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arXiv:1902.08753 · Quantum Information Processing

Quantum learning Boolean linear functions w.r.t. product distributions

Matthias C. Caro

Published Apr 20, 2020• 5 citations
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Abstract

Abstract The problem of learning Boolean linear functions from quantum examples w.r.t. the uniform distribution can be solved on a quantum computer using the Bernstein–Vazirani algorithm (Bernstein and Vazirani, in: Kosaraju (ed) Proceedings of the twenty-fifth annual ACM symposium on theory of computing, ACM, New York, 1993. 10.1145/167088.167097 ). A similar strategy can be applied in the case of noisy quantum training data, as was observed in Grilo et al. (Learning with errors is easy with quantum samples, 2017). However, extensions of these learning algorithms beyond the uniform distribution have not yet been studied. We employ the biased quantum Fourier transform introduced in Kanade et al. (Learning dnfs under product distributions via $$\mu $$ μ -biased quantum Fourier sampling, 2018) to develop efficient quantum algorithms for learning Boolean linear functions on n bits from quantum examples w.r.t. a biased product distribution. Our first procedure is applicable to any (except full) bias and requires $$\mathcal {O}(\ln (n))$$ O(ln(n)) quantum examples. The number of quantum examples used by our second algorithm is independent of n, but the strategy is applicable only for small bias. Moreover, we show that the second procedure is stable w.r.t. noisy training data and w.r.t. faulty quantum gates. This also enables us to solve a version of the learning problem in which the underlying distribution is not known in advance. Finally, we prove lower bounds on the classical and quantum sample complexities of the learning problem. Whereas classically, $$\varOmega (n)$$ Ω(n) examples are necessary independently of the bias, we are able to establish a quantum sample complexity lower bound of $$\varOmega (\ln (n))$$ Ω(ln(n)) only under an assumption of large bias. Nevertheless, this allows for a discussion of the performance of our suggested learning algorithms w.r.t. sample complexity. With our analysis, we contribute to a more quantitative understanding of the power and limitations of quantum training data for learning classical functions.

