Rigorous compression theory and memory bounds for influence matrices of chaotic quantum many-body dynamics

The influence matrix (IM, equivalently the process tensor or discrete Feynman-Vernon influence functional) encodes the multi-time action of a quantum many-body bath on a local probe; its temporal entanglement (TE) governs the bond dimension, and hence the classical cost, of simulating local observables. Vilkoviskiy et al. (2025) argued that chaotic one-dimensional baths have volume-law TE reflecting genuine non-Markovian memory - established via decoupling theorems only for Haar-random ensembles - but that this memory resides in high-order temporal correlations: restricting the allowed measurement frequency (temporal coarse-graining at density $n_{\mathrm{cg}}$) drives a transition to area-law TE, and numerically a modest-bond-dimension matrix product state already reproduces few-point correlators. All compressibility evidence beyond solvable random or dual-unitary circuits is heuristic or numerical. The open problem is a rigorous and constructive theory of this temporal complexity: provable bounds linking retained bond dimension $\chi$, coarse-graining density, and error on $N$-point correlators (including relative-error guarantees) for generic chaotic circuits; systematic algorithms that construct or time-evolve the compressed "fixed-point" IM with controlled error; a rigorous characterization of the critical coarse-graining density for structured baths; proof or refutation of the conjectured area-law TE for $d>1$ baths and $k>2$ circuits together with certified algorithms; and extension of the distillable-entanglement lower bounds beyond Haar-random ensembles. A 2026 rigorous logarithmic bound for the reduced transition matrix of random dual-unitary circuits resolves only a restricted slice and leaves this core open.

Background

Real-time dynamics of quantum many-body systems is the canonical regime where classical simulation fails: under generic (chaotic) evolution the spatial entanglement of the wavefunction grows linearly in time, so matrix-product-state (MPS) methods - one-dimensional tensor networks whose cost is set by the bond dimension, the size of their auxiliary indices - hit an "entanglement barrier" after a few scrambling times. A long-standing workaround, introduced by Bañuls, Hastings, Verstraete and Cirac and refined in Hastings and Mahajan's "folding algorithm", contracts the space-time tensor network transverse to the time axis: one evolves local observables in the Heisenberg picture and represents the effect of the rest of the system on a small subsystem instead of the full state.

That idea is formalized by the influence matrix (IM), the discrete-time analogue of the Feynman-Vernon influence functional, called a process tensor in quantum information: a multi-time object which, inserted between consecutive interventions on the subsystem, exactly reproduces all multi-time correlation functions of local observables. Because the IM lives in a temporal Hilbert space (one pair of $d$-level qudits per time step), it admits an MPS representation whose bond dimension is governed by temporal entanglement (TE): the entanglement entropy of the IM, viewed as a state via the Choi correspondence (the standard isomorphism that represents a linear map on operators as a quantum state), across a cut in time. TE thus measures the complexity of simulating local dynamics. Integrable and noninteracting baths, treatable through quasiparticles or free fermions, have area- or logarithmic-law TE (Sonner, Lerose and Abanin 2021; Giudice et al. 2022), whereas Foligno, Zhou and Bertini (2023) showed that chaotic dual-unitary (DU) circuits - brickwork circuits that are unitary in both the time and the space direction, hence exactly solvable yet maximally chaotic - have volume-law (linear-in-time) TE, apparently implying that simulating even local observables of chaotic systems is exponentially hard in the temporal direction.

The source paper (Vilkoviskiy et al. 2025) reconciles this apparent paradox for a local probe repeatedly coupled to a chaotic bath. First, in an exactly solvable random-unitary toy model, TE is extensive whenever the bath grows slowly enough relative to the temporal Hilbert space, and arguments based on decoupling theorems (quantum-information results stating that a random unitary acting on a large system leaves typical subsystems nearly uncorrelated) lower-bound the distillable entanglement (the number of Bell pairs extractable by local operations) between the probe's past and future, showing that volume-law TE reflects genuine non-Markovian memory. Second, that memory resides entirely in temporal correlations of very high order, with negligible effect on few-point correlators. Third, restricting the allowed frequency of probe measurements - coarse-graining the time axis, keeping a fraction $n_{\mathrm{cg}}$ of time steps - drives a sharp transition of the TE from volume-law to area-law at a critical density $n_{\mathrm{cg}}^{*}$, given in the toy models by $n_{\mathrm{cg}}^{*}=(2r_0)^{-1}$ for a static bath with ratio $r_0$ of temporal to bath degrees of freedom and by $n_{\mathrm{cg}}^{*}=v_B$ for a dynamically growing bath, where $v_B$ is the butterfly velocity, the speed of the operator-spreading light cone. Fourth, a generic (non-DU) kicked Ising circuit - a Floquet spin chain with alternating Ising interactions and spin rotations - shows the same phenomenology numerically, and an MPS compression of the coarse-grained IM obtained by simply discarding small Schmidt values (the singular values across temporal cuts) reproduces local correlators accurately at modest bond dimension $\chi$. Related empirical observations of high transverse-contraction accuracy at moderate $\chi$ include the temporal-entropy studies of Carignano et al.; dissipation-assisted operator evolution (DAOE, Rakovszky et al. 2022) achieves a comparable effect for operator spreading by artificially damping non-local operators; and influence-functional belief propagation (Park, Gray and Chan 2025) extends the framework to two spatial dimensions - all without error guarantees.

