Stochastic gradient ascent outperforms gamers in the Quantum Moves game

In a recent work on quantum state preparation, Sørensen and co-workers [Nature (London) 532, 210 (2016)] explore the possibility of using video games to help design quantum control protocols. The authors present a game called “Quantum Moves” (https://www.scienceathome.org/games/quantum-moves/) in which gamers have to move an atom from A to B by means of optical tweezers. They report that, “players succeed where purely numerical optimization fails.” Moreover, by harnessing the player strategies, they can “outperform the most prominent established numerical methods.” The aim of this Rapid Communication is to analyze the problem in detail and show that those claims are untenable. In fact, without any prior knowledge and starting from a random initial seed, a simple stochastic local optimization method finds near-optimal solutions which outperform all players. Counterdiabatic driving can even be used to generate protocols without resorting to numeric optimization. The analysis results in an accurate analytic estimate of the quantum speed limit which, apart from zero-point motion, is shown to be entirely classical in nature. The latter might explain why gamers are reasonably good at the game. A simple modification of the BringHomeWater challenge is proposed to test this hypothesis.

Figure 1

Quantum Moves interface. Screenshot of the BringHomeWater challenge developed by Sørensen et al. [8]. In the game, players move an optical tweezer (the blue circle is the cursus to move the tweezer) from an initial point on the left towards an atom trapped in a potential well on the right. The atom is depicted as the purple liquid filling up the potential well. Players have to find a route to bring this liquid to the target (purple shaded region on the left) as fast as possible without spilling it.

Figure 2

Effective potential for moving objects. The potential of a static optical tweezer is depicted by the dashed blue line. The atom, however, feels a different potential once the tweezer is moving. By going to the comoving frame, the atom picks up an additional pseudoforce if the tweezer accelerates, just as a person in an elevator. For a tweezer of finite depth, it unfortunately allows atoms to escape if the acceleration is too large.

Figure 3

Fidelity for different protocol durations. The green shaded region indicates the classical speed limit for moving a particle in a finite depth potential. The fidelity of the associated quantum counterdiabatic driving (dashed green) protocol shows a sharp rise of fidelity near $T_{\mathrm{CSL}}$. For the double tweezer, the protocol is adapted to account for an additional geometric contribution to the gauge potential. Up to $T_{\text{CSL}}$, its performance (blue line) is comparable to that obtained from numeric optimization (black dots). At later times, its performance starts to wane because it is derived from an approximate adiabatic gauge potential. Details on the numeric optimization algorithm can be found in Sec. 4. Gamers' performance and KASS results are shown for comparison and taken from Ref. [8]; for that reason we adopt the same units, $\hslash =m=1$.

Figure 4

Geometry and geodesic protocol. The left panel shows the effective adiabatic metric $g$ for moving the tweezer. In the middle there is a region where a small change in the position of a tweezer can cause a large change in the shape of the potential energy, at which point the metric becomes much larger than one. The resulting geodesic is shown in right panel; note how it slows down where the metric is large. All quantities are expressed in units of $\hslash =m=1$.

Figure 5

Fidelity of superposition. Instead of having to move the atom from one tweezer to the other, the goal is to make an equal superposition of the ground state in the tweezer at $L/2$ and at $-L/2$. The green dots show the final outcome of the stochastic ascend method for 200 random initial seeds and the black line shows the best out of that set. For comparison, the fidelity for the original BringHomeWater challenge is also shown in gray. In the BringHomeWater challenge, the difference between the worst and best fidelity, obtained from 200 random initial seeds, was only about 1%. In the present challenge, there is an intermediate regime where the spread is more than 20%, indicating that the problem is indeed much harder (see also Ref. [17]). All quantities are expressed in units of $\hslash =m=1$.

Figure 6

Stochastic fidelity traces for $T=0.1$. The figure shows how 100 initial random protocols move through the fidelity landscape. There are $M=128$ settings and $N=40$ time steps, such that a single sweep over the lattice results in 5120 fidelity evaluations. A large fraction of protocols has converged in about 20 000 steps, which is as little as four sweeps. The slowest protocol took 22 sweeps to end up in an optimum. A single sweep runs in just a few seconds on a laptop.