Cryptographic Conditions for Efficient Testing of Distributions and Quantum States

Bruno Cavalar; Eli Goldin; Matthew Gray; Min-Hsiu Hsieh; Taiga Hiroka; Tomoyuki Morimae

arxiv: 2510.05028 · v3 · pith:BHJJAOYBnew · submitted 2025-10-06 · 🪐 quant-ph · cs.CC· cs.CR

Cryptographic Conditions for Efficient Testing of Distributions and Quantum States

Bruno Cavalar , Eli Goldin , Matthew Gray , Taiga Hiroka , Min-Hsiu Hsieh , Tomoyuki Morimae This is my paper

Pith reviewed 2026-05-18 09:42 UTC · model grok-4.3

classification 🪐 quant-ph cs.CCcs.CR

keywords distribution testingidentity testingquantum state verificationKolmogorov complexitycertified randomnessquantum advantagesamplable distributions

0 comments

The pith

Promising that an unknown distribution is efficiently samplable allows polynomially many samples to verify it even when samples are adversarial and arbitrarily correlated.

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

The paper examines identity testing of distributions over bit strings when the target distribution is promised to be efficiently samplable. In this setting the usual requirement of exponentially many independent samples is replaced by a polynomial number that works even if the samples are generated by an adversary and may be correlated in any way. The authors characterize the computational complexity of the resulting verification task for both classically and quantumly samplable distributions. The same techniques are shown to apply to the verification of quantum states. Novel uses of Kolmogorov complexity support several of the results and yield side applications such as certified randomness from an efficient classical prover.

Core claim

Under the promise that the unknown distribution is efficiently samplable, polynomially many samples suffice to verify whether observed samples come from a target distribution, even when the samples are adversarially generated and arbitrarily correlated; the same conditions suffice for verifying quantum states and rest on a novel application of Kolmogorov complexity.

What carries the argument

The efficient sampler guaranteed by the promise, used together with Kolmogorov-complexity measures to certify that the observed samples are consistent with the target distribution.

If this is right

Polynomial sample complexity suffices for distribution verification under the samplability promise.
Verification is computationally efficient precisely when the sampler and the target distribution satisfy certain Kolmogorov-complexity conditions.
The same polynomial-sample protocol extends directly to verification of quantum states.
Certified randomness can be obtained from a classical efficient prover when the verifier is allowed to run in unbounded time.

Where Pith is reading between the lines

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

The model suggests that practical tests of quantum sampling devices become feasible once one assumes the device implements an efficiently describable distribution.
Kolmogorov-complexity benchmarks may serve as a concrete way to certify sampling-based quantum advantage without relying on unproven computational hardness.
Similar samplability promises could be explored in other adversarial testing settings such as property testing of graphs or functions.

Load-bearing premise

The unknown distribution must admit an efficient sampling algorithm that the verifier can run during the test.

What would settle it

An explicit efficiently samplable distribution together with a polynomial number of adversarial correlated samples that any polynomial-time tester accepts as coming from the target when in fact they do not.

read the original abstract

One of the most fundamental problems in distribution testing is the identity testing problem: given samples $x_1,\ldots,x_s$, the goal is to determine whether the samples are drawn from a target distribution $\mathcal{D}$. When $\mathcal{D}$ is a distribution over $\bit^n$, the optimal sample complexity of identity testing is known to be $\Omega(\sqrt{2^n})$. Furthermore, most existing results assume that the samples $x_1,\ldots,x_s$ are generated independently from an unknown distribution. In this work, we overcome both of these limitations by initiating study of distribution testing in a more realistic setting. In our model, the unknown distribution is promised to be efficiently samplable, while allowing the observed samples $x_1,\ldots,x_s$ to be adversarially generated and arbitrarily correlated. Under this model, we show that polynomially many samples suffice to verify distributions. We further characterize the computational complexity of verifying classically- and quantumly-samplable distributions. Our techniques also extend to verifications of quantum states. In establishing some of our results, we employ Kolmogorov complexity techniques in a novel manner. We also present multiple applications of Kolmogorov complexity that are of independent interest. In particular, we show that certified randomness with a classical efficient prover can be achieved without computational assumptions when inefficient verification is allowed. Furthermore, we also show that a natural quantum extension of a well-studied Kolmogorov complexity measure provides a good benchmark for certifying sampling-based quantum advantage.

