Bridging Classical and Quantum Information Scrambling with the Operator Entanglement Spectrum

TL;DR

The paper introduces the operator entanglement spectrum (OES) as a fine-grained probe of operator dynamics to distinguish classical automaton circuits from fully quantum chaotic evolution. By analyzing OES across Haar-random unitaries, random unitary circuits, and automaton dynamics, the authors uncover universal MP statistics for the DOS and WD level-spacing in quantum chaos, while automata align with a distinct Bernoulli spectrum with atoms at 0 and 1. They demonstrate a classical–quantum crossover by doping automata with superposition gates (Hadamard, Rx), showing that a finite density of such gates drives the OES toward quantum chaotic universality and induces level-spacing transitions reminiscent of Clifford-T gate behavior. The results establish OES as a powerful diagnostic for chaoticity and universality class in operator dynamics and propose efficient routes to simulate chaos-like operator dynamics via limited SG-gate doping. The work bridges classical and quantum information chaos and suggests applications in quantum complexity and cryptography.

Abstract

Universal features of chaotic quantum dynamics underlie our understanding of thermalization in closed quantum systems and the complexity of quantum computations. Reversible automaton circuits, comprised of classical logic gates, have emerged as a tractable means to study such dynamics. Despite generating no entanglement in the computational basis, these circuits nevertheless capture many features expected from fully quantum evolutions. In this work, we demonstrate that the differences between automaton dynamics and fully quantum dynamics are revealed by the operator entanglement spectrum, much like the entanglement spectrum of a quantum state distinguishes between the dynamics of states under Clifford and Haar random circuits. While the operator entanglement spectrum under random unitary dynamics is governed by the eigenvalue statistics of random Gaussian matrices, we show evidence that under random automaton dynamics it is described by the statistics of Bernoulli random matrices, whose entries are random variables taking values $0$ or $1$. We study the crossover between automaton and generic unitary operator dynamics as the automaton circuit is doped with gates that introduce superpositions, namely Hadamard or $R_x = e^{-i\fracπ{4}X}$ gates. We find that a constant number of superposition-generating gates is sufficient to drive the operator dynamics to the random-circuit universality class, similar to earlier results on Clifford circuits doped with $T$ gates. This establishes the operator entanglement spectrum as a useful tool for probing the chaoticity and universality class of quantum dynamics.

Figures (12)

• Figure 1: A cartoon demonstrating the operator entanglement spectrum for an operator $X$ (rectangular box) and a bipartition into regions $A$ (red) and $B$ (blue). Lines on the left and right sides of $X$ represent registers storing the input and output states, respectively, in the Hilbert spaces $\mathcal{H}_A$ and $\mathcal{H}_B$. Performing an appropriate singular value decomposition (SVD) results in the Schmidt decomposition of $X$ as a sum over products of operators acting in subregions $A$ and $B$, Eq. \ref{['eq:schmidt']}, as shown in the diagram on the right.
• Figure 2: Level-spacing ratios of "random" observables (points) compared to those of matrices drawn from classical random matrix ensembles (crosses). The system consists of 10 sites, and the data represent 256 matrix samples. $X$ is a Pauli-$X$ operator on a single site. $U$ is drawn from the ensemble of Haar-random unitary matrices (CUE) and $O$ is drawn from Haar-random orthogonal matrices (COE). The model distribution $p_{\text{GOE}}$ for the OES spacing ratios of GOE matrices is given in Eq. \ref{['eq:pgoe']}. Inset: the entanglement spectrum DOS of each ensemble approaches the MP distribution.
• Figure 3: Convergence and concentration of unitary-evolved OES to MP for different bipartitions on a system of $N = 10$ sites. $d_A = 2^{N_A}$ and $d_B = 2^{N_B}$ give the subsystem dimensions, and $\Delta N = N_A - N_B$. The scaling factor $\sqrt{d_B/d_A}$ originates in the identical fluctuations of the Pauli coefficients, which controls the average inner product of the columns of the matrix $\alpha_{ij}$ introduced above. The explicit derivation is given in Eq. \ref{['eq:normalization']}. The data represent 128 matrix samples, and the inset is a single sample on a system of size $n = 12$, demonstrating concentration of the spectrum.
• Figure 4: Convergence of DOS to MP and level spacings to WD as a function of circuit depth. The inset in the lower panel shows a circuit diagram of a local Pauli-$X$ gate evolved under a random unitary circuit on $N$ qubits with $l$ layers. The top left panel shows that the OES passes through a power-law with an apparently non-universal exponent $\eta$ at $l/N = \frac{1}{2}$, which is when the lightcone reaches the edge of the system. At the same time, the level spacings are nearly indistinguishable from WD. The WD distribution is not exact for large system sizes, and $10^{-4}$ is the minimum $D_{\text{KL}}$ value obtained numerically from directly sampling GUE matrices of the same system size, as shown in the figure by a horizontal gray dashed line. The top right inset shows the clear emergence of the semicircle in the OES at late times.
• Figure 5: The average OES of the Bernoulli matrix ensemble and the average OES of the ensembles $\{P\}_P$, where $P$ is a random automaton, and $\{P^tXP\}_P$, where $X$ is a Pauli-$X$, are seen to agree. The OES of $P^tXP$ for a single automaton sample $P$ also shows good agreement, suggesting that this is not just an average phenomenon. This defines another universal spectrum which is sharply distinct from the semicircle law for random unitary matrices. The prediction for the average entanglement rank is $d(1-p(\{0\}))$, where $p(\{0\}) \gtrsim \mathrm e^{-\frac{1}{\mathrm e}}+\frac{2}{\mathrm e} -1 \approx 0.43$. This prediction is made by using the independence assumption to estimate the average number of unique columns in a sample and is shown by the dashed line (rescaled by the bin size). The numerical value is $p(\{0\}) \approx 0.46$

Theorems & Definitions (8)

• Conjecture 1
• Lemma 1
• proof
• Lemma 2
• proof
• proof
• proof
• proof