Classical Simulability from Operator Entanglement Scaling

Abstract

Local-operator entanglement (LOE) quantifies the nonlocal structure of Heisenberg operators and serves as a diagnostic of many-body chaos. We provide rigorous bounds showing when an operator can be well-approximated by a matrix-product operator (MPO), given asymptotic scaling of its LOE $α$-Rényi entropies. Specifically, we prove that a volume law scaling for $α\geq 1$ implies that the operator cannot be approximated efficiently as an MPO while faithfully reproducing all expectation values. On the other hand, if we restrict to correlations over a relevant sub-class of (ensembles of) states, then logarithmic scaling of the $α< 1$ Rényi LOE entropies implies MPO simulability. This result covers a range of relevant quantities, including infinite temperature autocorrelation functions, out-of-time-ordered correlators, and average-case expectation values over ensembles of computational basis states. Beyond this regime, we provide numerical evidence together with a random matrix model to argue that, also for out-of-equilibrium expectation values, logarithmic scaling for $α< 1$ Rényi LOE typically guarantees simulability. Our results put on firm footing the heuristic expectation that a low operator entanglement implies efficient tensor network representability, extending celebrated foundational results from the theory of matrix-product states and providing a formal link between quantum chaos and classical simulability.

Figures (5)

• Figure 1: A schematic of the setting studied in this work. Anti-clockwise from the top-left: An operator $O$ on a chain of $N=N_A+N_B$ qudits is in one-to-one correspondence to a pure state $| O \rrangle$ in a doubled Hilbert space. Schmidt decomposition across some bipartition leads to a spectrum of singular values $\{ |\lambda_i|^2 \}_{i=1}^{d^{2N_A}}$, from which the (local-)operator entanglement (LOE) entropies are defined. Truncating and keeping only terms with the largest $\chi$ Schmidt values across each bipartition leads to an MPO approximation of $O$. We study when a sufficiently small [large] LOE implies the [non-] existence of an efficient and faithful MPO approximation.
• Figure 1: Results on the approximability of an operator's properties from the scaling of its LOE Rényi entropies, Eq. \ref{['eq:loe']}. Simulability ('sim.') is defined in Def. \ref{['def:sim']}, $\mathcal{E}_b$ refers to ensembles of states with a sufficiently mixed first moment (Def. \ref{['def:low-av']}), $X$ is some other normalized and bounded Hermitian operator, and OTOCs are defined in Eq. \ref{['eq:otocDef']}. The top row summarizes Thm. \ref{['thm:micro']} and the bottom Thms. \ref{['thm:itac']} and \ref{['thm:otoc']}.
• Figure 2: Numerical results on 1D brickwork dynamics of an initially local $\sigma_z$ on the center site, for the XXZ model [blue, circles] and the kicked Ising model (KIM) [red, squares]. On the left, for $N=12$ we plot the spectral [solid] and the Hilbert-Schmidt norm [dotted] of $\Delta O = O_N - \tilde{O}_{N,\chi}^{(N/2)}$, given a half-chain truncation with a cutoff of $\varepsilon = 0.02$. On the right, we plot the distribution of spectral norms of the matrices $A_i \otimes B_i$ from the corresponding operator Schmidt decomposition, after $t=20$ layers of evolution. The maximum values are shown as a star, and the quartiles are marked.
• Figure 3: A vectorized operator $| O \rrangle$ under the action of a brickwork circuit in the folded picture. Red bricks (foreground) are the two-site unitary $U$, and yellow bricks (background) denote its conjugate $U^*$. Here, we show how the lightcone in such circuits is exact for an initially local operator $O$. The shown circuit is on $N=8$ qudits, and $t=8$ layers of the Floquet evolution.
• Figure 4: Additional numerical results. (a) We first compute the truncation errors, as studied in the right panel of Fig. \ref{['fig:numerics']}, but for different system sizes $N=\{8,10,12\}$ and also including the integrable transverse field Ising model (TFIM). For different system sizes, the light cone of the initially local operator (on the center spin) hits the boundary at different depths, which are marked by suitably colored, vertical dotted lines. (b) We compute the exact von Neumann LOE of the half chain of the studied models for $N=2$ up to $t=20$ layers of evolution. (c) We provide data on the spectral norm distribution of $A_i \otimes B_i$ for the half-chain bipartition for the TFIM, complementary to the results on the right panel of Fig. \ref{['fig:numerics']}.

Theorems & Definitions (9)

• Definition 1
• Theorem 1
• Definition 2
• Theorem 2
• Theorem 3
• Theorem 4
• Lemma 5
• proof
• Lemma 6