The Space "Just Above" BQP

Aaronson, Scott; Bouland, Adam; Fitzsimons, Joseph F.; Lee, Mitchell

doi:10.1145/2840728.2840739

Cited by 11 publications

(36 citation statements)

References 19 publications

(45 reference statements)

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“…Conversely, we will show that regular languages not in AC 0 have quantum query complexity Ω(n). Thus, another way to state the trichotomy is that very roughly speaking regular languages in NC 0 have complexity O (1), regular languages in AC 0 but not NC 0 have complexityΘ( √ n), and everything else has complexity Ω(n). enjoy a myriad of equivalent characterizations.…”

Section: Generalization Of Grover's Algorithmmentioning

confidence: 99%

“…This will be important for the next logical step in the trichotomy theorem: proving lower bounds to match our upper bounds in Section 6. The core of this section is a dichotomy theorem for sensitivity, namely that the sensitivity is either O (1) or Ω(n). This implies an identical dichotomy for block sensitivity, from which the Ω( √ n) lower bound on approximate degree follows for all nontrivial languages.…”

Section: Dichotomy Theoremsmentioning

confidence: 99%

“…Notice that for a given length n, the sensitivity of L is exactly the weight of the maximum Hamming weight string in S L . Suppose the sensitivity is not O (1). Therefore, for any k, there exists a string y k ∈ S L with Hamming weight at least k.…”

Section: Corollary 27 Let L Be a Flat Regular Language The Sensitivmentioning

confidence: 99%

“…This implies that sensitivity is Ω(n) for infinitely many n. We can also pump down arbitrary blocks of p bits to decrease the length, so we can make sure sensitivity is Ω(n) for at least 1 p fraction of n. Finally, since L is flat, congruence classes contain strings of all length, which allows us to replace some substring of a Ω(n) sensitive string with a slightly longer or shorter string. In this way, we can construct strings of sensitivity Ω(n) for all n. (1) or Ω(n).…”

Section: Corollary 27 Let L Be a Flat Regular Language The Sensitivmentioning

confidence: 99%

“…If all L r have constant query complexity, then Q(L) = 0 or Q(L) = Θ(1). Therefore, assume there is some L r such that Q(L r ) = ω (1). We will show that Q(L) = Θ(max r Q(L r )).…”

Section: A1 Monotonic Query Complexitymentioning

confidence: 99%

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A Quantum Query Complexity Trichotomy for Regular Languages

Aaronson

Grier

Schaeffer

2019

2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS)

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We present a trichotomy theorem for the quantum query complexity of regular languages. Every regular language has quantum query complexity Θ(1),Θ( √ n), or Θ(n). The extreme uniformity of regular languages prevents them from taking any other asymptotic complexity. This is in contrast to even the context-free languages, which we show can have query complexity Θ(n c ) for all computable c ∈ [1/2, 1]. Our result implies an equivalent trichotomy for the approximate degree of regular languages, and a dichotomy-either Θ(1) or Θ(n)-for sensitivity, block sensitivity, certificate complexity, deterministic query complexity, and randomized query complexity.The heart of the classification theorem is an explicit quantum algorithm which decides membership in any star-free language inÕ( √ n) time. This well-studied family of the regular languages admits many interesting characterizations, for instance, as those languages expressible as sentences in first-order logic over the natural numbers with the less-than relation. Therefore, not only do the star-free languages capture functions such as OR, they can also express functions such as "there exist a pair of 2's such that everything between them is a 0." Thus, we view the algorithm for star-free languages as a nontrivial generalization of Grover's algorithm which extends the quantum quadratic speedup to a much wider range of stringprocessing algorithms than was previously known. We show a variety of applications-new quantum algorithms for dynamic constant-depth Boolean formulas, balanced parentheses nested constantly many levels deep, binary addition, a restricted word break problem, and pathdiscovery in narrow grids-all obtained as immediate consequences of our classification theorem.

