Quantum Algorithms for Learning Symmetric Juntas via Adversary Bound

Belovs, Aleksandrs

doi:10.1109/ccc.2014.11

Cited by 11 publications

(22 citation statements)

References 43 publications

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“…These results are only used in this section. The reader may refer to a textbook on the topic such as [50], or to the appendix of [10], where we briefly formulate the required notions and results.…”

Section: Efficient Implementation Of R λmentioning

confidence: 99%

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Efficient Quantum Algorithms for (Gapped) Group Testing and Junta Testing

Ambainis

Belovs

Regev

et al. 2015

Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms

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In the k-junta testing problem, a tester has to efficiently decide whether a given function f : {0, 1} n → {0, 1} is a k-junta (i.e., depends on at most k of its input bits) or is ε-far from any kjunta. Our main result is a quantum algorithm for this problem with query complexity O( k/ε) and time complexity O(n k/ε). This quadratically improves over the query complexity of the previous best quantum junta tester, due to Atıcı and Servedio. Our tester is based on a new quantum algorithm for a gapped version of the combinatorial group testing problem, with an up to quartic improvement over the query complexity of the best classical algorithm. For our upper bound on the time complexity we give a near-linear time implementation of a shallow variant of the quantum Fourier transform over the symmetric group, similar to the Schur-Weyl transform. We also prove a lower bound of Ω(k 1/3 ) queries for junta-testing (for constant ε).Time-efficient implementation. Our algorithm is one of the few quantum algorithms derived from the adversary bound with a time-efficient implementation, i.e., one that is efficient in total number of gates as well as in total number of queries (in general, the time complexity of the adversary-derived algorithm can be exponentially large in the number of input bits to the problem). Other examples are the formula-evaluation algorithm of Reichardt andŠpalek [49] and the algorithm for st-connectivity of Belovs and Reichardt [13].The time complexity of our algorithm is O(n k/d), roughly n times its query complexity. This is probably the best one can hope for: the oracle takes an n-qubit input register, so it takes Ω(n) gates just to touch all those qubits. Thus, any algorithm trying to beat our running time can only afford to change a small fraction of the string given to the input oracle. Also, realistic oracles will typically take time Ω(n) to answer the query.The key to our time-efficient algorithm is an efficient, O(n)-time, implementation of the quantum Fourier transform (QFT) on the linear space which we denote by M n . It is of dimension 2 n and has an orthonormal basis indexed by the set of all subsets of [n]. The symmetric group S n acts naturally on this space by permuting its basis elements, hence M n can be considered as an S n -module (a representation of S n ).Most of the previous work in this direction focused on the regular representation of S n , called the QFT over the symmetric group. The most efficient implementation of this kind is due to Kawano and Sekigawa [40] and can be implemented in depth O(n 3 ), improving over [7,45,39]. Our implementation, on the other hand, is close to the efficient quantum Schur-Weyl transform of Bacon, Chuang and Harrow [5,6], though their algorithm is defined for another group, namely a product of a general linear group and a symmetric group. To the best of our knowledge, this is the first "algorithmic" application of this transformation ([6] lists a number of applications of this transformation for quantum protocols). In order to ensure that the tran...

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Section: Efficient Implementation Of R λmentioning

confidence: 99%

“…The lemma follows from general theory [50, Sections 2.9 and 2.10]. The Appendix of [10] contains a short proof, see also Remark 5.12 below. Figure 1 depicts the different subspaces involved in the decomposition of M .…”

Section: Efficient Implementation Of R λmentioning

confidence: 99%

Efficient Quantum Algorithms for (Gapped) Group Testing and Junta Testing

Ambainis

Belovs

Regev

et al. 2015

Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms

Self Cite

View full text Add to dashboard Cite

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“…Also, using Fact 8, we have (9). Now let us prove (10). Note that L 2 ψ is the projector onto the 2-dimensional space spanned by e 0 and ψ, hence, we can write…”

Section: A Proof Of Theoremmentioning

confidence: 96%

“…Simple linear algebra shows O p − O q = O(α). (One way to see this is by using (10) and observing that L µp − L µq = µ p − µ q .) Let A O be a query algorithm making t queries to O and distinguishing O p from O q .…”

Section: Analysis In Model (Iii)mentioning

confidence: 99%

“…It is stated in the form of a relative γ 2 -norm and generalises the dual formulation of the general adversary bound [31,32] for function evaluation, as well as for other problems [5,27]. The dual adversary bound has been used rather successfully in construction of quantum algorithms, as in terms of span programs and learning graphs [9,26,13,8,24], as in an unrelated fashion [10,4]. Our work gives yet another application of these techniques for construction of quantum algorithms.…”

Section: Introductionmentioning

confidence: 99%

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Quantum Algorithms for Classical Probability Distributions

Belovs

2019

Preprint

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We study quantum algorithms working on classical probability distributions. We formulate four different models for accessing a classical probability distribution on a quantum computer, which are derived from previous work on the topic, and study their mutual relationships.Additionally, we prove that quantum query complexity of distinguishing two probability distributions is given by their inverse Hellinger distance, which gives a quadratic improvement over classical query complexity for any pair of distributions.The results are obtained by using the adversary method for state-generating input oracles and for distinguishing probability distributions on input strings.

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Quantum algorithms: an overview

2016

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Quantum computers are designed to outperform standard computers by running quantum algorithms. Areas in which quantum algorithms can be applied include cryptography, search and optimisation, simulation of quantum systems and solving large systems of linear equations. Here we briefly survey some known quantum algorithms, with an emphasis on a broad overview of their applications rather than their technical details. We include a discussion of recent developments and near-term applications of quantum algorithms. INTRODUCTIONA quantum computer is a machine designed to use quantum mechanics to do things which cannot be done by any machine based only on the laws of classical physics. Eventual applications of quantum computing range from breaking cryptographic systems to the design of new medicines. These applications are based on quantum algorithms-algorithms that run on a quantum computer and achieve a speedup, or other efficiency improvement, over any possible classical algorithm. Although large-scale general-purpose quantum computers do not yet exist, the theory of quantum algorithms has been an active area of study for over 20 years. Here we aim to give a broad overview of quantum algorithmics, focusing on algorithms with clear applications and rigorous performance bounds, and including recent progress in the field.Contrary to a rather widespread popular belief that quantum computers have few applications, the field of quantum algorithms has developed into an area of study large enough that a brief survey such as this cannot hope to be remotely comprehensive. Indeed, at the time of writing the 'Quantum Algorithm Zoo' website cites 262 papers on quantum algorithms. 1 There are now a number of excellent surveys about quantum algorithms, 2-5 and we defer to these for details of the algorithms we cover here, and many more. In particular, we omit all discussion of how the quantum algorithms mentioned work. We will also not cover the important topics of how to actually build a quantum computer 6 (in theory or in practice) and quantum error-correction, 7 nor quantum communication complexity 8 or quantum Shannon theory. 9

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Quantum Algorithms for Learning Symmetric Juntas via Adversary Bound

Cited by 11 publications

References 43 publications

Efficient Quantum Algorithms for (Gapped) Group Testing and Junta Testing

Efficient Quantum Algorithms for (Gapped) Group Testing and Junta Testing

Quantum Algorithms for Classical Probability Distributions

Quantum algorithms: an overview

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