Operator Spreading in Random Unitary Circuits

Abstract

Random quantum circuits yield minimally structured models for chaotic quantum dynamics, able to capture for example universal properties of entanglement growth. We provide exact results and coarse-grained models for the spreading of operators by quantum circuits made of Haar-random unitaries. We study both 1+1D and higher dimensions, and argue that the coarse-grained pictures carry over to operator spreading in generic many-body systems. In 1+1D, we demonstrate that the out-of-time-order correlator (OTOC) satisfies a biased diffusion equation, which gives exact results for the spatial profile of the OTOC, and the butterfly speed $v_B$. We find that in 1+1D the `front' of the OTOC broadens diffusively, with a width scaling in time as $t^{1/2}$. We address fluctuations in the OTOC between different realizations of the random circuit, arguing that they are negligible in comparison to the broadening of the front. Turning to higher D, we show that the averaged OTOC can be understood exactly via a remarkable correspondence with a classical droplet growth problem. This implies that the width of the front of the averaged OTOC scales as $t^{1/3}$ in 2+1D and $t^{0.24}$ in 3+1D (KPZ exponents). We support our analytic argument with simulations in 2+1D. We point out that, in a lattice model, the late time shape of the spreading operator is in general not spherical. However when full spatial rotational symmetry is present in 2+1D, our mapping implies an exact asymptotic form for the OTOC in terms of the Tracy-Widom distribution. For an alternative perspective on the OTOC in 1+1D, we map it to the partition function of an Ising-like model. As a result of special structure arising from unitarity, this partition function reduces to a random walk calculation which can be performed exactly. We also use this mapping to give exact results for entanglement growth in 1+1D circuits.

29 pages, 16 figures. v2: new appendix on 'mean field'

