Quantum speedup of branch-and-bound algorithms

Montanaro, Ashley

doi:10.1103/physrevresearch.2.013056

Cited by 49 publications

(53 citation statements)

References 96 publications

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“…Forthcoming work applying similar techniques to Max2SAT (Callison et al 2020b) will characterise the hardness of small random instances in more detail, and establish quantum walks as an effective tool for hard optimization problems more generally. While general methods are known to speed up the best classical algorithms (Hartwig et al 1984) for this type of problem (Montanaro 2018(Montanaro , 2019, further work is required to determine whether an optimal continuous-time quantum walk algorithm can be devised that fully leverages the advantage from the correlations. Nonetheless, our work represents a significant advance in developing continuous-time quantum walk computation for hard optimization problems, and provides key insights into the computational mechanisms that can be exploited over short timescales, well-suited to the limited coherence times of noisy, intermediate scale quantum hardware.…”

Section: Discussionmentioning

confidence: 99%

Finding spin glass ground states using quantum walks

et al. 2019

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Quantum computation using continuous-time evolution under a natural hardware Hamiltonian is a promising near-and mid-term direction toward powerful quantum computing hardware. We investigate the performance of continuous-time quantum walks as a tool for finding spin glass ground states, a problem that serves as a useful model for realistic optimization problems. By performing detailed numerics, we uncover significant ways in which solving spin glass problems differs from applying quantum walks to the search problem. Importantly, unlike for the search problem, parameters such as the hopping rate of the quantum walk do not need to be set precisely for the spin glass ground state problem. Heuristic values of the hopping rate determined from the energy scales in the problem Hamiltonian are sufficient for obtaining a better quantum advantage than for search. We uncover two general mechanisms that provide the quantum advantage: matching the driver Hamiltonian to the encoding in the problem Hamiltonian, and an energy redistribution principle that ensures a quantum walk will find a lower energy state in a short timescale. This makes it practical to use quantum walks for solving hard problems, and opens the door for a range of applications on suitable quantum hardware.

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Section: Discussionmentioning

confidence: 99%

Finding spin glass ground states using quantum walks

et al. 2019

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“…In this case, the best classical algorithm is based on a different method, backtracking which performs a depth-first search on a search tree of an unknown structure. The quantum algorithm follows by applying a result of Montanaro [Mon15] who constructed a quantum backtracking algorithm with a nearly quadratic advantage over its classical counterparts (with further developments in [AK17]). This work has no implications for TSP on general graphs.…”

Section: Prior Workmentioning

confidence: 99%

Quantum Speedups for Exponential-Time Dynamic Programming Algorithms

Ambainis

Balodis

Iraids

et al. 2019

Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms

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In this paper we study quantum algorithms for NP-complete problems whose best classical algorithm is an exponential time application of dynamic programming. We introduce the path in the hypercube problem that models many of these dynamic programming algorithms. In this problem we are asked whether there is a path from 0 n to 1 n in a given subgraph of the Boolean hypercube, where the edges are all directed from smaller to larger Hamming weight. We give a quantum algorithm that solves path in the hypercube in time O * (1.817 n ). The technique combines Grover's search with computing a partial dynamic programming table. We use this approach to solve a variety of vertex ordering problems on graphs in the same time O * (1.817 n ), and graph bandwidth in time O * (2.946 n ). Then we use similar ideas to solve the travelling salesman problem and minimum set cover in time O * (1.728 n ).

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“…They claim that it affects the security estimates of several lattice-based submissions to the NIST post-quantum standardization process [8]. In turn, Montanaro [24] has presented how to get a quantum speedup of branch-and-bound algorithms by using Ambainis-Kokainis' work and his own. Thus, Montanaro's algorithm is of high interest in computing science and cryptography.…”

Section: Introductionmentioning

confidence: 99%

Practical Implementation of a Quantum Backtracking Algorithm

Martiel

Remaud

2020

SOFSEM 2020: Theory and Practice of Computer Science

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In previous work, Montanaro presented a method to obtain quantum speedups for backtracking algorithms, a general meta-algorithm to solve constraint satisfaction problems (CSPs). In this work, we derive a space efficient implementation of this method. Assume that we want to solve a CSP with m constraints on n variables and that the union of the domains in which these variables take their value is of cardinality d. Then, we show that the implementation of Montanaro's backtracking algorithm can be done by using O(n log d) data qubits. We detail an implementation of the predicate associated to the CSP with an additional register of O(log m) qubits. We explicit our implementation for graph coloring and SAT problems, and present simulation results. Finally, we discuss the impact of the usage of static and dynamic variable ordering heuristics in the quantum setting.

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Quantum speedup of branch-and-bound algorithms

Cited by 49 publications

References 96 publications

Finding spin glass ground states using quantum walks

Finding spin glass ground states using quantum walks

Quantum Speedups for Exponential-Time Dynamic Programming Algorithms

Practical Implementation of a Quantum Backtracking Algorithm

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