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arXiv:2003.11777

Robust quantum minimum finding with an application to hypothesis selection

Yihui Quek, Clément L. Canonne, Patrick Rebentrost

Published Mar 26, 2020• 4 citations
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Abstract

We consider the problem of finding the minimum element in a list of length $N$ using a noisy comparator. The noise is modelled as follows: given two elements to compare, if the values of the elements differ by at least $α$ by some metric defined on the elements, then the comparison will be made correctly; if the values of the elements are closer than $α$, the outcome of the comparison is not subject to any guarantees. We demonstrate a quantum algorithm for noisy quantum minimum-finding that preserves the quadratic speedup of the noiseless case: our algorithm runs in time $\tilde O(\sqrt{N (1+Δ)})$, where $Δ$ is an upper-bound on the number of elements within the interval $α$, and outputs a good approximation of the true minimum with high probability. Our noisy comparator model is motivated by the problem of hypothesis selection, where given a set of $N$ known candidate probability distributions and samples from an unknown target distribution, one seeks to output some candidate distribution $O(\varepsilon)$-close to the unknown target. Much work on the classical front has been devoted to speeding up the run time of classical hypothesis selection from $O(N^2)$ to $O(N)$, in part by using statistical primitives such as the Scheffé test. Assuming a quantum oracle generalization of the classical data access and applying our noisy quantum minimum-finding algorithm, we take this run time into the sublinear regime. The final expected run time is $\tilde O( \sqrt{N(1+Δ)})$, with the same $O(\log N)$ sample complexity from the unknown distribution as the classical algorithm. We expect robust quantum minimum-finding to be a useful building block for algorithms in situations where the comparator (which may be another quantum or classical algorithm) is resolution-limited or subject to some uncertainty.

