Exact Quantum Query Complexity of $$\text {EXACT}_{k,l}^n$$

Ambainis, Andris; Iraids, Janis; Nagaj, Daniel

doi:10.1007/978-3-319-51963-0_19

Cited by 11 publications

(13 citation statements)

References 12 publications

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“…Ambainis, Iraids and Smotrovs [AIS13] showed a tight bound on the exact quantum query complexity of EXACT k (the Boolean function that outputs 1 iff the Hamming weight of the input equals k) for all k, and TH k (the Boolean threshold function that outputs 1 iff the Hamming weight of the input is at least k) for all k. Ambainis, Gruska and Zheng [AGZ15] showed that a function on n input variables has exact quantum query complexity n iff it is equal to AND n up to negations and permutations of the input variables and negation of the output. More recently, Ambainis, Iraids and Nagaj [AIN17] showed a tight bound for the exact quantum query complexity of EXACT k,ℓ (the Boolean function that outputs 1 iff the Hamming weight of the input equals k or ℓ) for all k, ℓ.…”

Section: Introductionmentioning

confidence: 99%

Exact quantum query complexity of computing Hamming weight modulo powers of two and three

Cornelissen¹,

Mande²,

Ozols³

et al. 2021

Preprint

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We study the problem of computing the Hamming weight of an n-bit string modulo m, for any positive integer m ≤ n whose only prime factors are 2 and 3. We show that the exact quantum query complexity of this problem is ⌈n(1 − 1/m)⌉. The upper bound is via an iterative query algorithm whose core components are:• the well-known 1-query quantum algorithm (essentially due to Deutsch) to compute the Hamming weight a 2-bit string mod 2 (i.e., the parity of the input bits), and• a new 2-query quantum algorithm to compute the Hamming weight of a 3-bit string mod 3. We show a matching lower bound (in fact for arbitrary moduli m) via a variant of the polynomial method [de Wolf, SIAM J. Comput., 32(3), 2003]. This bound is for the weaker task of deciding whether or not a given n-bit input has Hamming weight 0 modulo m, and it holds even in the stronger non-deterministic quantum query model where an algorithm must have positive acceptance probability iff its input evaluates to 1. For m > 2 our lower bound exceeds n/2, beating the best lower bound provable using the general polynomial method [Theorem 4.3, Beals et al., J. ACM 48(4), 2001].

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Section: Introductionmentioning

confidence: 99%

Exact quantum query complexity of computing Hamming weight modulo powers of two and three

Cornelissen¹,

Mande²,

Ozols³

et al. 2021

Preprint

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“…The exact quantum query complexity for partial functions was dealt with also in [10,15] and for total functions in [5][6][7]28].…”

Section: Exact Query Complexitymentioning

confidence: 99%

“…The exact quantum query complexity for partial functions was dealt with also in Brassard and Høyer (1997) and Deutsch and Jozsa (1992) and for total functions in Ambainis (2013), Ambainis et al (2013Ambainis et al ( , 2015 and Montanaro et al (2015).…”

Section: Exact Query Complexitymentioning

confidence: 99%

Generalizations of the distributed Deutsch–Jozsa promise problem

Gruska

Qiu

Zheng

2015

Math. Struct. Comp. Sci.

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In the distributed Deutsch-Jozsa promise problem, two parties are to determine whether their respective strings x, y ∈ {0, 1} n are at the Hamming distance H(x, y) = 0 or H(x, y) = n 2 . Buhrman et al. (STOC' 98) proved that the exact quantum communication complexity of this problem is O(log n) while the deterministic communication complexity is Ω(n). This was the first impressive (exponential) gap between quantum and classical communication complexity. In this paper, we generalize the above distributed Deutsch-Jozsa promise problem to determine, for any fixed n 2 ≤ k ≤ n, whether H(x, y) = 0 or H(x, y) = k, and show that an exponential gap between exact quantum and deterministic communication complexity still holds if k is an even such that 1 2 n ≤ k < (1 − λ)n, where 0 < λ < 1 2 is given. We also deal with a promise version of the well-known disjointness problem and show also that for this promise problem there exists an exponential gap between quantum (and also probabilistic) communication complexity and deterministic communication complexity of the promise version of such a disjointness problem. Finally, some applications to quantum, probabilistic and deterministic finite automata of the results obtained are demonstrated. (D. Qiu). 2We prove that quantum communication complexity of DISJ λ is not more than log 3 3λ (3 + 2 log n) while the deterministic communication complexity is Ω(n). For example, if λ = 1 4 , then the quantum communication complexity of DISJ λ is not more than 3 + 2log n while the deterministic communication complexity is more than 0.007n. We prove also that probabilistic communication complexity of DISJ λ is not more than log 3 λ log n. Therefore, there is an exponential gap between quantum (and also probabilistic) communication complexity and deterministic communication complexity of the above promise problem.Number of states is a natural complexity measure for all models of finite automata and state complexity of finite automata is one of the research fields with many applications [36]. There is a variety of methods how to prove lower bounds on the state complexity and methods as well as the results of communication complexity are among the main ones [23,25,26]. In this paper we also show how to make use of our new communication complexity results to get new state complexity bounds.The paper is structured as follows. In Section 2 basic needed concepts and notations are introduced and models involved are described in details. Communication complexities and query complexities of the promise problems EQ k and DJ k are investigated in Section 3. Communication complexity of the promise problem DISJ λ is dealt with in Section 4. Applications to finite automata are explored in Section 5. Some open problems are discussed in Section 6. PreliminariesIn this section, we recall some basic definitions about communication complexity, query complexity and quantum finite automata. Concerning basic concepts and notations of quantum information processing, we refer the reader to [18,29]. Communication comp...

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“…For a function f , quantum advantage can be investigated by comparing the exact quantum query complexity

and the deterministic decision tree complexity

[ 1 ], where

denotes the minimum number of queries used by any classical deterministic algorithm. Over the past decade, there have been many results on the quantum query model [ 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 ]. In particular, Ambainis et al [ 14 ] proved that exact quantum algorithms have advantage for almost all Boolean functions in 2015.…”

Section: Introductionmentioning

confidence: 99%

Partial Boolean Functions With Exact Quantum Query Complexity One

Qiu

2021

Entropy

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We provide two sufficient and necessary conditions to characterize any n-bit partial Boolean function with exact quantum query complexity 1. Using the first characterization, we present all n-bit partial Boolean functions that depend on n bits and can be computed exactly by a 1-query quantum algorithm. Due to the second characterization, we construct a function F that maps any n-bit partial Boolean function to some integer, and if an n-bit partial Boolean function f depends on k bits and can be computed exactly by a 1-query quantum algorithm, then F(f) is non-positive. In addition, we show that the number of all n-bit partial Boolean functions that depend on k bits and can be computed exactly by a 1-query quantum algorithm is not bigger than an upper bound depending on n and k. Most importantly, the upper bound is far less than the number of all n-bit partial Boolean functions for all efficiently big n.

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Exact Quantum Query Complexity of $$\text {EXACT}_{k,l}^n$$

Cited by 11 publications

References 12 publications

Exact quantum query complexity of computing Hamming weight modulo powers of two and three

Exact quantum query complexity of computing Hamming weight modulo powers of two and three

Generalizations of the distributed Deutsch–Jozsa promise problem

Partial Boolean Functions With Exact Quantum Query Complexity One

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