There has been much recent interest in understanding entanglement dynamics and operator spreading in the Haar random circuit and Brownian circuit, which can be unified as locally scrambled quantum evolutions. Here, the authors prove that the entanglement dynamics for such quantum dynamics is Markovian and introduce the entanglement feature formulation to describe entanglement dynamics with an imaginary-time Schr\"odinger equation. This work opens up new possibilities to apply the quantum many-body physics techniques to tackle the entanglement dynamics problem.
There has been much recent interest in understanding entanglement dynamics and operator spreading in the Haar random circuit and Brownian circuit, which can be unified as locally scrambled quantum evolutions. Here, the authors prove that the entanglement dynamics for such quantum dynamics is Markovian and introduce the entanglement feature formulation to describe entanglement dynamics with an imaginary-time Schr"odinger equation. This work opens up new possibilities to apply the quantum many-body physics techniques to tackle the entanglement dynamics problem.
We study the evolution of multiregion bipartite entanglement entropy under locally scrambled quantum dynamics. We show that the multiregion entanglement can significantly modify the growth of single-region entanglement, whose effect has been largely overlooked in the existing literature. We developed a theoretical framework, called the entanglement feature formalism, to organize all the multiregion entanglement systematically as a sign-free many-body state. We further propose a two-parameter matrix product state (MPS) ansatz to efficiently capture the exponentially many multiregion entanglement features. Using these tools, we are able to study the multiregion entanglement dynamics jointly and represent the evolution in the MPS parameter space. By comparing the dynamical constraints on the motion of entanglement cuts, we are able to identify different quantum dynamics models in a unifying entanglement feature Hamiltonian. Depending on the quantum dynamics model, we find that multiregion effects can dominate the single region entanglement growth and only vanish for Haar random circuits. We calculate the operator-averaged out-of-time-order correlator based on the entanglement feature Hamiltonian and extract the butterfly velocity from the result. We show that the previously conjectured bound between the entanglement velocity and the butterfly velocity holds true even under the influence of multiregion entanglement. These developments could enable more efficient numerical simulations and more systematic theoretical understandings of the multiregion entanglement dynamics in quantum many-body systems.
Interactions in quantum systems can spread initially localized quantum information into the exponentially many degrees of freedom of the entire system. Understanding this process, known as quantum scrambling, is key to resolving several open questions in physics. Here, by measuring the time-dependent evolution and fluctuation of out-of-time-order correlators, we experimentally investigate the dynamics of quantum scrambling on a 53-qubit quantum processor. We engineer quantum circuits that distinguish operator spreading and operator entanglement and experimentally observe their respective signatures. We show that whereas operator spreading is captured by an efficient classical model, operator entanglement in idealized circuits requires exponentially scaled computational resources to simulate. These results open the path to studying complex and practically relevant physical observables with near-term quantum processors.
Information scrambling, the process by which quantum information spreads and becomes effectively inaccessible, is central to modern quantum statistical physics and quantum chaos. These lecture notes provide an introduction to information scrambling from both static and dynamical perspectives. The spectral properties of reduced density matrices arising from Haar-random states are developed through the geometry of the unitary group and the universal results of random matrix theory. This geometric framework yields universal, model-independent predictions for entanglement and spectral statistics, capturing generic features of quantum chaos without reference to microscopic details. Dynamical diagnostics such as the spectral form factor and out-of-time-ordered correlators further reveal the onset of chaos in time-dependent evolution.
The notes are aimed at advanced undergraduate and graduate students in physics, mathematics, and computer science who are interested in the connections between quantum chaos, information dynamics, and quantum computing. Concepts such as entanglement growth, Haar randomness, random-matrix statistics, and unitary t-designs are introduced through their realization in random quantum circuits and circuit complexity. The same mathematical framework also underpins modern quantum-device benchmarking, where approximate unitary designs and random circuits translate geometric ideas into quantitative tools for assessing fidelity, noise, and scrambling efficiency in real quantum processors.
The evolution of entanglement entropy in quantum circuits composed of Haar-random gates and projective measurements shows versatile behavior, with connections to phase transitions and complexity theory. We reformulate the problem in terms of a classical Markov process for the dynamics of bipartition purities and establish a probabilistic cellular-automaton algorithm to compute entanglement entropy in monitored random circuits on arbitrary graphs. In one dimension, we further relate the evolution of the entropy to a simple classical spin model that naturally generalizes a two-dimensional lattice percolation problem. We also establish a Markov model for the evolution of the zeroth Rényi entropy and demonstrate that, in one dimension and in the limit of large local dimension, it coincides with the corresponding second-Rényi-entropy model. Finally, we extend the Markovian description to a more general setting that incorporates continuous-time dynamics, defined by stochastic Hamiltonians and weak local measurements continuously monitoring the system.
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 π 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.
The cost of classical simulations of quantum many-body dynamics is often determined by the amount of entanglement in the system. In this paper, we study entanglement in stochastic quantum trajectory approaches that solve master equations describing open quantum system dynamics. First, we introduce and compare adaptive trajectory unravelings of master equations. Specifically, building on [Phys. Rev. Lett. 128, 243601 (2022)], we study several greedy algorithms that generate trajectories with a low average entanglement entropy. Second, we consider various conventional unravelings of a one-dimensional open random Brownian circuit and locate the transition points from area- to volume-law-entangled trajectories. Third, we compare various trajectory unravelings using matrix product states with a direct integration of the master equation using matrix product operators. We provide concrete examples of dynamics, for which the simulation cost of stochastic trajectories is exponentially smaller than the one of matrix product operators.