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Section: Generalization Of Grover's Algorithmmentioning

confidence: 99%

Section: Dichotomy Theoremsmentioning

confidence: 99%

Section: Corollary 27 Let L Be a Flat Regular Language The Sensitivmentioning

confidence: 99%

Section: Corollary 27 Let L Be a Flat Regular Language The Sensitivmentioning

confidence: 99%

“…If all L r have constant query complexity, then Q(L) = 0 or Q(L) = Θ(1). Therefore, assume there is some L r such that Q(L r ) = ω (1). We will show that Q(L) = Θ(max r Q(L r )).…”

Section: A1 Monotonic Query Complexitymentioning

confidence: 99%

See 3 more Smart Citations

A Quantum Query Complexity Trichotomy for Regular Languages

Aaronson

Grier

Schaeffer

2019

2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS)

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Oracles and Query Lower Bounds in Generalised Probabilistic Theories

2018

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We investigate the connection between interference and computational power within the operationally defined framework of generalised probabilistic theories. To compare the computational abilities of different theories within this framework we show that any theory satisfying four natural physical principles possess a well-defined oracle model. Indeed, we prove a subroutine theorem for oracles in such theories which is a necessary condition for the oracle model to be well-defined. The four principles are: causality (roughly, no signalling from the future), purification (each mixed state arises as the marginal of a pure state of a larger system), strong symmetry (existence of a rich set of nontrivial reversible transformations), and informationally consistent composition (roughly: the information capacity of a composite system is the sum of the capacities of its constituent subsystems). Sorkin has defined a hierarchy of conceivable interference behaviours, where the order in the hierarchy corresponds to the number of paths that have an irreducible interaction in a multi-slit experiment. Given our oracle model, we show that if a classical computer requires at least n queries to solve a learning problem, because fewer queries provide no information about the solution, then the corresponding “no-information” lower bound in theories lying at the kth level of Sorkin’s hierarchy is . This lower bound leaves open the possibility that quantum oracles are less powerful than general probabilistic oracles, although it is not known whether the lower bound is achievable in general. Hence searches for higher-order interference are not only foundationally motivated, but constitute a search for a computational resource that might have power beyond that offered by quantum computation.

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Deriving Grover's lower bound from simple physical principles

Lee

Selby

2016

New J. Phys.

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Groverʼs algorithm constitutes the optimal quantum solution to the search problem and provides a quadratic speed-up over all possible classical search algorithms. Quantum interference between computational paths has been posited as a key resource behind this computational speed-up. However there is a limit to this interference, at most pairs of paths can ever interact in a fundamental way. Could more interference imply more computational power? Sorkin has defined a hierarchy of possible interference behaviours-currently under experimental investigation-where classical theory is at the first level of the hierarchy and quantum theory belongs to the second. Informally, the order in the hierarchy corresponds to the number of paths that have an irreducible interaction in a multi-slit experiment. In this work, we consider how Groverʼs speed-up depends on the order of interference in a theory. Surprisingly, we show that the quadratic lower bound holds regardless of the order of interference. Thus, at least from the point of view of the search problem, post-quantum interference does not imply a computational speed-up over quantum theory. Groverʼs algorithm [12] provides the optimal quantum solution to the search problem and is one of the most versatile and influential quantum algorithms. The search problem-in its simplest form-asks one to find a single 'marked' item from an unstructured list of N elements by querying an oracle which can recognise the marked item. The importance of Groverʼs algorithm stems from the ubiquitous nature of the search problem and its relation to solving NP-complete problems [6]. Classical computers require ( ) O N queries to solve this problem, but quantum computers-using Groverʼs algorithm-only require ( ) O N queries. Quantum interference between computational paths has been posited [32] as a key resource behind this computational 'speed-up'. However, as first noted by Sorkin [29,30], there is a limit to this interference-at most pairs of paths can ever interact in a fundamental way. Could more interference imply more computational power?Sorkin has defined a hierarchy of possible interference behaviours-currently under experimental investigation [24, 27, 28]-where classical theory is at the first level of the hierarchy and quantum theory belongs to the second. Informally, the order in the hierarchy corresponds to the number of paths that have an irreducible interaction in a multi-slit experiment. To get a greater understanding of the role of interference in computation, we consider how Groverʼs speed-up depends on the order of interference in a theory.Restriction to the second level of this hierarchy implies many 'quantum-like' features, which, at first glance, appear to be unrelated to interference. For example, such interference behaviour restricts correlations [11] to the 'almost quantum correlations' discussed in [21], and bounds contextuality in a manner similar to quantum theory [14,23]. This, in conjunction with interference being a key resource in the quantum speed-up, suggests that...

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The Space "Just Above" BQP

Cited by 11 publications

References 19 publications

A Quantum Query Complexity Trichotomy for Regular Languages

A Quantum Query Complexity Trichotomy for Regular Languages

Oracles and Query Lower Bounds in Generalised Probabilistic Theories

Deriving Grover's lower bound from simple physical principles

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