DOI
10.1007/s11128-020-02661-1

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Abstract The problem of learning Boolean linear functions from quantum examples w.r.t. the uniform distribution can be solved on a quantum computer using the Bernstein–Vazirani algorithm (Bernstein and Vazirani, in: Kosaraju (ed) Proceedings of the twenty-fifth annual ACM symposium on theory of computing, ACM, New York, 1993. 10.1145/167088.167097 ). A similar strategy can be applied in the case of noisy quantum training data, as was observed in Grilo et al. (Learning with errors is easy with quantum samples, 2017). However, extensions of these learning algorithms beyond the uniform distribution have not yet been studied. We employ the biased quantum Fourier transform introduced in Kanade et al. (Learning dnfs under product distributions via μ\mu μ μ -biased quantum Fourier sampling, 2018) to develop efficient quantum algorithms for learning Boolean linear functions on n bits from quantum examples w.r.t. a biased product distribution. Our first procedure is applicable to any (except full) bias and requires O(ln⁡(n))\mathcal {O}(\ln (n))O(ln(n)) O(ln(n)) quantum examples. The number of quantum examples used by our second algorithm is independent of n, but the strategy is applicable only for small bias. Moreover, we show that the second procedure is stable w.r.t. noisy training data and w.r.t. faulty quantum gates. This also enables us to solve a version of the learning problem in which the underlying distribution is not known in advance. Finally, we prove lower bounds on the classical and quantum sample complexities of the learning problem. Whereas classically, Ω(n)\varOmega (n)Ω(n) Ω(n) examples are necessary independently of the bias, we are able to establish a quantum sample complexity lower bound of Ω(ln⁡(n))\varOmega (\ln (n))Ω(ln(n)) Ω(ln(n)) only under an assumption of large bias. Nevertheless, this allows for a discussion of the performance of our suggested learning algorithms w.r.t. sample complexity. With our analysis, we contribute to a more quantitative understanding of the power and limitations of quantum training data for learning classical functions.
As quantum computing progresses toward fault-tolerant architectures, the question of which computational tasks admit provable quantum advantages and which do not has become increasingly central. Learning theory, and in particular learning from quantum examples, provides one of the few settings in which unconditional quantum-classical separations can be established. In distribution-free (i.e., worst-case) PAC learning, existing results show that quantum examples provide no asymptotic advantage in sample complexity. In contrast, under the uniform distribution, unbounded quantum-classical separations are known for learning Fourier-sparse Boolean functions. Together, these results reveal a striking dichotomy. However, this understanding has largely been developed in the context of learning Boolean functions in the batch setting, leaving open how these phenomena extend more broadly. This thesis develops the theory of learning with quantum examples beyond the batch Boolean setting along three directions: multiclass learning, online learning, and smoothed learning. In the multiclass PAC setting, we establish upper and lower bounds on quantum sample complexity in both the realizable and agnostic regimes, finding that quantum examples continue to yield no distribution-independent separation from classical examples, with learning rates governed by the Natarajan dimension up to logarithmic factors in the label-space size. We next study online learning, where no standard framework for learning with quantum examples existed prior to this work. We provide such a model by lifting the classical online framework to one in which the adversary provides distributions over labeled examples, and then by encoding these distributions as quantum examples. We establish expected regret guarantees for binary and multiclass classification in both the realizable and agnostic settings. The central finding is that unrestricted adversarial power permits highly concentrated distributions that dequantize the learning problem. Motivated by this dequantization phenomenon, we develop a smoothed learning framework that constrains distributions to be smooth, interpolating between the concentrated-distribution regime, in which no quantum advantage exists, and the uniform-distribution regime, in which unbounded separations are known. For the class of Fourier-sparse Boolean functions, we establish the robustness of quantum Fourier sampling throughout a nontrivial near-uniform regime in the batch setting and observe that these guarantees extend naturally to the online setting. We further analyze several approaches to recovering the target function once its Fourier support has been identified and, based on these observations, formulate conjectures supporting the persistence of the unbounded quantum advantage throughout this regime. Together, these results paint a coherent picture of learning with quantum examples beyond the batch Boolean setting, showing that quantum-classical separations depend on the interplay between hypothesis class structure, distributional assumptions, and the degree of adversarial control permitted in the learning process.
We study the number of uniform quantum examples needed to exactly learn a Boolean function with at most kkk nonzero Fourier coefficients. We give a learner with near-optimal O~(k)\widetilde{O}(k)O(k) sample complexity independent of the input length nnn. The learner uses only separate measurements of individual examples. This improves the previous O(k3/2log⁡2k)O(k^{3/2}\log^2 k)O(k3/2log2k) upper bound (Arunachalam, Chakraborty, Lee, Paraashar, and de Wolf, 2021). The main ingredient is a candidate set formed by symmetric differences of subsets from independent groups of Fourier samples. A multilinear majority polynomial combines unbiased random approximations into an accurate witness whose Fourier support lies in this set. Granularity then forces the set to contain the entire Fourier support, including subsets never directly sampled. For comparison, we derive the tight classical sample complexity Θ(nk)Θ(nk)Θ(nk) by combining known sparse-recovery and lower-bound results.
Many properties of Boolean functions can be tested far more efficiently than the function itself can be learned. However, this dramatic advantage often disappears when testers are limited to random samples of f instead of adaptively chosen queries to f. In this work we investigate the quantum version of this restriction: quantum algorithms that test properties of a Boolean function f solely from copies of either the function state |f⟩∝ ∑_x|x,f(x)⟩ or the phase state |(-1)^f⟩∝ ∑_x (-1)^{f(x)}|x⟩. Quantum advantage in testing from data. For monotonicity, symmetry, and triangle-freeness, we show passive quantum testers are unboundedly or super-polynomially better than their classical passive testing counterparts. They are competitive with classic query-based testers in each case. Inadequacy of Fourier sampling. Our new testers use techniques beyond quantum Fourier sampling, and it turns out this is necessary: we show a certain class of bent functions can be tested from 𝒪(1) function states but has a sample complexity lower bound of 2^{Ω(n)} for any tester relying exclusively on Fourier and classical samples. Classical queries vs. quantum data. Our passive quantum testers are competitive with classical query-based testers, but this isn't universal: we exhibit a testing problem that can be solved from 𝒪(1) classical queries but requires Ω(2^{n/2}) function state copies. The Forrelation problem provides a separation of the same magnitude in the opposite direction, so we conclude that quantum data and classical queries are "maximally incomparable" resources for testing. Towards lower bounds. We also begin the study of lower bounds for testing from quantum data. For quantum monotonicity testing, we prove that the ensembles of [Goldreich et al., 2000; Black, 2024], which give exponential lower bounds for classical sample-based testing, do not yield any nontrivial lower bounds for testing from quantum data. New insights specific to quantum data will be required for proving copy complexity lower bounds for testing in this model.
Quantum algorithms for solving noisy linear problems are reexamined, under the same assumptions taken from the existing literature. The findings of this work include on the one hand extended applicability of the quantum Fourier transform to the ring learning with errors problem which has been left open by Grilo et al., who first devised a polynomial-time quantum algorithm for solving noisy linear problems with quantum samples. On the other hand, this paper also shows there exist efficient classical algorithms for short integer solution and size-reduced learning with errors problems if the quantum samples used by the previous studies are given.
We present two new results about exact learning by quantum computers. First, we show how to exactly learn a k -Fourier-sparse n -bit Boolean function from O(k1.5(log⁡k)2) uniform quantum examples for that function. This improves over the bound of Θ~(kn) uniformly random classical examples (Haviv and Regev, CCC'15). Additionally, we provide a possible direction to improve our O~(k1.5) upper bound by proving an improvement of Chang's lemma for k -Fourier-sparse Boolean functions. Second, we show that if a concept class C can be exactly learned using Q quantum membership queries, then it can also be learned using O(Q2log⁡Qlog⁡|C|) classical membership queries. This improves the previous-best simulation result (Servedio and Gortler, SICOMP'04) by a log⁡Q -factor.