Universal distributions of overlaps from generic dynamics in quantum many-body systems

TL;DR

This work analyzes the distribution of overlaps with the computational basis for quantum states evolved under generic chaotic many-body dynamics without conserved quantities. It identifies a universal scaling regime with $x = L/L_{\rm Th}(t)$ in which the overlap distribution $\rho(w;x)$ becomes independent of microscopic details and depends only on dimension and boundary conditions, interpolating from Porter-Thomas to log-normal as $x$ grows. The universality is derived from a mapping to the Ginibre ensemble and validated by an exact Random Phase Model and numerical simulations of two random-circuit ensembles, yielding closed-form results for open boundaries and a spectral-sum representation for periodic boundaries. The framework extends to Floquet circuits with mild disorder and provides practical benchmarks for intermediate-depth scrambling, with implications for quantum-supremacy experiments.

Abstract

We study the distribution of overlaps with the computational basis of a quantum state generated under generic quantum many-body chaotic dynamics, without conserved quantities, for a finite time $t$. We argue that, scaling time logarithmically with the system size $t \propto \log L$, the overlap distribution converges to a universal form in the thermodynamic limit, forming a one-parameter family that generalizes the celebrated Porter-Thomas distribution. The form of the overlap distribution only depends on the spatial dimensionality and, remarkably, on the boundary conditions. This picture is justified in general by a mapping to Ginibre ensemble of random matrices and corroborated by the exact solution of a random quantum circuit. Our results derive from an analysis of arbitrary overlap moments, enabling the reconstruction of the distribution. Our predictions also apply to Floquet circuits, i.e., in the presence of mild quenched disorder. Finally, numerical simulations of two distinct random circuits show excellent agreement, thereby demonstrating universality.

Figures (7)

• Figure 1: The space-time of our problem parametrized as $(x,t)$ according to $L=x L_{\text{Th}}(t)$ coming from the definition of $x$. The variable $x$ defines a family of coordinate space-time curves, as indicated by the dashed lines. The solid curve represents the curve of $x=1$ and separates the regions of $x<1, x>1$, above and below, respectively. The scaling limit is taken at $t,L \to \infty$, implying that our results for $p(w;x)$ represent its behaviour at the upper right part of the graph. Moreover, $p(w;x)$ has two characteristic limits: in the region of space-time where $x \ll 1$ (blue coloured) it approaches the PT distribution, whereas in the region of $x\gg 1$ (red coloured), the distribution of $y=\log w$ approaches a log normal one.
• Figure 2: (a) A representation of the powers $w^k = | \braket{\mathbf{a} = a_1,\dots,a_L | W(t) | \Psi_0}|^{2 k}$, with $k=2$ for time $t=4$ and system size $L=9$. The transfer matrix is highlighted in red. The tensor product, $u\otimes u^*\otimes \dots \otimes u\otimes u^*$, is highlighted in green, which, upon ensemble-averaging, can be represented as a sum of operators of permutation states. (b) In the Thouless scaling limit, the overlap $\mathbb{E}[w^k]$ can be interpreted as the grand canonical partition function of a dilute gas of domain walls, corresponding to transpositions connecting two permutations and each carrying a fugacity $L_{\rm Th}^{-1}$. Correspondingly, the size of each domain is $\sim L_{\rm Th}(t)$.
• Figure 3: Comparison of the distribution of $y = \log w$ between numerical simulation and the theoretical prediction (black dashed line) for different values of $x$ and increasing value of the time $t$, which is indicated with darker shades of the same colour. For each $t$, the value of $L \sim L_{\rm Th}(t)$ (shown in the insets) is chosen so that $\mathbb{E}[y]$ matches the theoretical prediction. The sub-figure (a) demonstrates the results from the pbc, numerical simulation of RPM at $q=2,\epsilon=1$, whereas sub-figure (b); the obc, numerical simulation for a BWM where the local 2-site gate is chosen independently from the Haar distribution for $q=2$. For more details, see Appendixes \ref{['app:models']},\ref{['app:NumericalSim']}.
• Figure 4: Convergence of the numerical distributions (colored lines) to the theoretical ones (black-dashed line). (a): The obc numerical simulations of the RPM. For $x=0$, we provide data for $(t,L) \in \{ (10,6),(15,6),(20,6)\}$; for $x=1$, $(t,L) \in \{ (3,8),(5,10),(10,19)\}$; and for $x=1.5$, $(t,L) \in \{ (3,10),(5,13),(7,18)\}$. (b): Pbc numerical simulations for the BWM at $q=2$ and up to $L_{\text{max}}=20,\ t_{\text{max}}=20$. We provide data for $x=0$ at $(t,L) \in \{ (3,15),(5,15)\}$; for $x=0.5$ at $(t,L) \in \{ (1,5),(2,10)\}$; for $x=1$ at $(t,L) \in \{ (1,7),(2,16)\}$.
• Figure 5: Convergence of both of the RPM (blue curves) and BWM (coloured triangles) models to the same scaling limit (black dashed curve) for $x=1$. (a) The obc numerical simulations of RPM at $(t,L) \in \{ (3,8),(5,10) ,(10,19)\}$ and of BWM at $(t,L) \in \{ (1,8),(3,40),(4,88)\}$; (b) The pbc numerical simulations of RPM at $(t,L) \in \{ (3,6),(5,9),(10,17)\}$ and of BWM at $(t,L) \in \{ (1,7),(2,16)\}$.