Crowdsourcing human common sense for quantum control

Citizen science methodologies have over the past decade been applied with great success to help solve highly complex numerical challenges. Here, we take early steps in the quantum physics arena by introducing a citizen science game, Quantum Moves 2, and compare the performance of different optimization methods across three different quantum optimal control problems of varying difficulty. Inside the game, players can apply a gradient-based algorithm (running locally on their device) to optimize their solutions and we find that these results perform roughly on par with the best of the tested standard optimization methods performed on a computer cluster. In addition, cluster-optimized player seeds was the only method to exhibit roughly optimal performance across all three challenges. Finally, player seeds show significant statistical advantages over random seeds in the limit of sparse sampling. This highlights the potential for crowdsourcing the solution of future quantum research problems.

Figure 1

Abstract illustration of an optimization (control) landscape with two control parameters. Points in the plane correspond to a particular set of functions $\{u_1\left(t\right),u_2\left(t\right)\}$ and the height of the landscape is given by the associated fidelity. Only a small region of the control space is optimal (with respect to $F\approx 1$) and otherwise contains many local “traps”—the relative efficiency of a local “hillclimbing” optimizer is thus correlated to where it is seeded. Purple full region: Uniform random seeding. Obtaining $F\approx 1$ is possible, but with low probability. Green smaller region: Specialized seeding. Obtaining $F\approx 1$ is possible with high probability, but not guaranteed. Such a region is conventionally targeted by experts through their accumulated knowledge, problem insights, heuristics, and programming proficiency. Nonexpert citizen scientist players may be useful in uncovering this region without extensive prior training.

Figure 2

The three central state transfer problems in Quantum Moves 2. (a) Bring Home Water (single-particle), (b) Splitting (bec), and (c) Shake Up (bec). The instantaneous density ${\left|\psi (x,t)\right|}^2$ (red line with shaded area) must be transferred into the target density $|{\psi }_{\mathrm{tgt}}{\left(x\right)|}^2$ (yellow line) without residual excitation. The trapping potential (green line) is parametrized by the position of the round, draggable cursor, $\left\{u_1\left(t\right),u_2\left(t\right)\right\}=\left\{f_1\left(x_{\mathrm{cursor}}\left(t\right)\right),f_2(y_{\mathrm{cursor}}\left(t\right)\right\})$, which is unable to leave the turquoise bounding box (control boundaries). The functions $f_1$ and $f_2$ are linear. The wave function densities are offset by the potential, ${\left|\psi \left(x\right)\right|}^2+V\left(x\right)$, for illustrative purposes (e.g., the Splitting target density represents two equally sized wave packets trapped in a double well—the large center bump is the barrier and two smaller bumps are the wave packets).

Figure 3

Bring Home Water solution densities for different methods. (a) grape ps, (b) grape rs, (c) sa rs ($n_b=n_t$), and (d) sa rs ($n_b=40$). Each density is normalized for every individual $T$ and thus represents an estimate of the probability distribution for obtaining a particular $F$ for a given $T \left[\mathcal{P}\right(F\left|T\right)]$. The reference curve shows the best obtained results for grape ps. The crosses indicate solutions at densities lower than 0.002.

Figure 4

Aggregate, monotonically best optimization results (lower is better) for several methods in Bring Home Water. Solid lines with (without) symbols show the best results obtained with (without) player influence. The scattered green crosses show all results produced by 536 players seeding on average $\sim 32$ solutions and optimizing approximately $1/4$ of these for $\sim 131$ iterations on average. The scattered blue dots show the same, except the optimization is carried out on all player seeds with the computational resources described in the text (i.e., no player influence after seeding). The symbol translucency indicates the density distribution.

Figure 5

Bring Home Water: Clustering of controls. (a) Each cluster (0 and 1) corresponds to a strategy and the gap between them exhibits an exponential $1-F\left(T\right)$ behavior, leading to different estimates of minimal durations for a given threshold. The unclassified points ($-1$) are mostly populated by controls similar to either strategy, except for a few local defects that makes them appear distant to the cluster with respect to the Euclidean metric. (b) Lines correspond to cluster means and shaded areas to the standard deviation (cluster −1 has a dashed outline). The most populous cluster (0) corresponds to a front-swing strategy with the tweezer immediately being placed in front of the atom whereas the less populous cluster (1) corresponds to a back-swing strategy with the tweezer immediately being placed behind the atom.

Figure 6

(a) grape rs solution densities as a function of number of bins $n_b$ at $T=0.1045$. Each dot denotes a solution in the back-swing strategy. The density in the right black box shows the grape ps solution density from Fig. 3 at the same duration denoted by ps on the $x$ axis (the duration is on the edge of one of the bin limits and the density is therefore taken as the mean of the two neighboring bins). (b) Histogram of the number of back-swing solutions, counted as those with fidelity higher than the best front-swing result at $n_b=n_t$, for grape rs with variable $n_b$ (blue) and grape ps (turquoise) from Fig. 4. The total number of seeds per bin is indicated in the parentheses.

