A Quantum Query Complexity Trichotomy for Regular Languages

Aaronson, Scott; Grier, Daniel; Schaeffer, Luke

doi:10.1109/focs.2019.00061

Cited by 15 publications

(28 citation statements)

References 29 publications

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“…Active learning, which is a framework introduced by Angluin [7], applies to interactive data mining tools [22]- [24], quantum algorithms [25]- [27] and so on. Angluin showed that the class of regular languages, each of which is characterized by the set of strings accepted by a deterministic finite automaton, is learned from a Minimally Adequate Teacher (MAT) answering membership and equivalence queries.…”

Section: Practical Applications Of Learning Regular Pattern Languagesmentioning

confidence: 99%

An Efficient Learning Algorithm for Regular Pattern Languages Using One Positive Example and a Linear Number of Membership Queries

Matsumoto

Uchida

Shoudai

et al. 2020

IEICE Trans. Inf. & Syst.

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A regular pattern is a string consisting of constant symbols and distinct variable symbols. The language of a regular pattern is the set of all constant strings obtained by replacing all variable symbols in the regular pattern with non-empty strings. The present paper deals with the learning problem of languages of regular patterns within Angluin's query learning model, which is an established mathematical model of learning via queries in computational learning theory. The class of languages of regular patterns was known to be identifiable from one positive example using a polynomial number of membership queries, in the query learning model. In present paper, we show that the class of languages of regular patterns is identifiable from one positive example using a linear number of membership queries, with respect to the length of the positive example.

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Section: Practical Applications Of Learning Regular Pattern Languagesmentioning

confidence: 99%

An Efficient Learning Algorithm for Regular Pattern Languages Using One Positive Example and a Linear Number of Membership Queries

Matsumoto

Uchida

Shoudai

et al. 2020

IEICE Trans. Inf. & Syst.

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“…Aaronson, Grier and Schaeffer [1] have recently shown that any regular language L may have one of three possible quantum query complexities on inputs of length n: Θ(1) if the language can be decided by looking at O(1) first or last symbols of a word; Θ( √ n) if the best way to decide L is Grover's search (for example, for the language consisting of all words containing at least one letter a); Θ(n) for languages in which one can embed counting modulo some number p which has quantum query complexity Θ(n) (for example, the binary XOR function).…”

Section: Introductionmentioning

confidence: 99%

“…As shown in [1], a regular language being of complexity Õ( √ n) (which includes the first two cases of the list above) is equivalent to it being star-free. Star-free languages are defined as the languages which have regular expressions not containing the Kleene star (if it is allowed to use the complement operation).…”

Section: Introductionmentioning

confidence: 99%

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Quantum algorithm for Dyck Language with Multiple Types of Brackets

Khadiev¹,

Kravchenko²

2021

Preprint

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We consider the recognition problem of the Dyck Language generalized for multiple types of brackets. We provide an algorithm with quantum query complexity O( √ n(log n) 0.5k ), where n is the length of input and k is the maximal nesting depth of brackets. Additionally, we show the lower bound for this problem which is O( √ nc k ) for some constant c. Interestingly, classical algorithms solving the Dyck Language for multiple types of brackets substantially differ form the algorithm solving the original Dyck language. At the same time, quantum algorithms for solving both kinds of the Dyck language are of similar nature and requirements.

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“…Quantum computing has shown promise as an emerging technology to efficiently solve some instances of difficult computing tasks in fields ranging from optimisation (Gilyén et al 2019;Montanaro 2020), linear algebra (Harrow et al 2009;Berry et al 2017), number theory and pattern matching (Montanaro 2016;2017), language processing (Aaronson et al 2019;Wiebe et al 2019), machine learning (McClean et al 2016;Bausch 2018;Wang et al 2019;Li et al 2019), to quantum simulation (Lloyd 1996;Babbush et al 2018;Childs and Su 2019). While quantum computers are not yet robust enough to evaluate any of these applications on sample sizes large enough to claim an empirical advantage, a structured search problem such as language decoding is a prime candidate for a quantum speedup.…”

Section: Introductionmentioning

confidence: 99%

A quantum search decoder for natural language processing

Bausch

Subramanian

Piddock

2021

Quantum Mach. Intell.

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Probabilistic language models, e.g. those based on recurrent neural networks such as long short-term memory models (LSTMs), often face the problem of finding a high probability prediction from a sequence of random variables over a set of tokens. This is commonly addressed using a form of greedy decoding such as beam search, where a limited number of highest-likelihood paths (the beam width) of the decoder are kept, and at the end the maximum-likelihood path is chosen. In this work, we construct a quantum algorithm to find the globally optimal parse (i.e. for infinite beam width) with high constant success probability. When the input to the decoder follows a power law with exponent k > 0, our algorithm has runtime Rnf(R, k), where R is the alphabet size, n the input length; here f < 1/2, and $f\rightarrow 0$ f → 0 exponentially fast with increasing k, hence making our algorithm always more than quadratically faster than its classical counterpart. We further modify our procedure to recover a finite beam width variant, which enables an even stronger empirical speedup while still retaining higher accuracy than possible classically. Finally, we apply this quantum beam search decoder to Mozilla’s implementation of Baidu’s DeepSpeech neural net, which we show to exhibit such a power law word rank frequency.

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A Quantum Query Complexity Trichotomy for Regular Languages

Cited by 15 publications

References 29 publications

An Efficient Learning Algorithm for Regular Pattern Languages Using One Positive Example and a Linear Number of Membership Queries

An Efficient Learning Algorithm for Regular Pattern Languages Using One Positive Example and a Linear Number of Membership Queries

Quantum algorithm for Dyck Language with Multiple Types of Brackets

A quantum search decoder for natural language processing

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