Quantum jamming: Critical properties of a quantum mechanical perceptron

In this Letter, we analyze the quantum dynamics of the perceptron model: a particle is constrained on an $N$-dimensional sphere, with $N\to \infty$, and subjected to a set of randomly placed hard-wall potentials. This model has several applications, ranging from learning protocols to the effective description of the dynamics of an ensemble of infinite-dimensional hard spheres in Euclidean space. We find that the jamming transition with quantum dynamics shows critical exponents different from the classical case. We also find that the quantum jamming transition, unlike the typical quantum critical points, is not confined to the zero-temperature axis, and the classical results are recovered only at $T=\infty$. Our findings have implications for the theory of glasses at ultralow temperatures and for the study of quantum machine-learning algorithms.

Figure 1

Finite-dimensional representation of the perceptron model at $\sigma =0, N=3, M=4$. Each constraint is represented by a plane passing through the origin and cuts the sphere in half. The particle can move in the allowed (light orange) region on the sphere. The jamming transition is reached when the number of obstacles is such that (with probability 1 in the $N,M\to \infty$ limit) there is no light orange volume left anymore.

Figure 2

Edwards-Anderson order parameter as a function of the constraint density $\alpha$ for various temperatures. From bottom to top: infinite-temperature classical dynamics (red dotted line) to finite-temperature quantum dynamics ($\beta =2,4,8$). The $O\left(\alpha \right), \beta =\infty$ results are shown as dashed black lines (while the horizontal black line is a reference for the value $q=1$). Notice how, as soon as $\alpha \gtrsim 1,$ the temperature dependence of $q$ is effectively lost (it is $\sim e^{-c\beta /{(2-\alpha )}^2}$).

Figure 3

Edwards-Anderson order parameter close to the critical point $\alpha =2$. From top to bottom, increasing the number of Trotter slices $S=4,8,16,32$ for sufficiently large $\beta$, the slope increases. For reference, the classical value of the slope [from $(1-q)\sim (2-\alpha )$] is shown as the diagonal dotted black line. The inset shows the values of the slope with their errors and its extrapolation to $S\to \infty$ to the value $\kappa =2.0\pm 0.1$, quoted in the text.

Figure 4

Self-energy $\mathrm{\Sigma }\left(\omega \right)$ at $\alpha =1.7, \beta =1/2^3$ as a function of the Matsubara frequency $\omega$ for an increasing number of Trotter slices (accessing higher and higher frequencies). We see that $\mathrm{\Sigma }\left(\omega \right)$ develops a linear $\omega$ behavior (black dotted line) for intermediate $\omega$'s while retaining its analyticity in terms of ${\omega }^2$ around the origin for any $q<1$ (inset). The inset shows $\mathrm{\Sigma }\left(\omega \right)$ at small $\omega$'s for $\alpha =1.5, \beta =8$.