Barren Plateaus Preclude Learning Scramblers

Scrambling processes, which rapidly spread entanglement through many-body quantum systems, are difficult to investigate using standard techniques, but are relevant to quantum chaos and thermalization. In this Letter, we ask if quantum machine learning (QML) could be used to investigate such processes. We prove a no-go theorem for learning an unknown scrambling process with QML, showing that it is highly probable for any variational Ansatz to have a barren plateau landscape, i.e., cost gradients that vanish exponentially in the system size. This implies that the required resources scale exponentially even when strategies to avoid such scaling (e.g., from Ansatz-based barren plateaus or no-free-lunch theorems) are employed. Furthermore, we numerically and analytically extend our results to approximate scramblers. Hence, our work places generic limits on the learnability of unitaries when lacking prior information.

Figure 1

Learning a scrambling unitary. Panel (a) shows the setup of the classic Hayden-Preskill thought experiment where someone attempts to retrieve information (shown as a book) thrown into a black hole (a scrambler). If the scrambling unitary $U$ is known, then information can be retrieved. Panel (b) shows the process of attempting to learn $U$. This requires a time that is exponential in the number of quantum degrees of freedom (qubits) due to an exponentially vanishing cost gradient, see Letter below. This precludes the information retrieval shown in (a).

Figure 2

Numerical simulations of approximate scramblers. (a) A random cut of the landscape of the Local Hilbert-Schmidt Test cost function $C_{\mathrm{LHST}}(U,V)$ (defined in Appendix E [60]) where $V(g,t)$ is a randomly generated nine qubit scrambler ($n=9$) modeled via Eq. (12) and $U(g,t)$ is an Ansatz of the same form. Here, $\epsilon$ is a noise parameter that determines the deviation of the Ansatz parameters, $\mathit{\theta }$, from the target scrambler’s parameters, ${\mathit{\theta }}^{\text{target}}$. Specifically, we set ${\theta }_k={\theta }_k^{\text{target}}+\epsilon R$, where $R$ is a random number between -1 and 1. The landscape for a weak scrambler with $t=1$ and $g=0.1$ ($g=5$) is plotted in yellow (red). Stronger scramblers with $t=15$ and $g=0.1$ ($g=5$) are plotted in green (blue). (b) The variance in the gradient of a single local term of the LHST cost function, $C_{\mathrm{LHST}}^0(U,V)$ (defined in Appendix E [60]), where $V(g,t)$ is a scrambler modeled via Eq. (12) with $g=0.5$ (dashed) and $g=1$ (solid) and $U(g,t)$ is an Ansatz of the same form. In the first (second) row, the variance is calculated over an ensemble of random target (Ansatz) unitaries. In the first (second) column, the variance in the gradient is plotted as a function of $t$ ($n$). As indicated by the dotted line, for large $g$ and $t$ the variance vanishes exponentially as $\text{Var}[{\partial }_{{\theta }_k}C]\propto 2^{-2n}$.