Free Independence and Unitary Design from Random Matrix Product Unitaries

TL;DR

This work builds a bridge between unitary designs and free probability by introducing random matrix product unitaries (RMPUs) as a tractable model for chaotic quantum dynamics. It shows that, with bond dimension $\chi$ polynomial in system size, RMPUs reproduce Haar statistics for higher-order OTOCs of local, finite-trace observables, indicating emergent freeness at relatively shallow circuit depths; reproducing Haar values for traceless operators requires exponentially larger $\chi$, highlighting a sharp complexity boundary. The authors derive exact frame-potentials and show on-average freeness for global observables, while clarifying the role of locality and the geodesic non-crossing-m partition structures in the Weingarten calculus. Together, these results refine the understanding of what aspects of randomness are easy to realize in practical quantum devices and offer a new lens for studying thermalization, chaos, and potential quantum advantages in random circuit protocols.

Abstract

Unitary randomness underpins both fundamental tasks in quantum information and the modern theory of quantum chaos. On one side, a central concept is that of approximate unitary designs: circuits that look random according to small moments and for forward-in-time protocols. In a distinct setting, out-of-time-ordered correlators (OTOCs), intensely studied as a measure of information scrambling, have recently been shown to probe freeness between Heisenberg operators, the noncommutative generalization of statistical independence. Bridging these two concepts, we study the emergence of freeness in a random matrix product unitary ensemble. We prove that, with only polynomial bond dimension, these unitaries reproduce Haar values of higher-order OTOCs for local, finite-trace observables, while traceless observables instead require exponential resources. Indeed, local observables are precisely those predicted to thermalize in chaotic many-body systems according to the eigenstate thermalization hypothesis. Moreover, adding to previous literature, we show how random matrix product unitaries constitute approximate designs: we exactly compute the frame potential of the ensemble, showing convergence to the Haar value with polynomial deviations and so indicating that global observables are freely independent on-average. Our results highlight the need to refine previous notions of unitary design in the context of operator dynamics, guiding us towards protocols for quantum advantage and shedding light on the emergent complexity of chaotic many-body systems.

Figures (2)

• Figure 1: A depiction of the introduced random matrix ensemble and its properties. (a) A matrix product unitary on some $D=d^N$ dimensional system is constructed as a staircase of $n$ unitaries $\{ U_i\}_{i=1}^n$ overlapping on a space of (bond) dimension $\chi$. The random matrix product unitary (RMPU) ensemble $\mathcal{R}$ is generated by sampling each $U_i$ independently from the Haar measure on the unitary group, for a given $n$ and $\chi$. (b) Previous work shows that for polynomial bond dimension $\chi = \mathrm{poly}(N)$, this ensemble is a relative error unitary design schuster2024randomunlaracuente_approximate_2024 and exhibits anticoncentration in the computational basis Lami2025. Roughly speaking, these results mean that the Haar distribution of (possibly correlated) measurements of forward-in-time evolving states are well approximated by the ensemble $\mathcal{R}$. We extend these results and prove that also $\mathcal{R}$ is a unitary designs according to the frame potential. (c) We prove that polynomial bond dimension $\chi$ also leads to free independence for local, finite-trace observables, as characterized by the non-crossing partitions. Freeness is witnessed by out-of-time-ordered experiments [Eqs. \ref{['eq:otoc1']}-\ref{['eq:otoc']}] and is thus not accounted for by previous results on unitary designs. The red and blue non-crossing partition diagrams here denote the partitioned moments of $A$ [Eq. \ref{['eq:fact_moms']}] and free cumulants of $B$ [Eq. \ref{['eq:mom-cum']}], respectively. See Table \ref{['tab:example']} for precise results.
• Figure 2: The non-crossing partition lattice (Hasse diagram) for $k=4$. A diagrammatic representation of the integer partition is shown together with the corresponding element of the permutation group $S_k$ in cyclic notation. Solid lines connecting partitions indicate a unit distance on the lattice, i.e. permutations related via a single transposition, or equivalently integer partitions related by refinement. A multichain is defined as an ordered set of permutations lying on a path from the bottom to the top, satisfying the geodesic condition $\pi_1 \leq \sigma_1 \leq \dots \leq \gamma$ as detailed around Eq. \ref{['eq:geodesic']}. An example geodesic in shown in bold [red], where for instance, $\pi=(12)(3)(4)$ and $\sigma=(123)(4)$ satisfy the $2$-chain condition $\pi \leq \sigma \leq \gamma$. Inset: in the top right the one crossing partition of $k=4$ elements is given. Cumulants corresponding to this crossing partition do not appear in the leading order Haar-twirled OTOC [Eq. \ref{['eq:loOTOC']}].