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

With an efficient samplability promise, polynomially many adversarial samples suffice for distribution testing, but Kolmogorov complexity likely keeps the verifier non-explicit.

read the letter

The main takeaway is that promising the unknown distribution is efficiently samplable lets them claim polynomially many samples are enough to test identity, even when those samples are chosen by an adversary and can be arbitrarily correlated. This gets around the usual Ω(√2^n) lower bound that holds for independent samples over n-bit strings. They extend the same style of argument to verifying quantum states and give two side applications that stand on their own: certified randomness from a classical efficient prover when verification is allowed to be inefficient, and a quantum Kolmogorov complexity measure as a benchmark for sampling-based quantum advantage. The model itself is the clearest novelty; prior distribution testing work has stuck closely to the i.i.d. setting. The Kolmogorov complexity technique is used in a way that feels fresh for this area. The soft spot is exactly the one the stress-test flags. Because Kolmogorov complexity is uncomputable, the argument that “polynomially many samples suffice” reads as an existence result rather than a description of a concrete, efficient test the verifier can run. The paper does characterize the computational complexity of the verification task for both classical and quantum samplable distributions, which shows the authors see the distinction, but it is not obvious from the abstract whether the main testing claim comes with an explicit polynomial-time procedure or remains information-theoretic. If the latter, the practical payoff for verification protocols is narrower than the headline suggests. This is for people working on property testing, quantum verification, and certified randomness who are willing to explore models with strong promises on the sampler. A reader looking for new angles on realistic sampling assumptions will get something out of it. It deserves a serious referee because the model is new, the applications are concrete, and the computational characterization gives referees something definite to check.

Referee Report

3 major / 2 minor

Summary. The paper introduces a model for distribution testing where the unknown distribution over {0,1}^n is promised to be efficiently samplable, but the s observed samples may be generated adversarially and with arbitrary correlations. Under this promise, the authors claim that polynomially many samples suffice for identity testing, in contrast to the known Ω(√2^n) lower bound that holds for i.i.d. samples from an arbitrary distribution. They further characterize the computational complexity of verification for classically samplable and quantumly samplable distributions, extend the approach to verification of quantum states, and apply Kolmogorov complexity techniques both to obtain the main results and to derive independent applications such as certified randomness with a classical efficient prover (when inefficient verification is permitted) and a quantum extension of Kolmogorov complexity as a benchmark for sampling-based quantum advantage.

Significance. If the central claims are correct and the verification procedures are made explicit, the work would meaningfully relax the i.i.d. assumption that underlies most distribution-testing lower bounds and could enable practical testing in adversarial settings whenever an efficient sampler for the target distribution is available. The polynomial sample bound, the complexity characterizations, and the quantum-state extension would be notable contributions. The applications to certified randomness and quantum-advantage certification are of independent interest and illustrate the broader utility of the Kolmogorov-complexity techniques introduced.

major comments (3)

[Abstract and §1] Abstract and §1 (model definition): the claim that 'polynomially many samples suffice to verify distributions' is load-bearing for the entire paper, yet the abstract provides no indication whether the resulting tester is computationally efficient or merely information-theoretic. Because Kolmogorov complexity is uncomputable, any argument that invokes incompressibility with respect to the samplable distribution risks being an existence proof; the manuscript must clarify whether an explicit, efficient verifier is constructed or whether the polynomial bound holds only non-constructively.
[Kolmogorov complexity techniques section] Section on Kolmogorov complexity techniques (likely §3 or §4): the novel application of Kolmogorov complexity to handle arbitrarily correlated adversarial samples is central, but the paper must exhibit a concrete reduction or algorithm that the verifier can run on the given samples. If the argument relies solely on the non-existence of short programs without providing a search procedure or approximation, the claim that polynomially many samples suffice does not yet yield a usable testing algorithm under the stated model.
[Computational complexity characterization section] Computational complexity characterization (likely §5): the manuscript states that it 'characterizes the computational complexity of verifying classically- and quantumly-samplable distributions.' This section must explicitly relate the sample-complexity result to the computational complexity of the verifier; otherwise the polynomial bound and the complexity characterization appear disconnected.

minor comments (2)

