Machine-learning-inspired quantum control in many-body dynamics

Achieving precise preparation of quantum many-body states is crucial for the practical implementation of quantum computation and quantum simulation. However, the inherent challenges posed by unavoidable excitations at critical points during quench processes necessitate careful design of control fields. In this work, we introduce a promising and versatile dynamic control neural network tailored to optimize control fields. We address the problem of suppressing defect density and enhancing cat-state fidelity during the passage across the critical point in the quantum Ising model. Our method facilitates seamless transitions between different objective functions by adjusting the optimization strategy. In comparison to gradient-based power-law quench methods, our approach demonstrates significant advantages for both small system sizes and long-term evolutions. We provide a detailed analysis of the specific forms of control fields and summarize common features for experimental implementation. Furthermore, numerical simulations demonstrate the robustness of our proposal against random noise and spin number fluctuations. The optimized defect density and cat-state fidelity exhibit a transition at a critical ratio of the quench duration to the system size, coinciding with the quantum speed limit for quantum evolution.

Figure 1

Workflow of the DCNN and the visualization of the optimization landscape. (a) The structure of the DCNN. We choose two hidden layers with $n_1$ units in the first layer and $n_2$ in the second. $t$ is the input of the neural network, and $f\left(t\right)$ is the output. Each neuron in the hidden and output layers has a corresponding weight and bias. The blue dots are the original units in the neural network, the yellow dots are the newly increased units, and the dashed lines represent the corresponding increased connection. (b) The optimization of the combined objective function $Q$ for the observable $O$ and the loss $\mathcal{L}$. An initial trial form of the control field is used as a starting point (blue triangle). $Q$ moves downhill (darker red triangles) until the convergence is reached. To be noticed, the downhill movement signifies a simultaneous decline in both $O$ and $\mathcal{L}$. (c) The initial and final forms of the control field $g\left(t\right)$ with two fixed points (black crosses).

Figure 2

Learning results of the DCNN for (a) $T=0.5,N=50$ and (b) $T=50,N=50$. The number of neurons in the first hidden layer $n_1$ (orange dash-dot line) illustrates the change of the structure of the neural network with the learning iterations $\mathcal{N}$. The blue line shows the specific optimization process of the final defect density $\rho \left(T\right)$.

Figure 3

The comparison between the optimized forms of the control field $g\left(t\right)$ based on the DCNN (red solid line) and the gradient-based power-law quench (purple dash-dot line) for (a) $T=0.5,N=50$ and (b) $T=50,N=50$. In both panels, the horizontal gray dashed lines indicate QCPs, while the gray dashed vertical lines identify $t^{*}$.

Figure 4

(a) The final defect density $\rho \left(T\right)$ with respect to different $T$ and $N$ under the optimization by the DCNN. (b) The ratio $R_{\rho }$ of the final defect density optimized by the DCNN relative to that obtained by the gradient-based power-law quench method.

Figure 5

(a) The optimized control fields $g\left(t\right)$ obtained by the DCNN protocol with different initial values $g_i$ for $T=50,N=50$. Inset magnifies the control fields around $t^{*}$. (b) The absolute value of the relative offset of $g\left(t\right)$ to $g_c$ (dashed lines) and the energy gap $\mathrm{\Delta }E$ (solid lines with markers) with respect to $t$. Inset shows the corresponding final defect density $\rho \left(T\right)$ for different $g_i$.

Figure 6

Mean relative offsets $\delta \rho$ of the final defect density from the ideal value subjected to white Gaussian noise, as a function of the total number of numerical simulations $\mathcal{R}$, with $\text{SNR}=10$. (a) $T=0.5, N=50$, and (b) $T=50, N=50$. Here we define $\mathcal{R}=50p$, where $p=155$ for (a) and $p=438$ for (b). Dashed lines in both panels indicate the convergent values.

Figure 7

The fidelity $F\left(T\right)$ of the cat state is evaluated under various values of $T$ and $N$, optimized by the DCNN.

Figure 8

Comparison of the control fields $g\left(t\right)$ optimized for the cat-state fidelity $F\left(T\right)$ (purple dash-dot line) and the final defect density $\rho \left(T\right)$ (red solid lines) with (a) $T=0.5,N=50$ and (b) $T=50,N=50$. In both panels, the horizontal gray dashed lines indicate QCPs, and the gray dashed vertical lines identify $t^{*}$ of $g\left(t\right)$ guided by the cat-state fidelity.

Figure 9

(a) Computational time and (b) memory requirement for varying neural network scales with spin lengths of $N=24, N=50$, and $N=100$. The notation $n_1\times n_2$ denotes a network configuration with $n_1$ and $n_2$ neurons in the first and second hidden layers, respectively. Each point represents the average computational time or memory usage derived from ten iterations of neuron additions to networks of identical scale. The dashed lines indicate linear fits to the data points.