Everything beyond the solvable (random or DU) models in this picture is heuristic or numerical: no provable relation between $\chi$, $n_{\mathrm{cg}}$ and the error on $N$-point correlators is established; no systematic algorithm constructs or time-evolves the compressed IM; the critical density $n_{\mathrm{cg}}^{*}$ of structured circuits is known only numerically (for one kicked-Ising parameter set, $1/4<n_{\mathrm{cg}}^{*}<1/2$); and the memory lower bounds hold only for Haar-random (uniformly distributed over the unitary group) ensembles. The problem below asks to close these gaps.

Problem Statement

For chaotic quantum many-body dynamics in the setting of the source paper (Vilkoviskiy et al. 2025) - a local probe coupled to a spatially local one-dimensional bath, for example a kicked Ising circuit (a Floquet spin chain with alternating Ising interactions and spin rotations) away from its dual-unitary points - establish a rigorous and constructive theory of the temporal simulation complexity, replacing the heuristic and numerical evidence with provable statements. The common objective is a certified answer to the question: when is the influence matrix (IM) of chaotic dynamics genuinely complex (temporal memory), when is it provably compressible for few-point correlators, and how can that compression be realized algorithmically with controlled error. Five linked components serve this objective.

(1) Provable compression guarantees: establish bounds linking the retained bond dimension $\chi$ of an IM matrix-product-state approximation, the coarse-graining density $n_{\mathrm{cg}}$, and the error committed on $N$-point temporal correlation functions of local observables - including relative-error guarantees, not only absolute error - for generic non-random, non-dual-unitary chaotic circuits; and identify the regime in which finite-$\chi$ compression provably fails (persistence of volume-law temporal entanglement).

(2) Systematic construction and evolution algorithms: devise an algorithm that directly constructs the compressed "fixed-point" IM, or time-evolves a compressed representation without regenerating the exact IM at each time step, with error controlled by the bounds of component (1), going beyond ad hoc discarding of small Schmidt values. Routes explicitly anticipated by the source include generalizations of the temporal coarse-graining projection, dissipation-assisted evolution in the spirit of DAOE, and other systematic coarse-graining procedures.

(3) Critical coarse-graining density of structured baths: characterize rigorously the transition density $n_{\mathrm{cg}}^{*}$ for spatially local, non-random, non-dual-unitary baths - its dependence on the butterfly velocity $v_B$ (the speed of the operator-spreading light cone) and on other microscopic parameters - going beyond the solvable toy-model formulae $n_{\mathrm{cg}}^{*}=(2r_0)^{-1}$ (static bath with ratio $r_0$ of temporal to bath degrees of freedom) and $n_{\mathrm{cg}}^{*}=v_B$ (dynamically growing bath) and beyond purely numerical determination.

(4) Higher-dimensional and nonlocal regimes: prove or refute the source's conjectures that chaotic baths with spatial dimension $d>1$ and $k$-local circuits with $k>2$ (gates acting jointly on $k$ sites) built from sufficiently entangling gates obey area-law temporal entanglement (so that their IMs are compressible by moderate-$\chi$ tensor networks), and, wherever such area-law compressibility is established, develop efficient tensor-network IM approximations and algorithms for those settings, for instance building on influence-functional belief propagation.

(5) Memory lower bounds beyond random ensembles: extend the multi-time decoupling arguments used in the source to lower-bound the distillable entanglement between the probe's past and future - establishing genuine non-Markovian memory - from Haar-random unitary ensembles to structured chaotic dynamics such as dual-unitary or kicked Ising circuits, quantifying concentration-of-measure effects in those ensembles.

Components (1)-(4) form the sufficiency side (certified compressibility and its algorithmic exploitation), component (5) the matching necessity side (the uncompressed IM genuinely carries temporal memory), and component (3) the phase boundary between them. Components (1)-(3) and (5) are posed for generic one-dimensional circuits; component (4) fixes the corresponding statements in higher dimension and for nonlocal gates. A complete answer resolves all five components; partial advances on individual components are meaningful but do not by themselves close the problem.

References

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