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Exploring the quantum speed limit with computer games

This article was retracted on 22 July 2020

An Addendum to this article was published on 05 May 2020


Humans routinely solve problems of immense computational complexity by intuitively forming simple, low-dimensional heuristic strategies1,2. Citizen science (or crowd sourcing) is a way of exploiting this ability by presenting scientific research problems to non-experts. ‘Gamification’—the application of game elements in a non-game context—is an effective tool with which to enable citizen scientists to provide solutions to research problems. The citizen science games Foldit3, EteRNA4 and EyeWire5 have been used successfully to study protein and RNA folding and neuron mapping, but so far gamification has not been applied to problems in quantum physics. Here we report on Quantum Moves, an online platform gamifying optimization problems in quantum physics. We show that human players are able to find solutions to difficult problems associated with the task of quantum computing6. Players succeed where purely numerical optimization fails, and analyses of their solutions provide insights into the problem of optimization of a more profound and general nature. Using player strategies, we have thus developed a few-parameter heuristic optimization method that efficiently outperforms the most prominent established numerical methods. The numerical complexity associated with time-optimal solutions increases for shorter process durations. To understand this better, we produced a low-dimensional rendering of the optimization landscape. This rendering reveals why traditional optimization methods fail near the quantum speed limit (that is, the shortest process duration with perfect fidelity)7,8,9. Combined analyses of optimization landscapes and heuristic solution strategies may benefit wider classes of optimization problems in quantum physics and beyond.

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Figure 1: The BringHomeWater challenge as seen by the player.
Figure 2: Fidelities of the transport problem for different solution durations.
Figure 3: Shovelling and tunnelling clans.
Figure 4: Optimization landscapes.

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We thank the Quantum Moves players, without whom this work would have been impossible. We thank J. Rafner for graphical support and J. Jarecki, O. Vuculescu and C. Bergenholtz for discussions. This work was supported by the European Research Council, the Lundbeck Foundation, the Aarhus University Research Foundation, the Templeton Foundation, the Danish Council for Independent Research, the Villum Foundation and the Carlsberg Foundation.

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Authors and Affiliations



All authors contributed to the construction of the online game platform and the effort to enlist users. J.J.W.H.S., M.K.P., T.P., M.G.A., M.G., K.M. and J.F.S. participated in the numerical analysis of the player and computer results. All authors contributed to the writing of the manuscript.

Corresponding author

Correspondence to Jacob F. Sherson.

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The authors declare no competing financial interests.

Extended data figures and tables

Extended Data Figure 1 Deviations from sin2(•) behaviour.

Colouring as in Fig. 2. a, The fidelity as a function of duration for the optimal families rescaled by the apparent QSL. The TQSL is found for each solution by fitting sin2(aT + b) where a and b are fitting parameters. The black line shows sin2(π/2T) for reference. The inset shows the infidelity (1 − F) as a function of duration close to the . b, The direct Hilbert velocity 〈QT for the best solutions found by the HILO (purple), tunnelling (yellow) and shovelling (blue) clans respectively. Note that 〈QT is only defined for durations with F < 1, so the curves end at different durations. The varying direct Hilbert velocity explains the deviation from in a.

Extended Data Figure 2 The reachability plot for Fig. 3a.

The distance between subsequent solutions as calculated by equation (1) in a list sorted according to the pairwise distance between solutions. The index is the position of the solutions in the list. Valleys identify closely spaced solutions, denoted clans. The valleys corresponding to tunnelling and shovelling clans are marked with yellow and blue respectively. The black line marks the threshold for the distance between consecutive solutions in a clan at 0.05.

Extended Data Figure 3 Solutions from CHOP and HILO.

The amplitude and the position of the tweezer as a function of time for the best player solutions in the tunnelling (yellow) and shovelling (blue) clans and the best HILO (purple). The dashed purple line shows the initial seed used by the best HILO solution (note that the dashed and solid purple lines in the top panel overlap). The total duration is T = 0.15.

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Supplementary information

Supplementary Information

The initial version of Quantum Moves: This .jar version includes all the levels used in the original Quantum Moves, on which the results presented here are based. For the new and improved version of Quantum Moves, see (ZIP 20131 kb)

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Sørensen, J., Pedersen, M., Munch, M. et al. Exploring the quantum speed limit with computer games. Nature 532, 210–213 (2016).

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