Any AND-OR Formula of Size <i>N</i> Can Be Evaluated in Time $N^{1/2+o(1)}$ on a Quantum Computer

Ambainis, Andris; Childs, Andrew M.; Reichardt, Ben W.; Špalek, Robert; Zhang, S.

doi:10.1137/080712167

Cited by 129 publications

(164 citation statements)

References 24 publications

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“…This suggests that the balanced ternary majority function is significantly different from the balanced k-ary NAND function, for which randomized alphabeta pruning is known to be optimal [40]. In contrast, we show that the optimal quantum algorithm of [4] does extend to give an optimal O(2 d )-query algorithm for evaluating the balanced ternary majority formula. Moreover, the algorithm also generalizes to a significantly larger gate set S.…”

Section: Introductionmentioning

confidence: 80%

“…Although the relationship between span programs, the general adversary bound and quantum query complexity has been resolved, there remain numerous open problems, including most of the problems from our earlier work [4]. Specifically, for evaluating formulas, the optimal quantum query algorithm may or may not also have a time-efficient implementation.…”

Section: Extensions and Open Problemsmentioning

confidence: 99%

“…x N . An optimal, O( √ N)-query quantum algorithm is known to evaluate "approximately balanced" formulas over the gates S = {AND, OR, NOT} [4]. We develop an optimal quantum algorithm for evaluating balanced formulas over a greatly extended gate set.…”

Section: Introductionmentioning

confidence: 99%

“…The classical complexity of evaluating this formula is Ω((5/2) d ), and the previous best quantum algorithm, from [4], uses O( √ 5 d ) queries. The graph gadgets themselves are derived from "span programs" [26], establishing a new connection between span programs and quantum algorithms.…”

Section: Introductionmentioning

confidence: 99%

“…In the other extreme case, Farhi, Goldstone and Gutmann devised a breakthrough algorithm for evaluating the depth-(log 2 N) balanced binary AND-OR formula using N 1/2+o (1) queries [17,15]. Ambainis et al [4] improved this to O( √ N) queries, even for "approximately balanced" formulas, and also extended the algorithm to arbitrary {AND, OR, NOT} formulas with N 1 2 +o(1) queries and time after a preprocessing step. More recent work has given a general O( √ N)-query upper bound [37,36].…”

Section: Introductionmentioning

confidence: 99%

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Reichardt¹,

Špalek²

2012

Theory of Comput.

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Section: Introductionmentioning

confidence: 80%

Section: Extensions and Open Problemsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Reichardt¹,

Špalek²

2012

Theory of Comput.

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A New Type of Classical Logic Circuit with Exponential Speedup

Ying-ji

Chen

Tong

et al. 2023

Advanced Intelligent Systems

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Compared with classical algorithms, quantum algorithms can show the advantage of exponential speedup in solving some problems. Solving the NAND‐Tree problem is a typical example, which can be speed up exponentially by quantum walk. This scheme is groundbreaking due to the universality of the negative‐AND (NAND) gate. In fact, the NAND gate is also universal for classical computation, making it an important target for development. Herein, the quantum NAND‐Tree onto the classical circuit network is mapped and a new type of classical logic circuit with the NAND gate that can perform the quantum algorithms is designed. These classical logic gates that are designed have exponential speedup functions compared with conventional ones. Because classical circuit networks possess good scalability and stability, the present scheme is expected to have potential applications for information processing in the era of big data.

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Quantum Combinational Logics and their Realizations with Circuits

Tong,

Chen,

Pan

et al. 2023

Adv Quantum Tech

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Classical combinational logic circuits (CCLCs) are widely used in various fields. Corresponding to the CCLCs, here schemes are given for some quantum combinational logic circuits (QCLCs) based on the quantum NAND tree. Three typical circuits, adder, comparator, and seven‐segment display decoder, are discussed in detail as examples. All the designs of the schemes are based on the quantum random walk theory. Furthermore, these QCLCs are mapped onto the classical circuit networks and design new types of CCLCs, and take advantage of the fact that there is a good correspondence between the voltage in the circuit satisfying Kirchhoff's law and the system wave function satisfying the Schrodinger equation. These CCLCs that are designed have exponential speedup functions compared with conventional ones, which have been demonstrated experimentally. Because classical circuit networks possess good scalability and stability, the realization of QCLCs on classical circuits is expected to have potential applications for information processing in the era of big data.

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Any AND-OR Formula of Size N Can Be Evaluated in Time $N^{1/2+o(1)}$ on a Quantum Computer

Cited by 129 publications

References 24 publications

Untitled

Untitled

A New Type of Classical Logic Circuit with Exponential Speedup

Quantum Combinational Logics and their Realizations with Circuits

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