Temporal Complexity Hierarchies in Solvable Quantum Many-Body Dynamics

TL;DR

This work analyzes the computational complexity of the influence matrix (IM) for nonequilibrium quantum dynamics in solvable brickwork circuits. By deriving an exact MPO representation in which inner bonds form the group algebra of $G=PU(q)$ and linking the bond-dimension growth $\chi(T)$ to the geometric-group-theoretic growth function of the reachable set $H\subset G$, it reveals a three-class hierarchy of temporal entanglement entropy: area-law saturation, logarithmic growth, and linear growth corresponding to finite, virtually nilpotent, and free group structures. It further distinguishes classical versus quantum memory in IMs, showing that some models admit efficient Monte Carlo simulations for multitime statistics, while others require a quantum-memory diagnostic, implemented via a teleportation-inspired protocol that detects long-lived quantum correlations. Together, these results connect quantum dynamical behavior to algebraic properties of the generating set, offering a unified framework to assess IM complexity and memory across integrable to chaotic regimes, with potential extensions to noisy dynamics and higher dimensions.

Abstract

The influence matrix (IM) provides a powerful framework for characterizing nonequilibrium quantum many-body dynamics by encoding multitime correlations into tensor-network states. Understanding how its computational complexity relates to underlying dynamics is crucial for both theoretical insight and practical utility, yet remains largely unexplored despite a few case studies. Here, we address this question for a family of brickwork quantum circuits ranging from integrable to chaotic regimes. Using tools from geometric group theory, we identify three qualitatively distinct scalings of temporal entanglement entropy, establishing a hierarchy of computational resources required for accurate tensor-network representations of the IM for these models. We further analyze the memory structure of the IM and distinguish between classical and quantum temporal correlations. In particular, for certain examples, we identify effectively classical IMs that admit an efficient Monte Carlo algorithm for computing multitime correlations. In more generic settings without an explicit classical description of the IM, we introduce an operational measure of quantum memory with an experimental protocol, and discuss examples exhibiting long-time genuinely quantum correlations. Our results establish a new connection between quantum many-body dynamics and group theory, providing fresh insights into the complexity of the IM and its intricate connection to the physical characteristics of the dynamics.

Figures (8)

• Figure 1: Graphical representations of the quantum-circuit dynamics and the influence matrix. Total number of time steps $T=3$. (a) Folded representation of the spatiotemporal correlation function in 1+1 D quantum circuits. The evolution of a matrix product initial state [Eq. \ref{['eq:MPSinitial']}] is generated by two-site gates [Eq. \ref{['eq:swap_control']}] arranged in the brickwork architecture. Local operators $\hat{O}_0$ and $\hat{O}_1$ are inserted at different time steps. Red arrows indicate the direction of tensor-network contractions leading to influence matrices. (b) Illustration of the left and right influence matrices. The partial-traced components are compressed into MPS expanding along the time direction. (c) Representation of the observable expectation value in impurity dynamics. An impurity site (green tensor) is coupled to the quantum-circuit dynamics on the left. Red round tensors represent sequential quantum operations acting only on the impurity site.
• Figure 2: (a) The spacetime mapping in Eq. \ref{['eq:leftIM']} represented in terms of controlled and ${\mathrm{SWAP}}$ gates, according to Eq. \ref{['eq:swap_control_tensor']}. (b) Rotating the deforming the triangular tensor to a linear structure. Open legs originally at the bottom are relocated to the left boundary, as indicated by the arrows. (c) The MPO representation and the definition of local tensors. The inner bonds carry the linear space of the group algebra $\mathbb{C}[G]$, with $G=\mathrm{PU}(q)$.
• Figure 3: Temporal entanglement entropy for Model B as a function of time for different values of $K$ from Eq. \ref{['eq:dihedral']}. Each subplot includes an irrational value of $K$ shown in dotted lines, along with a sequence of rational approximations shown in solid lines. The initial state is set to the products of $(|0\rangle+|1\rangle)/\sqrt{2}$. (a) Values of $K$: $\ln 2$, $7/10$, $9/13$, and $61/88$. (b) Values of $K$: $(\sqrt{5}+1)/2$, $13/8$, $21/13$, and $34/21$.
• Figure 4: Temporal entanglement entropy for Model C as a function of time for different values of $\theta$ in Eq. \ref{['eq:ModelIII']}, with varied truncated bond dimensions $\chi$. (a) $\theta=\pi/2+0.05$, close to an area-law parameter point $\pi/2$. (b) $\theta=\pi/3$, far from area-law points.
• Figure 5: Histograms of level-spacing ratios for Floquet operators defined in Eq. \ref{['eq:unitary_OBC']}, compared with the Wigner-Dyson distribution of circular orthogonal ensembles and the Poisson distribution (solid lines). The mean values of ratios are also shown. System size $L=14$. (a,b) Model B defined by Eq. \ref{['eq:dihedral']}, deformed by $v=e^{-0.01i\sigma^y}$ as in Eq. \ref{['eq:deformation']}. $K:\ln 2, (\sqrt{5}+1)/2$. (c,d) Model C defined by Eq. \ref{['eq:ModelIII']}, with $\theta=\pi/2+0.05$ and $\pi/3$.