Operator dynamics in a Brownian quantum circuit

Key issues review: useful autonomous quantum machines

Controlled quantum machines have matured significantly. A natural next step is to increasingly grant them autonomy, freeing them from time-dependent external control. For example, autonomy could pare down the classical control wires that heat and decohere quantum circuits; and an autonomous quantum refrigerator recently reset a superconducting qubit to near its ground state, as is necessary before a computation. Which fundamental conditions are necessary for realizing useful autonomous quantum machines? Inspired by recent quantum thermodynamics and chemistry, we posit conditions analogous to DiVincenzo's criteria for quantum computing. Furthermore, we illustrate the criteria with multiple autonomous quantum machines (refrigerators, circuits, clocks, etc) and multiple candidate platforms (neutral atoms, molecules, superconducting qubits, etc). Our criteria are intended to foment and guide the development of useful autonomous quantum machines.

Critical Spontaneous Breaking of U(1) Symmetry at Zero Temperature in One Dimension

The Hohenberg-Mermin-Wagner theorem states that there is no spontaneous breaking of continuous internal symmetries in spatial dimensions d≤2 at finite temperature. At zero temperature, the quantum-to-classical mapping further implies the absence of such symmetry breaking in one dimension, which is also known as Coleman's theorem in the context of relativistic quantum field theories. One route to violate this "folklore" is requiring an order parameter to commute with a Hamiltonian, as in the classic example of the Heisenberg ferromagnet and its variations. However, a systematic understanding has been lacking. In this Letter, we propose a family of one-dimensional models that display spontaneous breaking of a U(1) symmetry at zero temperature, where the order parameter does not commute with the Hamiltonian. While our models can be deformed continuously within the same phase, there exist symmetry-preserving perturbations that render the observed symmetry breaking fragile. We argue that a more general condition for this behavior is that the Hamiltonian is frustration-free.

Signature of Scramblon Effective Field Theory in Random Spin Models

Information scrambling refers to the propagation of information throughout a quantum system. Its study not only contributes to our understanding of thermalization but also has wide implications in quantum information and black hole physics. Recent studies suggest that information scrambling in large-N systems with all-to-all interactions is mediated by collective modes called scramblons. However, a criterion for the validity of scramblon theory in a specific model is still missing. In this work, we address this issue by investigating the signature of the scramblon effective theory in random spin models with all-to-all interactions. We demonstrate that, in scenarios where the scramblon description holds, the late-time operator size distribution can be predicted from its early-time value, requiring no free parameters. As an illustration, we examine whether Brownian circuits exhibit a scramblon description and obtain a positive confirmation both analytically and numerically. Our findings provide a concrete experimental framework for unraveling the scramblon field theory in random spin models using quantum simulators.

Relationship between Information Scrambling and Quantum Darwinism

A quantum system interacting with a multipartite environment can induce redundant encoding of the information of a system into the environment, which is the essence of quantum Darwinism. At the same time, the environment may scramble the initially localized information about the system. Based on a collision model, we mainly investigate the relationship between information scrambling in an environment and the emergence of quantum Darwinism. Our results show that when the mutual information between the system and environmental fragment is a linear increasing function of the fragment size, the tripartite mutual information (TMI) is zero, which can be proved generally beyond the collision model; when the system exhibits Darwinistic behavior, the TMI is positive (i.e., scrambling does not occur); when we see the behavior of an "encoding" environment, the TMI is negative (i.e., scrambling occurs). Additionally, we give a physical explanation for the above results by considering two simple but illustrative examples. Moreover, depending on the nature of system and environment interactions, it is also shown that the single qubit and two-qubit systems behave differently for the emergence of quantum Darwinism, and hence the scrambling, while their relationship is consistent with the above conclusion.

Unifying Emergent Hydrodynamics and Lindbladian Low-Energy Spectra across Symmetries, Constraints, and Long-Range Interactions

We identify emergent hydrodynamics governing charge transport in Brownian random time evolution with various symmetries, constraints, and ranges of interactions. This is accomplished via a mapping between the averaged dynamics and the low-energy spectrum of a Lindblad operator, which acts as an effective Hamiltonian in a doubled Hilbert space. By explicitly constructing dispersive excited states of this effective Hamiltonian using a single-mode approximation, we provide a comprehensive understanding of diffusive, subdiffusive, and superdiffusive relaxation in many-body systems with conserved multipole moments and variable interaction ranges. Our approach further allows us to identify exotic Krylov-space-resolved diffusive relaxation despite the presence of dipole conservation, which we verify numerically. Therefore, we provide a general and versatile framework to qualitatively understand the dynamics of conserved operators under random unitary time evolution.

Dynamical Transition of Operator Size Growth in Quantum Systems Embedded in an Environment

In closed generic many-body systems, unitary evolution disperses local quantum information into highly nonlocal objects, resulting in thermalization. Such a process is called information scrambling, whose swiftness is quantified by the operator size growth. However, the impact of couplings to the environment on the process of information scrambling remains unexplored for quantum systems embedded within an environment. Here we predict a dynamical transition in quantum systems with all-to-all interactions accompanied by an environment, which separates two phases. In the dissipative phase, information scrambling halts as the operator size decays with time, while in the scrambling phase, dispersion of information persists, and the operator size grows and saturates to an O(N) value in the long-time limit with N the number of degrees of freedom of the systems. The transition is driven by the competition between the system's intrinsic and environment propelled scramblings and the environment-induced dissipation. Our prediction is derived from a general argument based on epidemiological models and demonstrated analytically via solvable Brownian Sachdev-Ye-Kitaev models. We provide further evidence which suggests that the transition is generic to quantum chaotic systems when coupled to an environment. Our study sheds light on the fundamental behavior of quantum systems in the presence of an environment.

