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1. Complexity of High-Dimensional Identity Testing with Coordinate Conditional Sampling

   Blanca, Antonio; Chen, Zongchen; Štefankovič, Daniel; Vigoda, Eric (August 2023, COLT 2023)

   We study the identity testing problem for high-dimensional distributions. Given as input an explicit distribution μ, an ε>0, and access to sampling oracle(s) for a hidden distribution π, the goal in identity testing is to distinguish whether the two distributions μ and π are identical or are at least ε-far apart. When there is only access to full samples from the hidden distribution π, it is known that exponentially many samples (in the dimension) may be needed for identity testing, and hence previous works have studied identity testing with additional access to various “conditional” sampling oracles. We consider a significantly weaker conditional sampling oracle, which we call the Coordinate Oracle, and provide a computational and statistical characterization of the identity testing problem in this new model. We prove that if an analytic property known as approximate tensorization of entropy holds for an n-dimensional visible distribution μ, then there is an efficient identity testing algorithm for any hidden distribution π using O˜(n/ε) queries to the Coordinate Oracle. Approximate tensorization of entropy is a pertinent condition as recent works have established it for a large class of high-dimensional distributions. We also prove a computational phase transition: for a well-studied class of n-dimensional distributions, specifically sparse antiferromagnetic Ising models over {+1,−1}^n, we show that in the regime where approximate tensorization of entropy fails, there is no efficient identity testing algorithm unless RP=NP. We complement our results with a matching Ω(n/ε) statistical lower bound for the sample complexity of identity testing in the model. 

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2. A Distributional-Lifting Theorem for Learning

   Blanc, Guy; Lange, Jane; Strassle, Carmen; Tan, Li-Yang (June 2025, 38th Annual Conference on Learning Theory (COLT 2025))

   The apparent difficulty of efficient distribution-free PAC learning has led to a large body of work on distribution-specific learning. Distributional assumptions facilitate the design of efficient algorithms but also limit their reach and relevance. Towards addressing this, we prove a distributional-lifting theorem: This upgrades a learner that succeeds with respect to a limited distribution family \mathcal{D} to one that succeeds with respect to any distribution D^\star, with an efficiency overhead that scales with the complexity of expressing D^\star as a mixture of distributions in \mathcal{D}. Recent work of Blanc, Lange, Malik, and Tan considered the special case of lifting uniform-distribution learners and designed a lifter that uses a conditional sample oracle for D^\star, a strong form of access not afforded by the standard PAC model. Their approach, which draws on ideas from semi-supervised learning, first learns D^\star and then uses this information to lift. We show that their approach is information-theoretically intractable with access only to random examples, thereby giving formal justification for their use of the conditional sample oracle. We then take a different approach that sidesteps the need to learn D^\star, yielding a lifter that works in the standard PAC model and enjoys additional advantages: it works for all base distribution families, preserves the noise tolerance of learners, has better sample complexity, and is simpler. 

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3. Data Structures for Density Estimation

   Aamand, Anders; Andoni, Alexandr; Chen, Justin; Indyk, Piotr; Narayanan, Shyam; Silwal, Sandeep (January 2023, International Conference on Machine Learning)

   We study statistical/computational tradeoffs for the following density estimation problem: given kdistributionsv1,...,vk overadiscretedomain of size n, and sampling access to a distribution p, identify vi that is “close” to p. Our main result is the first data structure that, given a sublinear (in n) number of samples from p, identifies vi in time sublinear in k. We also give an improved version of the algorithm of (Acharya et al., 2018) that reports vi in time linear in k. The experimental evaluation of the latter algorithm shows that it achieves a significant reduction in the number of operations needed to achieve a given accuracy compared to prior work. 

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4. Separating QMA from QCMA with a Classical Oracle

   https://doi.org/10.1145/3798129.3800776  

   Bostanci, John; Haferkamp, Jonas; Nirkhe, Chinmay; Zhandry, Mark (June 2026, ACM)

   We construct a classical oracle proving that, in a relativized setting, the set of languages decidable by an efficient quantum verifier with a quantum witness (QMA) is strictly bigger than those decidable with access only to a classical witness (QCMA). The separating classical oracle we construct is for a decision problem we coin spec­tral Forrelation - the oracle describes two subsets of the boolean hypercube, and the computational task is to decide if there exists a quantum state whose standard basis measurement distribution is well supported on one subset while its Fourier basis measurement distribution is well supported on the other subset. This is equivalent to estimating the spectral norm of a "Forrelation" matrix between two sets that are accessible through membership queries. Our lower bound derives from a simple observation that a query algorithm with a classical witness can be run multiple times to gen­erate many samples from a distribution, while a quantum witness is a "use once" object. This observation allows us to reduce proving a QCMA lower bound to proving a sampling hardness result which does not simultaneously prove a QMA lower bound. To prove said sampling hardness result for QCMA, we observe that quantum access to the oracle can be compressed by expressing the problem in terms of bosons - a novel "second quantization" perspective on compressed oracle techniques, which may be of independent inter­est. Using this compressed perspective on the sampling problem, we prove the sampling hardness result, completing the proof. 

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5. Sublinear quantum algorithms for training linear and kernel-based classifiers

   Li, Tongyang; Chakrabarti, Shouvanik; Wu, Xiaodi (June 2019, Proceedings of Machine Learning Research)

   We investigate quantum algorithms for classification, a fundamental problem in machine learning, with provable guarantees. Given n d-dimensional data points, the state-of-the-art (and optimal) classical algorithm for training classifiers with constant margin by Clarkson et al. runs in Õ (n+d), which is also optimal in its input/output model. We design sublinear quantum algorithms for the same task running in Õ (\sqrt{n}+\sqrt{d}), a quadratic improvement in both n and d. Moreover, our algorithms use the standard quantization of the classical input and generate the same classical output, suggesting minimal overheads when used as subroutines for end-to-end applications. We also demonstrate a tight lower bound (up to poly-log factors) and discuss the possibility of implementation on near-term quantum machines. 

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