Coarse-grained spectral projection: A deep learning assisted approach to quantum unitary dynamics

We propose the coarse-grained spectral projection method (CGSP), a deep learning assisted approach for tackling quantum unitary dynamic problems with an emphasis on quench dynamics. We show that CGSP can extract spectral components of many-body quantum states systematically with a sophisticated neural network quantum ansatz. CGSP fully exploits the linear unitary nature of the quantum dynamics and is potentially superior to other quantum Monte Carlo methods for ergodic dynamics. Preliminary numerical results on one-dimensional XXZ models with periodic boundary conditions are carried out to demonstrate the practicality of CGSP.

Figure 1

(a) Dynamics of local magnetization $\langle {\sigma }_k^z\left(t\right)\rangle$ for spins adjacent to the domain wall and spins in the middle of the ferromagnetic domain (the scheme below the legend specifies the labeling). The scattered data are calculated by CGSP with $(M,N)=(32,32)$ (left column) and $(M,N)=(32,64)$ (right column). Solid lines are results from the converged TDVP-MPS. The color of the solid line matches the color of scattered data for the same spin. Due to the mirror symmetry of the initial state, the solid lines associated with $k=9$, 16, 25, 32 are covered by the others. (b) The amplitude $c_i^2$ of projected states for $(M,N)=(32,32)$ (left column) and $(M,N)=(32,64)$ (right column). Because $c_i^2$ vanishes for all ${\mathrm{\Lambda }}_i>0$, we plot only the lower section of the energy spectrum.

Figure 2

(a) Dynamics of local magnetization $\langle {\sigma }_k^z\left(t\right)\rangle$ after a sudden quench calculated by CGSP for the 32-spin XXZ model. (b) Exact dynamics of $\langle {\sigma }_k^z\left(t\right)\rangle$ obtained with the TDVP-MPS. The right column is a duplicate of the left column for easier comparison. (c) The correlation function ${\langle {\sigma }_k^z{\sigma }_{\overline{k}}^z\rangle }_c$ between spin $k$ and its counterpart spin $\overline{k}$ ($\overline{k}=\frac{l}{2}+1-k$), averaged with ${\langle {\sigma }_{l+1-k}^z{\sigma }_{\overline{l+1-k}}^z\rangle }_c$. (d) Exact dynamics of $({\langle {\sigma }_k^z{\sigma }_{\overline{k}}^z\rangle }_c+{\langle {\sigma }_{l+1-k}^z{\sigma }_{\overline{l+1-k}}^z\rangle }_c)/2$ obtained with the TDVP-MPS. The right column is a duplicate of the left column.

Figure 3

Structure of a CGSP-adapted NAQS and the workflow of forward propagation in the evaluation mode. The original input is a batch of 1D spin configurations. The output is a batch of complex $N$ vectors, i.e., ${\left\{{\mathrm{\Theta }}_k(s_1,...,s_l)\right\}}_{k\in [0,N-1]}$. The forward propagation consists of four steps. (i) Preprocessing: Reordering of the original spin configuration and regrouping of the outcome into its hexadecimal equivalent. (ii) Embedding: Dilated convolution layers. Only one end of the convolution input is padded with 0 such that the convolution output obeys the conditional dependence required by the NAQS. (iii) Projection: Fully connected layers applied to each input node, respectively. The connection is sparse for the entire input tensor. Let $N_B$ denote the batch size. The output is a 4D (5D) real tensor of size ($M, N_B, l/4$, 16, (2)). The fourth dimension is associated with ${[0,1]}^4$, i.e., the possible outcomes of $h_{{\nu }_j}^{\prime }$. The optional fifth dimension is devoted to representing the real part and the imaginary part of a complex wave function separately. (iv) Postprocessing: Conservation law enforced by multiplication of the tensor element corresponding to illegal spin configurations with 0. $L_2$ normalization $x\to x/\sqrt{{\left|\right|x\left|\right|}_2}$ is imposed for evaluating the conditional wave function ${\tilde{\varphi }}_j^{\left(m\right)}$ in possible configurations. Then with the input spin configuration, one can select the corresponding outcomes from the conditional wave function and compute the total wave function ${\mathrm{{\rm Y}}}_m(s_1,...,s_l)$ ($m\in [1,M]$). Finally, with Eq. (10), ${\left\{{\mathrm{{\rm Y}}}_m(s_1,...,s_l)\right\}}_{m\in [1,M]}$ is recombined together with the initial condition to produce the final results.

Figure 4

Schematic of the direct sampling process (sampling mode) of the CGSP-adapted NAQS. Forward propagation is different in the sampling mode in two aspects. First, between enforcing the conservation law and enforcing $L_2$ normalization, there is an extra operation devoted to softening the distribution to be sampled. The softening acts only on the (real) amplitude part of the unnormalized wave function through $x\to {x\left|x\right|}^{\gamma -1}$. The purpose is to prevent mode collapse, i.e., the neural network continues to generate a small set of samples. The resultant normalized conditional wave function is denoted $\widehat{{\tilde{\varphi }}_j^m}$, to distinguished it from the original ${\tilde{\varphi }}_j^m$. The second difference in the sampling mode is that the postprocessing procedure is terminated immediately after obtaining $\widehat{{\tilde{\varphi }}_j^m}$.

Figure 5

(a) Estimated loss $L$ versus iterations. (b) Estimated $logL$ versus iterations. (c) Estimated ${\sum }_{i=0}^{N-1}{c}_i^2$ versus iterations.

Figure 6

A parallel scheme for CGSP breakdown. The label of each state implies a tree structure. The label $\iota =o$ denotes the root node, $\iota =1\_\mathrm{i}$ denotes the $i\mathrm{th}$ child of the root node, and $\iota =1\_\mathrm{i}\_\mathrm{j}$ denotes the $j\mathrm{th}$ child of $\iota =1\_\mathrm{i}$.