[Abstract] The abstract lists applications of Kolmogorov complexity but does not name them; a brief enumeration in the introduction would improve readability.
[§2] Notation for the efficient sampler and the adversarial sample generation process should be introduced with a formal definition early in §2 to avoid ambiguity when the Kolmogorov-complexity argument is later invoked.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful and constructive review. The comments highlight important points about the information-theoretic versus computational nature of our results and the need for explicit connections in the manuscript. We address each major comment below and will make corresponding revisions.

read point-by-point responses

Referee: [Abstract and §1] Abstract and §1 (model definition): the claim that 'polynomially many samples suffice to verify distributions' is load-bearing for the entire paper, yet the abstract provides no indication whether the resulting tester is computationally efficient or merely information-theoretic. Because Kolmogorov complexity is uncomputable, any argument that invokes incompressibility with respect to the samplable distribution risks being an existence proof; the manuscript must clarify whether an explicit, efficient verifier is constructed or whether the polynomial bound holds only non-constructively.

Authors: We agree that the abstract and introduction require clarification on this point. Our main result is information-theoretic: using Kolmogorov complexity, we prove that polynomially many (possibly adversarially correlated) samples suffice for the existence of a verifier that can certify consistency with an efficiently samplable distribution. We do not construct or claim an explicit computationally efficient verifier, as any such procedure would require approximating or deciding Kolmogorov complexity, which is uncomputable. We will revise the abstract and §1 to state explicitly that the polynomial sample bound is information-theoretic and non-constructive. revision: yes
Referee: [Kolmogorov complexity techniques section] Section on Kolmogorov complexity techniques (likely §3 or §4): the novel application of Kolmogorov complexity to handle arbitrarily correlated adversarial samples is central, but the paper must exhibit a concrete reduction or algorithm that the verifier can run on the given samples. If the argument relies solely on the non-existence of short programs without providing a search procedure or approximation, the claim that polynomially many samples suffice does not yet yield a usable testing algorithm under the stated model.

Authors: The argument proceeds by contradiction via incompressibility: with sufficiently many samples, the observed sequence cannot be generated by any short program consistent with the efficient sampler for the target distribution. This yields an existential guarantee rather than an algorithmic procedure, as searching for short programs is impossible. We will expand the relevant section to spell out this logical reduction in detail, including how the sample count forces the incompressibility contradiction, while explicitly noting the non-constructive character of the resulting tester. revision: yes
Referee: [Computational complexity characterization section] Computational complexity characterization (likely §5): the manuscript states that it 'characterizes the computational complexity of verifying classically- and quantumly-samplable distributions.' This section must explicitly relate the sample-complexity result to the computational complexity of the verifier; otherwise the polynomial bound and the complexity characterization appear disconnected.

Authors: We will add a dedicated paragraph in the computational complexity section that directly connects the two aspects: the polynomial sample complexity is established information-theoretically via Kolmogorov complexity, while the computational complexity of any verifier is characterized separately (typically high or undecidable in the limit, owing to the need to reason about program lengths). This will make the relationship between the sample bound and the complexity results explicit. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation relies on external Kolmogorov complexity and new model promise

full rationale

The paper defines a new testing model that explicitly promises the unknown distribution is efficiently samplable, then invokes standard (uncomputable) Kolmogorov complexity to argue that polynomially many adversarial samples suffice for verification. This is a direct application of an external mathematical tool to the newly stated promise; no equation or claim reduces to a fitted parameter, self-defined quantity, or load-bearing self-citation. The computational-complexity characterization is presented as a separate result rather than presupposing the sample bound. The derivation is therefore self-contained against external benchmarks and does not exhibit any of the enumerated circular patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Central claims rest on the new model definition and properties of Kolmogorov complexity; no free parameters or invented entities are apparent from the abstract.

axioms (1)

standard math Standard properties of Kolmogorov complexity
Invoked for verification and randomness certification results.

pith-pipeline@v0.9.0 · 5819 in / 1114 out tokens · 24826 ms · 2026-05-18T09:42:17.850212+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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

We introduce a quantum variant of time-bounded Kolmogorov complexity, and show a coding theorem and incompressibility for it... Every QPT-samplable distribution is adaptively-verifiable with polynomially-many samples and with a classical deterministic polynomial-time algorithm querying a PP oracle.
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

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

If infinitely-often OWFs exist, then any classically-efficiently-samplable distribution D(1^n) with H(D(1^n))=n^Ω(1) is not selectively-verifiable

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supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.