review A104, 022417 (2021)

2021

[15] [15]

Haferkamp, Random quantum circuits are approxi- mate unitaryt-designs in depthO(nt 5+o(1)), Quantum 6, 795 (2022)

J. Haferkamp, Random quantum circuits are approxi- mate unitaryt-designs in depthO(nt 5+o(1)), Quantum 6, 795 (2022)

2022

[16] [16]

C.-F. Chen, J. Haah, J. Haferkamp, Y. Liu, T. Metger, and X. Tan, Incompressibility and spectral gaps of ran- dom circuits, arXiv preprint arXiv:2406.07478 (2024)

Pith/arXiv arXiv 2024

[17] [17]

Schuster, J

T. Schuster, J. Haferkamp, and H.-Y. Huang, Random unitaries in extremely low depth, Science389, 92 (2025)

2025

[18] [18]

LaRacuente and F

N. LaRacuente and F. Leditzky, Approximate Unitary k- Designs from Shallow, Low-Communication Circuits, in 16th Innovations in Theoretical Computer Science Con- ference (ITCS 2025), Leibniz International Proceedings 5 in Informatics (LIPIcs), Vol. 325, edited by R. Meka (Schloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl, Germany, 2025) pp. 69:1–...

2025

[19] [19]

A. M. Dalzell, N. Hunter-Jones, and F. G. S. L. Brandão, Random quantum circuits anti-concentrate in log depth, PRX Quantum3, 010333, 2011.12277

Pith/arXiv arXiv 2011

[20] [20]

Barak, C.-N

B. Barak, C.-N. Chou, and X. Gao, Spoofing Linear Cross-Entropy Benchmarking in Shallow Quantum Cir- cuits, in12th Innovations in Theoretical Computer Sci- ence Conference (ITCS 2021), Leibniz International Pro- ceedings in Informatics (LIPIcs), Vol. 185, edited by J. R. Lee (Schloss Dagstuhl–Leibniz-Zentrum für Informatik, Dagstuhl, Germany, 2021) pp. 30...

2021

[21] [21]

Hangleiter and J

D. Hangleiter and J. Eisert, Computational advantage of quantum random sampling, Reviews of Modern Physics 95, 035001 (2023)

2023

[22] [22]

Nahum, S

A. Nahum, S. Vijay, and J. Haah, Operator Spreading in Random Unitary Circuits, Physical Review X8, 021014 (2018)

2018

[23] [23]

Hunter-Jones, Unitary designs from statistical me- chanics in random quantum circuits, arXiv:1905.12053 [cond-mat, physics:hep-th, physics:quant-ph] (2019), 1905.12053

N. Hunter-Jones, Unitary designs from statistical me- chanics in random quantum circuits, arXiv:1905.12053 [cond-mat, physics:hep-th, physics:quant-ph] (2019), 1905.12053

Pith/arXiv arXiv 1905

[24] [24]

Zhou and A

T. Zhou and A. Nahum, Emergent statistical mechan- ics of entanglement in random unitary circuits, Physical Review B99, 174205 (2019)

2019

[25] [25]

Belkin, J

D. Belkin, J. Allen, and B. K. Clark, Absence of cen- soring inequalities in random quantum circuits (2025), arXiv:2502.15995

Pith/arXiv arXiv 2025

[26] [26]

Hangleiter, J

D. Hangleiter, J. Bermejo-Vega, M. Schwarz, and J. Eis- ert, Anticoncentration theorems for schemes showing a quantum speedup, Quantum2, 65 (2018)

2018

[27] [27]

Belkin, J

D. Belkin, J. Allen, and B. K. Clark, Apparent universal behavior in 2nd moments of random quantum circuits (2025), to appear on arXiv on 29 October 2025

2025

[28] [28]

Laracuente, Quantum relative entropy decay com- position yields shallow, unstructured k-designs, arXiv preprint arXiv:2510.08537 (2025)

N. Laracuente, Quantum relative entropy decay com- position yields shallow, unstructured k-designs, arXiv preprint arXiv:2510.08537 (2025)

arXiv 2025

[29] [29]

Suzuki, H

R. Suzuki, H. Katsura, Y. Mitsuhashi, T. Soejima, J. Eis- ert, and N. Yoshioka, More global randomness from less random local gates, arXiv preprint arXiv:2410.24127 (2024)

Pith/arXiv arXiv 2024

[30] [30]

Sauliere, B

A. Sauliere, B. Magni, G. Lami, X. Turkeshi, and J. D. Nardis, Universality in the Anticoncentration of Chaotic Quantum Circuits (2025), arXiv:2503.00119

Pith/arXiv arXiv 2025

[31] [31]

Grevink, J

L. Grevink, J. Haferkamp, M. Heinrich, J. Helsen, M. Hinsche, T. Schuster, and Z. Zimborás, Will it glue? On short-depth designs beyond the unitary group (2025), arXiv:2506.23925. 6 Anti-concentration is (almost) all you need – Supplemental material – EXP ANSIONS IN THE LOCAL PERMUT A TION BASIS Using Schur’s lemma and the projectors onto the globally sym...

arXiv 2025

[32] [32]

In particular, we have the non-negativity of the principal minor 0≤ ˜m0,0 ˜ma+c,b+d ˜ma+c,b+d ˜m0,0 = ˜m2 0,0 −˜m2 a+c,b+d , thus˜m0,0 ≥˜ma,b for alla, b

We can thus invoke Bochner’s theorem to conclude that the matrix˜Aa,b c,d := ˜ma+c,b+d is psd. In particular, we have the non-negativity of the principal minor 0≤ ˜m0,0 ˜ma+c,b+d ˜ma+c,b+d ˜m0,0 = ˜m2 0,0 −˜m2 a+c,b+d , thus˜m0,0 ≥˜ma,b for alla, b. We can then write Eq. (7) of the main text as ε= qn 2 max a ˜ma,a qn + (−1)|a| 2Da −1. As we have shown abo...