A Quantum N-Queens Solver

Valentin Torggler1, Philipp Aumann1, Helmut Ritsch1, and Wolfgang Lechner1,2

1Institute for Theoretical Physics, University of Innsbruck, A-6020 Innsbruck, Austria
2Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences, A-6020 Innsbruck, Austria

The $N$-queens problem is to find the position of $N$ queens on an $N$ by $N$ chess board such that no queens attack each other. The excluded diagonals $N$-queens problem is a variation where queens cannot be placed on some predefined fields along diagonals. This variation is proven NP-complete and the parameter regime to generate hard instances that are intractable with current classical algorithms is known. We propose a special purpose quantum simulator that implements the excluded diagonals $N$-queens completion problem using atoms in an optical lattice and cavity-mediated long-range interactions. Our implementation has no overhead from the embedding allowing to directly probe for a possible quantum advantage in near term devices for optimization problems.

Solving puzzles can be a difficult task even for the fastest digital computers. A generic example already studied by the famous mathematician Gauss is the so-called N-queens problem. The task is to find all possibilities to place N queens on an NxN chess board, such that they cannot attack each other according to chess rules. Recently it was shown that variations of the problem with extra constraints are not solvable with classical algorithms within reasonable time.

In the last decades the idea emerged that special purpose quantum computers might find a solution faster. However, whether quantum effects can indeed speed up such calculations is still an open question and thus it is crucial to provide experimental possibilities to gauge an advantages of quantum computing.

In this work we propose a special purpose analog computer serving as a test bed to study speedup for the N-queens problem in the quantum regime. The idea is to build a miniaturized chess board from atoms and light at nearly zero temperature, where the rules of quantum mechanics apply. Ultra-cold atoms moving via quantum tunneling in a lattice created by laser light, take the role of the queens on the chess board. The chess rules are imposed by placing the atoms in an optical resonator and shining in specially structured light. Due to the one-to-one natural design of the experiment there is no overhead in atoms and fairly small system sizes are sufficient to enter the classically intractable regime.

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Cited by

[1] Ronen M. Kroeze, Yudan Guo, Varun D. Vaidya, Jonathan Keeling, and Benjamin L. Lev, "Spinor Self-Ordering of a Quantum Gas in a Cavity", Physical Review Letters 121 16, 163601 (2018).

[2] François Damanet, Andrew J. Daley, and Jonathan Keeling, "Atom-only descriptions of the driven-dissipative Dicke model", Physical Review A 99 3, 033845 (2019).

[3] Rounak Jha, Debaiudh Das, Avinash Dash, Sandhya Jayaraman, Bikash K. Behera, and Prasanta K. Panigrahi, "A Novel Quantum N-Queens Solver Algorithm and its Simulation and Application to Satellite Communication Using IBM Quantum Experience", arXiv:1806.10221.

[4] Philipp Hauke, Helmut G. Katzgraber, Wolfgang Lechner, Hidetoshi Nishimori, and William D. Oliver, "Perspectives of quantum annealing: Methods and implementations", arXiv:1903.06559.

[5] Peter Kirton, Mor M. Roses, Jonathan Keeling, and Emanuele G. Dalla Torre, "Introduction to the Dicke model: from equilibrium to nonequilibrium, and vice versa", arXiv:1805.09828.

[6] Emily J. Davis, Gregory Bentsen, Lukas Homeier, Tracy Li, and Monika H. Schleier-Smith, "Photon-Mediated Spin-Exchange Dynamics of Spin-1 Atoms", Physical Review Letters 122 1, 010405 (2019).

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