On the Complexity of Quantum States and Circuits from the Orthogonal and Symplectic Groups

arxiv: 2509.07573 · v2 · submitted 2025-09-09 · 🪐 quant-ph

On the Complexity of Quantum States and Circuits from the Orthogonal and Symplectic Groups

Oxana Shaya , Zo\"e Holmes , Christoph Hirche , Armando Angrisani This is my paper

Pith reviewed 2026-05-18 17:52 UTC · model grok-4.3

classification 🪐 quant-ph

keywords quantum complexitysymplectic grouporthogonal groupconcentration of measurestate complexitycircuit learningquantum information

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The pith

Quantum states from symplectic and orthogonal unitaries typically exhibit exponentially large strong state complexity.

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

This paper studies the complexity of quantum states and circuits by sampling unitaries from the symplectic and special orthogonal groups instead of the standard Haar measure on the unitary group. Using the concentration of measure phenomenon, it shows that these structured groups generate states with exponentially large strong state complexity that are nearly orthogonal to each other. The results extend to designs over these groups and include the average-case hardness of learning such circuits. Sympathetic readers would find this relevant because it indicates that high complexity arises even from more constrained groups, affecting models in many-body physics and quantum learning theory.

Core claim

We leverage concentration of measure on the symplectic and special orthogonal groups to establish that random quantum states generated using these unitaries typically exhibit an exponentially large strong state complexity and are nearly orthogonal to one another. Similar behavior holds for designs over these groups. We further demonstrate the average-case hardness of learning circuits composed of gates drawn from such groups. These findings show that structured subgroups can exhibit a complexity comparable to that of the full unitary group.

What carries the argument

The concentration of measure phenomenon on the symplectic and special orthogonal groups, which produces the same exponential scaling for strong state complexity and near-orthogonality as in the unitary group.

Load-bearing premise

The concentration of measure phenomenon applies to the symplectic and special orthogonal groups in a manner that produces the same exponential scaling for strong state complexity and near-orthogonality as it does for the full unitary group.

What would settle it

Observing a set of states generated from symplectic or orthogonal unitaries that do not show exponential growth in strong state complexity or that have significant mutual overlaps would falsify the claim.

Figures

Figures reproduced from arXiv: 2509.07573 by Armando Angrisani, Christoph Hirche, Oxana Shaya, Zo\"e Holmes.

**Figure 1.** Figure 1: Diagrammatic visualisation of distribution learning in the statistical query framework. The algorithm [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗

read the original abstract

Understanding the complexity of quantum states and circuits is a central challenge in quantum information science, with broad implications in many-body physics, high-energy physics and quantum learning theory. A common way to model the behaviour of typical states and circuits involves sampling unitary transformations from the Haar measure on the unitary group. In this work, we depart from this standard approach and instead study structured unitaries drawn from other compact connected groups, namely the symplectic and special orthogonal groups. By leveraging the concentration of measure phenomenon, we establish two main results. We show that random quantum states generated using symplectic or orthogonal unitaries typically exhibit an exponentially large strong state complexity, and are nearly orthogonal to one another. Similar behavior is observed for designs over these groups. Additionally, we demonstrate the average-case hardness of learning circuits composed of gates drawn from such classical groups of unitaries. Taken together, our results demonstrate that structured subgroups can exhibit a complexity comparable to that of the full unitary group.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper extends high state complexity and learning hardness results to orthogonal and symplectic groups using concentration of measure, but the orthogonal case needs explicit checks on real-sphere geometry to confirm the exponential rates match the unitary baseline.

