Principal component analysis of wavefunction snapshots in non-equilibrium dynamics

Abstract

We study non-equilibrium quantum dynamics by performing principal component analysis on the data sets of wavefunction snapshots. We show that a specific transformation of the data sets maximizes the information content in the largest principal component and further enables its connection to certain observables. This connection enables us to explain the dynamical features revealed by such a dimensionality-reduction scheme. We demonstrate this using quantum dynamics of the Heisenberg spin chain, starting from different initial states, and further extend the approach to extract higher-order correlations. Our framework should also be applicable to other unsupervised machine-learning methods based on dimensionality-reduction schemes and is highly relevant to experiments with quantum simulators, including those in higher dimensions.

Figures (5)

• Figure 1: (a) Time evolution of the scaled eigenvalue $\lambda_1$ (scaled by $S=\sum_{i} \lambda_i$) for three different initial states: (i) domain wall (DW) state which have two halves of the lattice with opposite spin polarization, (ii) Néel state, characterized by alternating down and up spins, and (iii) XZ-type multi-periodic domain wall (MPDW) state, which has a helical mixed-basis form composed of $x$ and $z$ bases with wavelength $\lambda_{XZ}$supp. (b) Decay of PCA eigenvalues $\lambda_k$ versus $k$ at a fixed time $t=4$ for the same three initial states. (c)-(e) Dynamics of $\Delta \langle{X}\rangle_C(t) = |\langle{X}\rangle_C(t) - \langle{X}\rangle_C(0)|$, defined in Eq. \ref{['x-eq']}, and the principal component $\Delta \lambda_1(t)= |\lambda_1(t)-\lambda_1(0)|$ for the DW, Néel, and the XZ-type MPDW states, respectively. For the DW case, $\Delta \lambda_1(t)$ approximates the dynamics of $\Delta \langle{X}\rangle_C(t)$ quite accurately, whereas for the other cases, they carry distinct dynamical features. For the numerics, we consider $\Delta=1$, $L=100$, $\lambda_{XZ}=100$, and $N_r=1000$ for DW and Néel, whereas $N_r=4000$ for XZ-type MPDW state. In plots (c)-(e) the quantity $\Delta \langle{X}\rangle_C(t)$ is multiplied by a constant factor $d$ for plotting it on the same scale as $\Delta \lambda_1(t)$. The value of $d$ is $1$ for (c), $40$ for (d), and $6.5$ for (e). Note that the mismatch between $\Delta\langle{X}\rangle_C(t)$ and $\Delta \lambda_1(t)$ in (d) and (e) is due to the contribution arising from higher eigenvalues of the snapshot matrix.
• Figure 2: Plot of leading PCA eigenvalue $\bar{\lambda}_1$ for three different choices of $a_i \in \left[+1,\, \mathrm{sgn}[\langle S_i^z(0)\rangle], \, [-1,1] \right]$, starting with the Néel state in (a) and XZ-type MPDW state in (c). The insets in (a) and (c) show the dynamics of $\Delta S(t)$, as defined in Eq. \ref{['Universal_equation']}, for the same three choices of $a_i$. In all cases, the choice $a_i = \mathrm{sgn}[\langle S_i^z(0) \rangle]$ leads to maximum weightage to $\bar{\lambda}_1$. Plots (b), (d) shows growth of the exact $\Delta \langle{O(t)}\rangle_{\textrm{Exact}}=|\langle{O(t)}\rangle-\langle{O(0)}\rangle|$) (green solid line) and the leading PCA eigenvalue $\Delta \bar{\lambda}_1 =|\bar{\lambda}_1(t)-\bar{\lambda}_1(0)|$) (blue solid line) for the Néel and XZ-type MPDW initial states, respectively, where the form of $O$ is defined in Eq. \ref{['adjusted_magnetization_defn']} with $a_i = \mathrm{sgn}[\langle S_i^z(0)\rangle]$. $\Delta \bar{\lambda}_1$ captures the exact behaviour of the observable $O$ for both initial states and furthermore gives the correct dynamical exponent $z=2$ for the XZ-type MPDW initial states. The parameters chosen are the same as in Fig. \ref{['fig:Bare_PCA']}.