DOI
10.48550/arxiv.2003.11777

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We consider the problem of finding the minimum element in a list of length NNN using a noisy comparator. The noise is modelled as follows: given two elements to compare, if the values of the elements differ by at least ααα by some metric defined on the elements, then the comparison will be made correctly; if the values of the elements are closer than ααα, the outcome of the comparison is not subject to any guarantees. We demonstrate a quantum algorithm for noisy quantum minimum-finding that preserves the quadratic speedup of the noiseless case: our algorithm runs in time O~(N(1+Δ))\tilde O(\sqrt{N (1+Δ)})O~(N(1+Δ)​), where ΔΔΔ is an upper-bound on the number of elements within the interval ααα, and outputs a good approximation of the true minimum with high probability. Our noisy comparator model is motivated by the problem of hypothesis selection, where given a set of NNN known candidate probability distributions and samples from an unknown target distribution, one seeks to output some candidate distribution O(ε)O(\varepsilon)O(ε)-close to the unknown target. Much work on the classical front has been devoted to speeding up the run time of classical hypothesis selection from O(N2)O(N^2)O(N2) to O(N)O(N)O(N), in part by using statistical primitives such as the Scheffé test. Assuming a quantum oracle generalization of the classical data access and applying our noisy quantum minimum-finding algorithm, we take this run time into the sublinear regime. The final expected run time is O~(N(1+Δ))\tilde O( \sqrt{N(1+Δ)})O~(N(1+Δ)​), with the same O(log⁡N)O(\log N)O(logN) sample complexity from the unknown distribution as the classical algorithm. We expect robust quantum minimum-finding to be a useful building block for algorithms in situations where the comparator (which may be another quantum or classical algorithm) is resolution-limited or subject to some uncertainty.
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Quantum machine learning algorithms could provide significant speed-ups over their classical counterparts; however, whether they could also achieve good generalization remains unclear. Recently, two quantum perceptron models which give a quadratic improvement over the classical perceptron algorithm using Grover's search have been proposed by Wiebe et al. arXiv:1602.04799 . While the first model reduces the complexity with respect to the size of the training set, the second one improves the bound on the number of mistakes made by the perceptron. In this paper, we introduce a hybrid quantum-classical perceptron algorithm with lower complexity and better generalization ability than the classical perceptron. We show a quadratic improvement over the classical perceptron in both the number of samples and the margin of the data. We derive a bound on the expected error of the hypothesis returned by our algorithm, which compares favorably to the one obtained with the classical online perceptron. We use numerical experiments to illustrate the trade-off between computational complexity and statistical accuracy in quantum perceptron learning and discuss some of the key practical issues surrounding the implementation of quantum perceptron models into near-term quantum devices, whose practical implementation represents a serious challenge due to inherent noise. However, the potential benefits make correcting this worthwhile.
Suppose one has access to oracles generating samples from two unknown probability distributions p and q on some N-element set. How many samples does one need to test whether the two distributions are close or far from each other in the L1-norm? This and related questions have been extensively studied during the last years in the field of property testing. In the present paper we study quantum algorithms for testing properties of distributions. It is shown that the L1-distance ∥p-q∥1can be estimated with a constant precision using only O(N1/2) queries in the quantum settings, whereas classical computers need Ω(N1-o(1)) queries. We also describe quantum algorithms for testing uniformity and orthogonality with query complexity O(N1/3). The classical query complexity of these problems is known to be Ω(N1/2).
Quantum counting is a fundamental quantum algorithm that estimates the fraction of marked elements using a membership oracle, achieving a quadratic speedup over classical sampling. The membership oracle, however, assumes exact labeling of each element, but this assumption fails when the labels are inherently probabilistic. We study quantum counting with \emph{Bernoulli oracles}, where given mmm Bernoulli distributions with unknown biases p1,…,pmp_1,\dots,p_mp1​,…,pm​ and a gap parameter ΔΔΔ, the goal is to estimate the fraction ρρρ of \emph{positive} distributions (pi≥1/2+Δp_i\ge1/2+Δpi​≥1/2+Δ) to within additive error εεε. We prove an upper bound of O~ ⁣(ρΔε+1Δε)\tilde{O}\!\big(\frac{\sqrtρ}{Δε}+\frac{1}{Δ\sqrtε}\big)O~(Δερ​​+Δε​1​) queries, achieving a quadratic speedup over the classical sample complexity. % of Θ(ρ/Δ2ε2)Θ(ρ/Δ^2ε^2)Θ(ρ/Δ2ε2). Our algorithm first uses the Quantum Singular Value Transformation (QSVT) to coherently amplify the bias gap without collapsing the superposition over distributions, and then applies two-stage adaptive amplitude estimation. We complement this upper bound with a near-matching lower bound of Ω(ρ/(Δε))Ω(\sqrtρ/(Δε))Ω(ρ​/(Δε)) via a new composition theorem for the quantum adversary method in the Boolean-over-average-case direction. For the special case of a constant gap Δ=Θ(1)Δ=Θ(1)Δ=Θ(1), which corresponds to the bounded-error oracle where each query returns the correct label with constant probability, our bounds specialize to O~(ρε+1ε)\tilde{O}\big(\frac{\sqrtρ}ε+\frac{1}{\sqrtε}\big)O~(ερ​​+ε​1​) and Ω(ρε)Ω(\frac{\sqrtρ}ε)Ω(ερ​​), thereby characterizing the query complexity of quantum counting with bounded-error oracles.
Unstructured database search is a fundamental problem that requires finding one or more elements satisfying a specific property within a set of candidates. While Grover's algorithm provides a quadratic speedup for finding a marked element when the number of solutions is known, finding all solutions when their number is unknown presents additional challenges. In this paper, we thoroughly analyze this problem by considering all the different regimes of combinations of classical, Grover and generalized fixed-point quantum search instances. We unify these into an adaptive Bayesian approach that maximizes the expected success probability per unit execution time at each searching instance. For a problem with a known number of solutions, much lower than the database size, we prove that our proposed approach is asymptotically optimal within the considered search model. Furthermore, we prove that for any probability distribution characterizing the likely number of solutions, in regimes where the classical verification time is at most 25% of the quantum amplitude amplification time, hybrid classical and Grover approaches are preferred to generalized fixed-point. Outside of this regime, we derive an upper bound on the maximal improvement achievable by fixed-point search compared to the best Grover-based score. Additionally, we determine the optimal transition point between classical sampling and quantum searching using an integral approximation of the probability distribution of the number of solutions. The analytical results are also supported by numerical simulations of the proposed algorithm and by comparisons between the generalized fixed-point and Grover-based search policies.
Quantum machine learning algorithms could provide significant speed-ups over their classical counterparts; however, whether they could also achieve good generalization remains unclear. Recently, two quantum perceptron models which give a quadratic improvement over the classical perceptron algorithm using Grover's search have been proposed by Wiebe et al. arXiv:1602.04799 . While the first model reduces the complexity with respect to the size of the training set, the second one improves the bound on the number of mistakes made by the perceptron. In this paper, we introduce a hybrid quantum-classical perceptron algorithm with lower complexity and better generalization ability than the classical perceptron. We show a quadratic improvement over the classical perceptron in both the number of samples and the margin of the data. We derive a bound on the expected error of the hypothesis returned by our algorithm, which compares favorably to the one obtained with the classical online perceptron. We use numerical experiments to illustrate the trade-off between computational complexity and statistical accuracy in quantum perceptron learning and discuss some of the key practical issues surrounding the implementation of quantum perceptron models into near-term quantum devices, whose practical implementation represents a serious challenge due to inherent noise. However, the potential benefits make correcting this worthwhile.