Quantum Speedups for Exponential-Time Dynamic Programming Algorithms

Ambainis, Andris; Balodis, Kaspars; Iraids, Janis; Kokainis, Martins; Prūsis, Krišjānis; Vihrovs, Jevgēnijs

doi:10.1137/1.9781611975482.107

Cited by 54 publications

(68 citation statements)

References 19 publications

(15 reference statements)

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“…3SAT is the canonical example of the so-called NP-complete problems, believed to be exponentially difficult even for quantum computers. Nevertheless, quantum computers can still accelerate their solving [1,2], and given their ubiquity, they may become one of the most important applications of quantum computers. However, the best quantum algorithms, which "quantum-enhance" classical SAT solvers [1], require many qubits and are not directly applicable given small quantum computers.…”

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confidence: 99%

Computational Speedups Using Small Quantum Devices

Dunjko

Cirac

2018

Phys. Rev. Lett.

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Suppose we have a small quantum computer with only M qubits. Can such a device genuinely speed up certain algorithms, even when the problem size is much larger than M ? Here we answer this question to the affirmative. We present a hybrid quantum-classical algorithm to solve 3SAT problems involving n ≫ M variables that significantly speeds up its fully classical counterpart. This question may be relevant in view of the current quest to build small quantum computers.

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confidence: 99%

Computational Speedups Using Small Quantum Devices

Dunjko

Cirac

2018

Phys. Rev. Lett.

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“…Quantum algorithms for GBP and some other NP-hard problems were studied in [38]. Theoretical results of speedups were presented, albeit without any numerical results.…”

Section: Approximation Results About the Bandwidthmentioning

confidence: 99%

On the Embed and Project Algorithm for the Graph Bandwidth Problem

Povh

2021

Mathematics

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The graph bandwidth problem, where one looks for a labeling of graph vertices that gives the minimum difference between the labels over all edges, is a classical NP-hard problem that has drawn a lot of attention in recent decades. In this paper, we focus on the so-called Embed and Project Algorithm (EPA) introduced by Blum et al. in 2000,which in the main part has to solve a semidefinite programming relaxation with exponentially many linear constraints. We present several theoretical properties of this special semidefinite programming problem (SDP) and a cutting-plane-like algorithm to solve it, which works very efficiently in combination with interior-point methods or with the bundle method. Extensive numerical results demonstrate that this algorithm, which has only been studied theoretically so far, in practice gives very good labeling for graphs with n≤1000.

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“…While exhaustive search is believed to be the fastest classical approach for several NP-complete problems including satisfiability and hitting-set [8], there are much better classical algorithms using dynamic programming, inclusionexclusion and other structural approaches for problems such as graph coloring [4], [10], [18], the traveling salesman problem [13], [24], set cover [15] etc. Several authors have obtained quantum speedup on these classical algorithms [2], [30], [33]; however, all of these algorithms have the limitation that they cannot be easily generalized to the list coloring problem.…”

Section: Introductionmentioning

confidence: 99%