Classical simulation of quantum circuits by dynamical localization: Analytic results for Pauli-observable scrambling in time-dependent disorder

We extend the concept of Anderson localization, the confinement of quantum information in a spatially irregular potential, to quantum circuits. Considering matchgate circuits, generated by time-dependent spin-$\frac{1}{2}$ $XY$ Hamiltonians, we give an analytic formula for the out-of-time-ordered correlator of a local observable and show that it can be efficiently evaluated by a classical computer even when the explicit Heisenberg time evolution cannot. Because this quantity bounds the average error incurred by truncating the evolution to a spatially limited region, we demonstrate dynamical localization as a means for classically simulating quantum computation and give examples of localized phases under certain spatiotemporal disordered Hamiltonians.

Figure 1

Typical light cones for $Z_{50}$ (left) and $X_{50}$ (right) propagating through the time-dependent disordered $XY$ model (Model 1) with mean disorder strength $\nu$ and fluctuation strength $\mathrm{\Delta }$, for $n=100$ qubits (time in units of $4\delta t$). Light cones are taken as the geometric mean of the determinant over 10 disorder realizations to preserve exponential decay. Representatives of the ballistic ($\nu =0$, $\mathrm{\Delta }=0$, top), diffusive ($\nu =0.75$, $\mathrm{\Delta }=1$, middle), and localized ($\nu =2$, $\mathrm{\Delta }=0$, bottom) phases are shown. We note that, while the light cone envelopes for $Z_{50}$ and $X_{50}$ are always nearly the same, the interior of that for $X_{50}$ generally has a higher value.

Figure 2

Phase diagrams for the propagation of $Z$ (left) and $X$ (right) in the presence of locally fluctuating disorder (top) and fluctuating interactions (bottom). These are given by taking the fitted slopes $m$ of the light cone envelopes. Although this parameter is continuous, we see that several natural regions emerge: ballistic ($m\approx 1$), diffusive ($m\approx 0.5$), and localized ($m\approx 0$). Numbered points correspond to the corresponding light cones in Fig. 1, where points (1) and (3) on the $y$ axis are meant to be located at $\mathrm{\Delta }=0$. The extended localized phase demonstrates that localization survives under weak fluctuations.

Figure 3

(top, left) Envelopes of the $X$ light cones in Fig. 1, given by the principal temporal components of their singular value decompositions (time in units of $4\delta t$). (top, right) Envelopes on a log-log plot, with fitted slopes displayed. Note the saturation of the ballistic case around $t\simeq 6$ is a boundary effect; otherwise, it is expected to be a straight slope. (bottom, left) Singular values in these decompositions on log-log scale, which demonstrate the error in truncating to a product function; we see that these decay by several orders of magnitude over the first ten. (bottom, right) Light cone decay profiles, given by the principal spatial vectors; these profiles are very nearly Gaussian, rather than exponential.