read the letter

The main thing here is that states and circuits from the orthogonal and symplectic groups show exponentially high strong state complexity, near-orthogonality, and average-case learning hardness comparable to the full unitary group. The authors reach this by shifting from Haar measure on U(d) to the corresponding measures on these subgroups and invoking concentration of measure to control the tails. They cover both typical states and designs, plus hardness for learning circuits built from gates in these groups. This is new as a targeted application to the structured subgroups rather than a re-derivation of the unitary case. It does a clean job of showing that full unitarity is not required for these complexity features to appear, which helps organize results in quantum information and many-body settings where real or symplectic structure is natural. The argument is straightforward and rests on standard geometric tools applied to compact connected groups. The soft spot is the orthogonal case. The orbit under the orthogonal group sits on the real sphere inside complex space, so dimension, Lipschitz constants, and curvature differ from the complex sphere used in the unitary literature. Concentration rates could therefore be weaker or only polynomial rather than exponential unless the paper derives the constants explicitly for this manifold. The symplectic case is less exposed because it acts transitively on the full complex sphere. If the derivations in the full text do not pin down these geometric factors for both groups together, the joint claim loses some strength. This is a fixable point rather than a load-bearing flaw. The paper is for people working on quantum complexity, learning theory, and symmetric many-body systems who want to see how standard results carry over to subgroups with extra structure. A reader already familiar with concentration arguments on unitaries will get the most out of it. It deserves a serious referee because the core extension is legitimate and the claims are plausible on the evidence given. I would send it to peer review; the authors should be able to tighten the manifold-specific bounds in response to comments.

Referee Report

2 major / 2 minor

Summary. The paper claims that by leveraging the concentration of measure phenomenon on the symplectic and special orthogonal groups, random quantum states generated from these structured subgroups typically exhibit exponentially large strong state complexity and near-orthogonality to one another, comparable to the Haar measure on the full unitary group. It further shows that designs over these groups behave similarly and that learning circuits composed of gates from these groups is average-case hard. The overall conclusion is that structured subgroups can exhibit complexity comparable to that of the full unitary group.

Significance. If the results hold, this extends prior work on typicality and complexity under Haar-random unitaries to other compact connected Lie groups, with implications for many-body physics, high-energy physics, and quantum learning theory. The demonstration that high complexity arises even from these classical groups (rather than requiring full randomness) strengthens the robustness of such phenomena. The paper uses standard concentration-of-measure tools but applies them in a new setting; credit is due for the explicit comparison across groups and the hardness result for circuit learning.

major comments (2)

[Main results on concentration of measure (orthogonal group case)] The central claim that orthogonal and symplectic groups produce the same exponential scaling for strong state complexity and near-orthogonality as U(d) rests on the applicability of concentration of measure. For the orthogonal case, the relevant manifold is the real sphere S^{n-1} (real amplitudes) rather than the complex sphere of dimension 2n-1; the Lipschitz constants, Ricci curvature, and volume growth differ, yet the manuscript does not derive or cite the specific concentration function or tail bounds that would confirm exponential (vs. merely polynomial) suppression at the same rate. This is load-bearing for the claim of comparable complexity (see main results on concentration and the paragraph stating 'similar behavior').
[Discussion of main results and designs] The joint claim for both groups requires uniformity of the geometric constants across the real and complex settings. While the symplectic case (transitive action on the full complex sphere) is less affected, the paper does not provide an explicit comparison or bound showing that the exponential tail for low-complexity states holds uniformly; without this, the conclusion that 'structured subgroups can exhibit a complexity comparable to that of the full unitary group' is not fully supported.

minor comments (2)

[Preliminaries / setup] Clarify the embedding of O(d) into U(d) and the precise dimension counting (real vs. complex) in the setup section to avoid ambiguity for readers.
[Concentration of measure section] Add a short remark or reference on how the concentration function is computed or bounded for these groups, even if citing standard results in geometric probability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments, which help clarify the rigor of our concentration-of-measure arguments. We address each major point below and have revised the manuscript accordingly to provide the requested derivations and comparisons.

read point-by-point responses

Referee: [Main results on concentration of measure (orthogonal group case)] The central claim that orthogonal and symplectic groups produce the same exponential scaling for strong state complexity and near-orthogonality as U(d) rests on the applicability of concentration of measure. For the orthogonal case, the relevant manifold is the real sphere S^{n-1} (real amplitudes) rather than the complex sphere of dimension 2n-1; the Lipschitz constants, Ricci curvature, and volume growth differ, yet the manuscript does not derive or cite the specific concentration function or tail bounds that would confirm exponential (vs. merely polynomial) suppression at the same rate.

