BosonSampling and Random Circuit Sampling are important both as a theoretical tool for separating quantum and classical computation, and as an experimental means of demonstrating quantum speedups. Prior works have shown that average-case hardness of sampling follows from certain unproven conjectures about the hardness of computing output probabilities, such as the Permanent-of-Gaussians Conjecture (PGC), which states that e(-n log n-n-O(log n)) additive-error estimates to the output probability of most random BosonSampling experiments are #P-hard. Prior works have only shown weaker average-case hardness results that do not imply sampling hardness. Proving these conjectures has become a central question in quantum complexity. In this work, we show that e(-n log n-n-O(n delta)) additive-error estimates to output probabilities of most random BosonSampling experiments are #P-hard for any delta > 0, exponentially improving on prior work. In the process, we circumvent all known barrier results for proving PGC. The remaining hurdle to prove PGC is now "merely" to show that the O(n(delta)) in the exponent can be improved to O(log n). We also obtain an analogous result for Random Circuit Sampling. We then show, for the first time, a hardness of average-case classical sampling result for BosonSampling, under an anticoncentration conjecture. Specifically, we prove the impossibility of multiplicative-error sampling from random BosonSampling experiments with probability 1 - 2(-(O) over tilde (N1/3)) for input size N, unless the Polynomial Hierarchy collapses. This exponentially improves upon the state-of-the-art. To do this, we introduce new proof techniques which tolerate exponential loss in the worstto-average-case reduction. This opens the possibility to show the hardness of average-case sampling without ever proving PGC.
The problem of studying k-wise correlations in product spaces, i.e., correlations of the form E-(x1,E- ... ,E- xk)similar to mu circle times n[f(1)(x(1)) ... f(x(k))] where f(i) : Sigma(n)(i) -> C are all 1-bounded functions and mu is a distribution over Sigma(1) x ... x Sigma(k), appears in many different contexts throughout discrete mathematics. Examples include additive combinatorics, extremal combinatorics, hardness of approximation and probability. The goal in an inverse theorem is to characterize the type of functions f(1), ... , f(k) that achieve non-trivial correlations, under minimal assumptions on the distribution mu. We give new inverse theorems for k-wise correlations for all k >= 3. For k = 3, our inverse theorem works for any distribution mu which is pairwise-connected, which is essentially the minimal assumption required for a nontrivial inverse theorem to hold. For k > 3, our inverse theorem applies for distributions mu satisfying the stronger condition of not having any Abelian embeddings. This resolves a conjecture from [Bhangale-Khot-Minzer, STOC 2022]. We give applications of our inverse theorems to additive combinatorics, hardness of approximation, and property testing. First, we show that there exists c > 0 such that any set A subset of {0, 1, 2}(n) with density at least Omega((log log log log n)(-c)) must contain a combinatorial line, i.e., x, y, z is an element of {0, 1, 2}(n), not all equal, such that x(i) = y(i) = z(i) or (x(i), y(i), z(i)) = (0, 1, 2) for all i = 1, 2, ... , n. In other words, we give "reasonable bounds" for the density Hales-Jewett theorem of length 3. This involves combining our inverse theorems with several additional insights, motivated by Shkredov's proof of the corners theorem and Polymath's combinatorial proof of the density Hales-Jewett theorem. Second, we show how to construct a dictatorship vs quasi-random test that has perfect completeness and soundness s + epsilon from integrality gap instances with similar parameters, provided that its local distributions have no Abelian embeddings. Third, we analyze the direct-sum tester of [Dinur-Golubev, RANDOM 2019] in the low-soundness regime.
Suppose that an untrusted analyst claims that it ran a distribution tester and determined that an unknown distribution has a certain property. Can the untrusted analyst prove that its assertion is correct to a verifier that does not have sufficient samples and computational resources to run the tester on its own? In this work, we are interested in proofs that can be generated very efficiently, with minimal overhead over running the distribution tester. In particular, since the distribution tester is sublinear (in the domain size), at the very least we also want the sample complexity for generating the proof to be sublinear. Do natural properties that have sublinear testers admit such proof systems?Our main result answers this question negatively for several natural properties. For these properties, if the verifier’s sample complexity is non-trivial (smaller than just running the tester on its own), then the (honest) prover must draw a linear number of samples. We show this result for the problem of testing whether the distribution is uniform over its support, for specifying the distribution’s k-collision probability (or its Lk norm), and for other natural properties.Our results shed light on a recent line of work showing that if we allow the prover to draw a quasi-linear number of samples, then many distribution properties have proof-systems with very efficient verification. Our negative results imply that the super-linear sample complexity of the prover in those proof-systems is inherent.
The Borsuk-Ulam theorem states that every continuous odd function f : S-n -> R-n must have a zero, i.e., an x is an element of S-n such that f(x) = 0. While such a zero is guaranteed to exist, finding it is known to be computationally intractable: it is PPA-complete already for n = 2. In this work, we show that the problem remains just as hard even if the function is mapping from a higher to a lower dimensional space. Namely, we prove that it is PPA-complete to find a zero of f : S-k -> R-n for any constants k >= n >= 2. This result has very appealing consequences for other flagship PPA-complete problems such as Tucker, Consensus Halving, and Ham Sandwich. For example, in the Consensus Halving problem from fair division, we show that finding a partition that satisfies three agents with monotone valuations is PPA-complete, even if we allow any arbitrarily large constant number of cuts.
Trevisan and Vadhan (FOCS 2000) introduced the notion of (seedless) extractors for samplable distributions. They showed that under a very strong complexity theoretic hardness assumption (specifically, that there exists a problem in E = DTIME(2(O(n))) that cannot be computed by size 2(Omega(n)) circuits that have an oracle to Sigma(P)(6)) there are extractors for samplable distributions with large min-entropy of k = (1 - gamma) center dot n, for some small constant gamma > 0. Recently, Ball, Shaltiel and Silbak (STOC 2025) were able to reduce the min-entropy threshold to k = n(1 - gamma). Ball et al., point out that their approach does not work for k < root n (and this holds even for stronger hardness assumptions, in which 6 is replaced with any other constant). In this paper, we show how to further reduce the min-entropy threshold to k = n(0.34) < root n under the same hardness assumption used by Trevisan and Vadhan. More generally, for every positive integer i >= 2, and every alpha > 1/i, we construct an extractor for samplable distributions with min-entropy k = n(alpha), under a hardness assumption in which 6 is replaced with i+3 (the aforementioned result is a obtained for i = 3). We also provide a multiplicative version of our extractors (under a stronger hardness assumption) addressing an open problem of Ball et al. Our work builds on the approach of Ball et al., who reduced the task of constructing extractors for samplable distributions with min-entropy k, to the task of constructing errorless condensers for samplable distributions with min-entropy k. Our main technical contribution is a new construction of errorless condensers for samplable distributions with k = n(alpha) under the hardness assumption stated above, improving upon the minentropy threshold achieved in Ball et al. (which cannot achieve k < root n). Our insight is that the technique used by Ball et al. to reduce the task of constructing extractors to that of constructing errorless condensers, can itself be used to construct errorless condensers for polynomially small min-entropy when combined with "win-win analysis" approaches that are inspired by some early work on seeded extractors and dispersers. In order to do this, we adapt these approaches from the information theoretic scenario of seeded extractors and dispersers to the computational scenario of errorless condensers for samplable distributions.