Operator growth from global out-of-time-order correlators

In chaotic many-body systems, scrambling or the operator growth can be diagnosed by out-of-time-order correlators of local operators. We show that operator growth also has a sharp imprint in out-of-time-order correlators of global operators. In particular, the characteristic spacetime shape of growing local operators can be accessed using global measurements without any local control or readout. Building on an earlier conjectured phase diagram for operator growth in chaotic systems with power-law interactions, we show that existing nuclear spin data for out-of-time-order correlators of global operators are well fit by our theory. We also predict super-polynomial operator growth in dipolar systems in 3d and discuss the potential observation of this physics in future experiments with nuclear spins and ultra-cold polar molecules.

Operator Relaxation and the Optimal Depth of Classical Shadows

Classical shadows are a powerful method for learning many properties of quantum states in a sample-efficient manner, by making use of randomized measurements. Here we study the sample complexity of learning the expectation value of Pauli operators via "shallow shadows," a recently proposed version of classical shadows in which the randomization step is effected by a local unitary circuit of variable depth t. We show that the shadow norm (the quantity controlling the sample complexity) is expressed in terms of properties of the Heisenberg time evolution of operators under the randomizing ("twirling") circuit-namely the evolution of the weight distribution characterizing the number of sites on which an operator acts nontrivially. For spatially contiguous Pauli operators of weight k, this entails a competition between two processes: operator spreading (whereby the support of an operator grows over time, increasing its weight) and operator relaxation (whereby the bulk of the operator develops an equilibrium density of identity operators, decreasing its weight). From this simple picture we derive (i) an upper bound on the shadow norm which, for depth t∼log(k), guarantees an exponential gain in sample complexity over the t=0 protocol in any spatial dimension, and (ii) quantitative results in one dimension within a mean-field approximation, including a universal subleading correction to the optimal depth, found to be in excellent agreement with infinite matrix product state numerical simulations. Our Letter connects fundamental ideas in quantum many-body dynamics to applications in quantum information science, and paves the way to highly optimized protocols for learning different properties of quantum states.

Entanglement distribution in the quantum symmetric simple exclusion process

We study the probability distribution of entanglement in the quantum symmetric simple exclusion process, a model of fermions hopping with random Brownian amplitudes between neighboring sites. We consider a protocol where the system is initialized in a pure product state of M particles, and we focus on the late-time distribution of Rényi-q entropies for a subsystem of size ℓ. By means of a Coulomb gas approach from random matrix theory, we compute analytically the large-deviation function of the entropy in the thermodynamic limit. For q>1, we show that, depending on the value of the ratio ℓ/M, the entropy distribution displays either two or three distinct regimes, ranging from low to high entanglement. These are connected by points where the probability density features singularities in its third derivative, which can be understood in terms of a transition in the corresponding charge density of the Coulomb gas. Our analytic results are supported by numerical Monte Carlo simulations.

Minimal Model for Fast Scrambling

We study quantum information scrambling in spin models with both long-range all-to-all and short-range interactions. We argue that a simple global, spatially homogeneous interaction together with local chaotic dynamics is sufficient to give rise to fast scrambling, which describes the spread of quantum information over the entire system in a time that is logarithmic in the system size. This is illustrated in two tractable models: (1) a random circuit with Haar random local unitaries and a global interaction and (2) a classical model of globally coupled nonlinear oscillators. We use exact numerics to provide further evidence by studying the time evolution of an out-of-time-order correlator and entanglement entropy in spin chains of intermediate sizes. Our results pave the way towards experimental investigations of fast scrambling and aspects of quantum gravity with quantum simulators.

Operator Lévy Flight: Light Cones in Chaotic Long-Range Interacting Systems

We argue that chaotic power-law interacting systems have emergent limits on information propagation, analogous to relativistic light cones, which depend on the spatial dimension d and the exponent α governing the decay of interactions. Using the dephasing nature of quantum chaos, we map the problem to a stochastic model with a known phase diagram. A linear light cone results for α≥d+1/2. We also provide a Lévy flight (long-range random walk) interpretation of the results and show consistent numerical data for 1D long-range spin models with 200 sites.

Quantum Autoencoders to Denoise Quantum Data

Entangled states are an important resource for quantum computation, communication, metrology, and the simulation of many-body systems. However, noise limits the experimental preparation of such states. Classical data can be efficiently denoised by autoencoders-neural networks trained in unsupervised manner. We develop a novel quantum autoencoder that successfully denoises Greenberger-Horne-Zeilinger, W, Dicke, and cluster states subject to spin-flip errors and random unitary noise. Various emergent quantum technologies could benefit from the proposed unsupervised quantum neural networks.