Figure 7

Splitting solution densities for different methods. (a) grape ps, (b) grape rs, (c) sa rs ($n_b=n_t$), and (d) sa rs ($n_b=40$). Each density is normalized for every individual $T$ and thus represents an estimate of the probability distribution for obtaining a particular $F$ for a given $T$ ($\mathcal{P}\left(F\right|T)$). The blue reference curve shows the best obtained results for grape ps. Crosses indicate solutions at densities lower than 0.002.

Figure 8

Aggregate, monotonically best optimization results (lower is better) for several methods in Splitting. Solid lines with (without) symbols show the best results obtained with (without) player influence. The scattered green crosses show all results produced by 193 players seeding on average $\sim 28$ solutions and optimizing approximately $1/3$ of these for $\sim 127$ iterations on average. The scattered blue dots show the same, except the optimization is carried out on all player seeds with the computational resources described in the text (i.e., no player influence after seeding). The symbol translucency indicates the density distribution.

Figure 9

Splitting: mean of (near) optimal solutions ${\langle u_2(t/T)\rangle }_{\mathrm{opt}}$, corresponding to the height of the potential barrier. For a given $T$ ($y$ axis), the color indicates the mean optimal control value at a given $t/T$ ($x$ axis). Clustering analysis identified only a single cluster.

Figure 10

Shake Up solution densities for different methods. (a) grape ps, (b) grape rs, (c) sa rs ($n_b=n_t$), and (d) sa rs ($n_b=40$). Each density is normalized for every individual $T$ and thus represents an estimate of the probability distribution for obtaining a particular $F$ for a given $T$ ($\mathcal{P}\left(F\right|T)$). The reference curve shows the best obtained results for grape ps. Crosses indicate solutions at densities lower than 0.002.

Figure 11

Aggregate, monotonically best optimization results (lower is better) for several methods in Shake Up. Solid lines with (without) symbols show the best results obtained with (without) player influence. The scattered green crosses show all results produced by 266 players seeding on average $\sim 37$ solutions and optimizing approximately $1/3$ of these for $\sim 116$ iterations on average. The scattered blue dots show the same, except the optimization is carried out on all player seeds with the computational resources described in the text (i.e., no player influence after seeding). The symbol translucency indicates the density distribution.

Figure 12

Shake Up clustering results. (a) Component weights $c_k$ with colors corresponding to the different clusters. The bars denote the mean and standard deviation whereas the different lines only help to guide the eye. (b) Cluster means in real space color-coded as above. (c) Fidelity distribution color-coded as above.

Figure 13

Estimated $\langle T_{\min }^{\mathrm{fit}}\rangle$ in units of $T_{\min }^{F=0.99}$ and empirical success rate (estimated probability of success) $\mathcal{P}(T_{\mathrm{QSL}}^{{}^{\mathrm{fit}}}\in {\mathcal{T}}_{\bigsqcup })$ as a function of sample size. (a), (b) Bring Home Water, (c), (d) Splitting, and (e), (f) Shake Up. Strong statistical performance is signified by low $N_{\mathrm{samples}}$ having simultaneously low $\langle T_{\min }^{\mathrm{fit}}\rangle$ and high $\mathcal{P}$.

Figure 14

Results (lower is better) for grape (dark blue), sa $(n_b=n_t)$ (turquoise), and sa $(n_b=40)$ (mint green) using the same 100 rs seeds at $T_{\min }^{F=0.99}\approx 0.0973,0.92,0.89\mathrm{ms}$ for the three levels, respectively. (a) Bring Home Water, (b) Splitting, and (c) Shake Up. Each line denotes an optimization trajectory as a function of wall time. The translucency is related to the density distribution and the three dotted lines indicate the $25\%,50\%,$ and $75\%$ quantiles for all active optimizations.

Figure 15

Partial screen shots of the two main views in Quantum Moves 2. Much of the UI has been hidden for simplicity. (a) Dynamic play view with dynamic potential (green), instantaneous wave function density (red line with dark green area), and target state (yellow outline). (b) Graph view. Each solution corresponds to a point in the graph. The duration axis is normalized by an approximate $T_{\min }^{F=0.99}$ bound and is divided into 12 blocks. The uppermost green line (the challenge curve) corresponds to the best solution found by conventional methods prior to the launch of the game. Any point can be selected for optimization. (c) The player can invoke the embedded optimization algorithm (pgrape) by toggling the switch.