Machine Learns Quantum Complexity

TL;DR

This work demonstrates that a convolutional neural network can learn the Krylov spread complexity $C(t)$ for quantum systems with $N\times N$ Gaussian unitary ensemble Hamiltonians from time-evolved $TFD$ states, across all times including late-time plateaus. The results reveal a strong dependence on the basis in which the state is represented: energy eigenbasis and Krylov basis yield accurate predictions, while the original basis fails, with a pseudo-random basis offering intermediate performance. Moreover, the model can distinguish temperature-dependent features and shows that the system time variable is not a meaningful descriptor for $C(t)$ at late times. These findings suggest that Krylov spread complexity captures essential quantum-state structure in chaotic systems and motivate future applications to other ensembles and nonthermal states.

Abstract

We study how a machine based on deep learning algorithms learns Krylov spread complexity in quantum systems with N x N random Hamiltonians drawn from the Gaussian unitary ensemble. Using thermofield double states as initial conditions, we demonstrate that a convolutional neural network-based algorithm successfully learns the Krylov spread complexity across all timescales, including the late-time plateaus where states appear nearly featureless and random. Performance strongly depends on the basis choice, performing well with the energy eigenbasis or the Krylov basis but failing in the original basis of the random Hamiltonian. The algorithm also effectively distinguishes temperature-dependent features of thermofield double states. Furthermore, we show that the system time variable of state predicted by deep learning is an irrelevant quantity, reinforcing that the Krylov spread complexity well captures the essential features of the quantum state, even at late times.

Figures (8)

• Figure 1: Krylov spread complexity $C(t)/N$ for $N\times N$ random Hermitian matrices $H$ as a function of time $t/N$ with a $\beta=0$ TFD initial state and $N=1024$. As indicated on the plot we distinguish four phases: a ramp, a peak, a slope, and a plateau. The error bar denotes the standard deviation of samples.
• Figure 2: The magnitudes of the time-dependent amplitudes $|\text{Re}(\psi^{\cal B}_n(t)) + \text{Im}(\psi^{\cal B}_n(t))|$ for the intervals $0 \le n/N \le 1$ and $0 \le t/N \le 3$. Here we considered four different choices of the basis, (a) the energy, (b) the Krylov, (c) the original, and (d) the pseudo-random basis.
• Figure 3: Our 1D CNN architecture. Starting from the left, this schematic diagram shows the input layer with 4 channels, representing the real and imaginary components of the $N =1024$ dimensional state vectors $|\psi(0)\rangle$ and $|\psi(t)\rangle$. Next are three convolutional layers with 256, 512, and 1024 channels, respectively, followed by an AveragePooling layer with $N^{(3)}_{ch}=1024$ pooling nodes ($h^{AP}_k=\frac{1}{X_{(3)}}\sum_{n=1}^{X_{(3)}}h^{(3)}_{k,n},k=1,\dots,N^{(3)}_{ch}$ : the red and light blue outlines in the third convolutional layer indicate the averaging over the entire region of the channel.). This is followed by two fully connected layers with 256 and 128 nodes, respectively, and finally, an output layer consisting of a single node. The light blue, light green, and navy blue elements displayed as three vertically stacked blocks in the input layer and the first two convolutional layers represent convolution kernels (e.g., in the first convolutional layer, $W_{i,j,m}^{(1)},\, i=1,...,256,\,j=1,2,3,4,\,m=0,1,2$) with $K=3$ where the kernel size $K=3$ is chosen only for the sake of illustration.
• Figure 4: Loss function versus epoch for (a) training and (b) validation during training. Circles, squares, diamonds, and triangles are for the energy, Krylov, original, and pseudo-random bases, respectively.
• Figure 5: The comparison of $C(t)/N$ determined by CNN and truth values for (a) the energy and (b) the Krylov basis. The error bars denote the RMSE, $\Delta = \sqrt{\frac{1}{M_{test}}\sum_{i=1}^{M_{test}} (p_i/N - t_i/N)^2}$, where $p_i/N$ and $t_i/N$ are the $i$-th element of the predicted and truth values divided by $N$, respectively. Here we considered the TFD state with $N=1024$ for $\beta=0$. Note that $\langle \Delta \rangle$ is the time-averaged RMSE. The dotted lines are guides to the eye.