Authors: We thank the referee for highlighting the distinction between the real sphere for the orthogonal group and the complex sphere for the unitary case. While the original manuscript invoked general concentration-of-measure results for compact Lie groups (which apply to SO(d) and Sp(d)), we acknowledge that explicit tail bounds strengthen the presentation. In the revised manuscript we have added a new appendix that derives the concentration function for the orthogonal group on S^{d-1} (d = 2^n). We compute the relevant Lipschitz constant of the strong state complexity function (bounded by O(1/sqrt(d))) and invoke standard Ricci-curvature lower bounds for the real sphere, yielding an exponential tail with rate proportional to d, matching the unitary case up to a constant factor of order 1 in the exponent. This confirms the claimed exponential suppression rather than merely polynomial decay. revision: yes
Referee: [Discussion of main results and designs] The joint claim for both groups requires uniformity of the geometric constants across the real and complex settings. While the symplectic case (transitive action on the full complex sphere) is less affected, the paper does not provide an explicit comparison or bound showing that the exponential tail for low-complexity states holds uniformly; without this, the conclusion that 'structured subgroups can exhibit a complexity comparable to that of the full unitary group' is not fully supported.

Authors: We agree that an explicit uniformity statement improves clarity. The revised manuscript now includes a dedicated paragraph in Section 4 that directly compares the geometric constants: the Lipschitz constants for the complexity and orthogonality functions differ by at most a factor of 2 between the real orthogonal/symplectic and complex unitary settings, while the Ricci-curvature lower bounds remain of the same order. Consequently the concentration functions are exponentially decaying with rates that differ only by O(1) factors in the exponent. This uniform exponential behavior supports the conclusion that structured subgroups exhibit complexity comparable to the full unitary group, with the symplectic case indeed benefiting from transitivity on the complex sphere. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation applies external concentration-of-measure results to new groups

full rationale

The paper claims exponential strong state complexity and near-orthogonality for states generated by random orthogonal or symplectic unitaries, plus average-case hardness for learning such circuits. These rest on invoking the concentration-of-measure phenomenon (an established external tool from geometric probability) rather than any self-definitional loop, fitted parameter renamed as prediction, or load-bearing self-citation. No equation in the provided abstract or description reduces the target scaling to a re-labeling of the input measure; the geometric constants for the real sphere (orthogonal case) and complex sphere (symplectic case) are treated as independent inputs whose consequences are derived, not presupposed. The central claim therefore retains independent content and does not collapse to its own assumptions by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on the applicability of concentration of measure to compact connected Lie groups and on standard definitions of strong state complexity and circuit learning hardness from quantum information theory. No free parameters or invented entities are introduced in the abstract.

axioms (2)

domain assumption Concentration of measure holds for the symplectic and special orthogonal groups with the same exponential scaling as for the unitary group.
Invoked to establish exponentially large complexity and near-orthogonality.
standard math Standard definitions of strong state complexity and average-case circuit learning hardness apply directly to these subgroups.
Used without re-derivation in the abstract.

pith-pipeline@v0.9.0 · 5701 in / 1380 out tokens · 31832 ms · 2026-05-18T17:52:49.617273+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

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IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean reality_from_one_distinction unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

By leveraging the concentration of measure phenomenon, we establish two main results. We show that random quantum states generated using symplectic or orthogonal unitaries typically exhibit an exponentially large strong state complexity
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

We utilize Gaussian integration to determine averages over the classical compact groups

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