• Figure 3: Plot for second order cumulant $\sigma_{\textrm{Exact}}^2(t)=\langle O^2(t)\rangle-\langle O(t) \rangle^2$, with $a_i=\mathrm{sgn}[\langle S_i^z(0)\rangle]$. Both PCA approximated dynamics $\Delta \sigma^2_{\textrm{PCA}}(t)=|\sigma^2_{\textrm{PCA}}(t)-\sigma^2_{\textrm{PCA}}(0)|$ [Eq. \ref{['PCA_second_order_cumulant']}] and the exact dynamics $\Delta \sigma^2_{\textrm{Exact}}(t)=|\sigma^2_{\textrm{Exact}}(t)-\sigma^2_{\textrm{Exact}}(0)|$ are shown for DW initial state in (a) and for Néel state in (b). The PCA approach captures the $z=3/2$ dynamical exponent in the DW case, whereas no exact behaviour can be extracted from the Néel initial state. Exact growth of $\Delta w^2(M,t)= |w^2(M,t) - w^2(M,0)|$, and the first principal component $\Delta \widetilde{\lambda}_1(t)=| \widetilde{\lambda}_1(t)-\widetilde{\lambda}_1(0)|$ obtained from the snapshot matrix defined in Eq. \ref{['Snapshot_matrix_Surface_Roughness']} are plotted for the DW initial state in (c) and the Néel state in (d), respectively. The first principal component $\widetilde{\lambda}_1(t)$ captures the same dynamical exponent as the exact $w^2(M,t)$, given by $z=3/4$ for the DW case, and $z=3/2$ for the Néel case. The parameters used here are $L=100$, $N_r=1000$, and the domain $M$ is chosen from the 10th to the 90th site of the lattice chain. In all plots, the quantity $\Delta\sigma^2_{\textrm{Exact}}(t)$ and $\Delta w^2(M,t)$ is multiplied by a constant factor $d$ for plotting it on the same scale as $\Delta\sigma^2_{\textrm{PCA}}(t)$ and $\Delta \widetilde{{\lambda}}_1(t)$ respectively. The value of $d$ is $100$ for (a) and (b), $0.1$ for (c), and $40$ for (d).
• Figure S1: Plot for the dynamics of (a) $\bar{\lambda}_1$, and (b) $\Delta S$ for the Domain wall (DW) initial state. We observe that $\bar{\lambda}_1$ corresponding to $a_i=+1$ has the highest weightage compared to other choices of $a_i$. The $\Delta S$, however, in all cases grows identically. This confirms that $\bar{\lambda}_1$ best approximates the dynamics of observable $\langle O \rangle$. For the numerics we consider, $\Delta=1$, $L=100$, and $N_r=1000$.
• Figure S2: (a)-(b) Plot for scaled eigenvalue $\bar{\lambda}_1$ (scaled by $S=\sum_{i} \bar{\lambda}_i$) for three different choices of $a_i$ for the XXZ spin chain for $\Delta = 0.6$ (easy-plane) in (a) and $\Delta =1.1$ (easy-axis) in (b), starting from the DW initial state. (c)-(d) Plot for $\Delta S$ which evolves almost identically for all the three choices of $a_i$ for both the $\Delta$ values. (e)-(f) Exact dynamics of $\Delta \langle O(t) \rangle_{\textrm{Exact}} =|\langle O(t) \rangle -\langle O(0) \rangle|$ (scaled by a scalar $d$) obtained via TEBD and first principal component $\Delta \bar{\lambda}_1(t) = |\bar{\lambda}_1(t) - \bar{\lambda}_1(0)|$ of the snapshot matrix $\overline{\mathbf{{X}}}$. The correct transport exponents are captured in $\Delta \lambda_1(t)$ as in the $\Delta \langle O(t) \rangle_{\textrm{Exact}}$, which are ballistic for $\Delta=0.6$ as shown in (e), whereas diffusive for $\Delta=1.1$ as shown in (f). In (g) and (h), the dynamics of the second-order cumulant of observable $O$ defined as $\sigma^2_{\textrm{Exact}}(t) = \langle{O^2(t)}\rangle - \langle{O(t)}\rangle^2$, when $a_i=+1$ is shown. Both PCA $\Delta \sigma^2_{\textrm{PCA}}(t)=|\sigma^2_{\textrm{PCA}}(t)-\sigma^2_{\textrm{PCA}}(0)|$ and exact $\Delta \sigma^2_{\textrm{Exact}}(t)=|\sigma^2_{\textrm{Exact}}(t)-\sigma^2_{\textrm{Exact}}(0)|$ (scaled by $d$) are shown in (g) and (h), where $\sigma^2_{\textrm{PCA}}(t)$ is defined in Eq. \ref{['PCA_second_order_cumulant']}. $\bar{\Lambda}_1$ corresponding to $\mathbf{\bar{Z}}$ captures the ballistic scaling in (g) and the diffusive scaling in (h). For the numerics, we use $L=100$, and the number of realizations $N_r=1000$. The scaling factor $d$ is $0.5$ for (e) and (f), $50$ for (g), and $80$ for (h).