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• Signatures of fragmentation for periodically driven fermions
• Phase transitions in sampling and error correction in local Brownian circuits
• Free fermions under adaptive quantum dynamics
• Dynamical phase transitions of information flow in random quantum circuits
• Scheme to Detect the Strong-to-weak Symmetry Breaking via Randomized Measurements
• Entanglement of Disjoint Intervals in Dual-Unitary Circuits: Exact Results
• Out-of-Time Ordered Correlators in Kicked Coupled Tops: Information Scrambling in Mixed Phase Space and the Role of Conserved Quantities
• Measurement-Induced Phase Transition in State Estimation of Chaotic Systems and the Directed Polymer
• Universal distributions of overlaps from generic dynamics in quantum many-body systems
• Operator spreading in quantum hardcore gases
• Dilution of error in digital Hamiltonian simulation
• Symmetry protected entanglement in random mixed states
• Dynamics of operator size distribution in q-local quantum Brownian SYK and spin models
• Accelerated Decay due to Operator Spreading in Bulk-Dissipated Quantum Systems
• Quantum reservoir probing: an inverse paradigm of quantum reservoir computing for exploring quantum many-body physics
• Non-Randomness of Google's Quantum Supremacy Benchmark
• Superiority in dense coding through non-Markovian stochasticity
• Real-time dynamics of 1D and 2D bosonic quantum matter deep in the many-body localized phase
• Multi-particle interferometry in the time-energy domain with localized topological quasiparticles
• Hamiltonian Dynamics of a Sum of Interacting Random Matrices
• Fast scrambling dynamics and many-body localization transition in an all-to-all disordered quantum spin model
• Effective field theory of random quantum circuits
• Tunable Geometries in Sparse Clifford Circuits
• Quantum chaos in PT symmetric quantum systems
• Subsystem symmetry, spin glass order, and criticality from random measurements in a two-dimensional Bacon-Shor circuit
• Local spreading of stabilizer Rényi entropy in a brickwork random Clifford circuit
• Structural Stability Hypothesis of Dual Unitary Quantum Chaos
• Classification of same-gate quantum circuits and their space-time symmetries with application to the level-spacing distribution
• Relaxation exponents of OTOCs and overlap with local Hamiltonians
• Non-Haar random circuits form unitary designs as fast as Haar random circuits
• Nontrivial damping of quantum many-body dynamics
• Efficient post-selection in light-cone correlations of monitored quantum circuits
• Long-ranged spectral correlations in eigenstate phases
• Exact quench dynamics of symmetry resolved entanglement in a free fermion chain
• Reaction-diffusion dynamics in a Fibonacci chain: Interplay between classical and quantum behavior
• Classical simulation of bosonic linear-optical random circuits beyond linear light cone
• Exotic equilibration dynamics on a 1-D quantum CNOT gate lattice
• Slow melting of a disordered quantum crystal
• Traveling discontinuity at the quantum butterfly front
• A Theory of Direct Randomized Benchmarking
• Noise Induced Universal Diffusive Transport in Fermionic Chains
• Purification Dynamics in a Continuous-time Hybrid Quantum Circuit Model
• Scrambling Transition in Free Fermion Systems Induced by a Single Impurity
• Fidelity decay and error accumulation in random quantum circuits
• Architectures and random properties of symplectic quantum circuits
• Emergence of Navier-Stokes hydrodynamics in chaotic quantum circuits
• Exact spectral gaps of random one-dimensional quantum circuits
• Coherent Information Phase Transition in a Noisy Quantum Circuit
• Random Quantum Circuits with Time-Reversal Symmetry
• Solvable Quantum Circuits in Tree+1 Dimensions
• Emergent random matrix universality in quantum operator dynamics
• Entanglement dynamics and Page curves in random permutation circuits
• Decoding across transversal Clifford gates in the surface code
• Machine learning, quantum chaos, and pseudorandom evolution
• Entanglement growth and information capacity in a quasiperiodic system with a single-particle mobility edge
• Dynamics of entanglement fluctuations and quantum Mpemba effect in the $\nu =1$ QSSEP model
• Charge-conserving unitaries typically generate optimal covariant quantum error-correcting codes
• Quantum many-body physics from a gravitational lens
• Void formation in operator growth, entanglement, and unitarity
• Spectral chaos bounds from scaling theory of maximally efficient quantum-dynamical scrambling
• Operator Spreading in Random Unitary Circuits with Unitary-invariant Gate Distributions
• Hierarchical analytical approach to universal spectral correlations in Brownian Quantum Chaos
• Multipartite Greenberger-Horne-Zeilinger Entanglement in Monitored Random Clifford Circuits
• A recipe for local simulation of strongly-correlated fermionic matter on quantum computers: the 2D Fermi-Hubbard model
• Dynamically generated concatenated codes and their phase diagrams
• Loschmidt echo and scrambling of systematic errors in tomography -- a quantum signature of chaos
• Tangent Space Excitation Ansatz for Quantum Circuits
• Directional scrambling of quantum information in helical multiferroics
• Random Permutation Circuits Beyond Qubits are Quantum Chaotic
• Many-body chaos and anomalous diffusion across thermal phase transitions in two dimensions
• Absorption to Fluctuating Bunching States in Non-Unitary Boson Dynamics
• Error Mitigation Thresholds in Noisy Random Quantum Circuits
• Logarithmic entanglement lightcone from eigenstate correlations in the many-body localised phase
• Nonlinear entanglement growth in inhomogeneous spacetimes
• Dual-unitary shadow tomography
• Kinetics of information scrambling in correlated electrons: disorder-driven transition from shock-wave to FKPP dynamics
• Information scrambling in quantum walks: Discrete-time formulation of Krylov complexity
• Measurement-invisible quantum correlations in scrambling dynamics
• Monte Carlo Simulation of Operator Dynamics and Entanglement in Dual-Unitary Circuits
• Polynomially restricted operator growth in dynamically integrable models
• Entanglement dynamics of multi-parametric random states: a single parametric formulation
• Lieb-Robinson Bounds on Entanglement Gaps from Symmetry-Protected Topology
• Relaxation Critical Dynamics in Measurement-induced Phase Transitions
• Bridging Classical and Quantum Information Scrambling with the Operator Entanglement Spectrum
• Estimating time in quantum chaotic systems and black holes
• Anatomy of information scrambling and decoherence in the integrable Sachdev-Ye-Kitaev model
• Out-of-time-ordered commutators in Dirac--Weyl systems
• Rate of cluster decomposition via Fermat-Steiner point
• Markovian to non-Markovian phase transition in the operator dynamics of a mobile impurity
• Anticoncentration in Clifford Circuits and Beyond: From Random Tensor Networks to Pseudo-Magic States
• Lieb-Robinson bounds with exponential-in-volume tails
• Revealing quantum operator scrambling via measuring Holevo information on digital quantum simulators
• Superdiffusion resilience in Heisenberg Chains with 2D interactions on a quantum processor
• Field theory of charge sharpening in symmetric monitored quantum circuits
• Boundary Chaos: Spectral Form Factor
• Dynamics of charge-imbalance-resolved entanglement negativity after a quench in a free-fermion model
• Mechanisms of scrambling and unscrambling of quantum information in the ground state in spin chains: domain-walls, spin-flips and scattering phase shifts
• Emergent quantum state designs and biunitarity in dual-unitary circuit dynamics
• Integrals of motion as slow modes in dissipative many-body operator dynamics
• Dynamics of Loschmidt echoes from operator growth in noisy quantum many-body systems
• Optimal quantum reservoir learning in proximity to universality
• Pinned, locked, pushed, and pulled traveling waves in structured environments
• Open-system dynamics in local Lindbladians with chaotic spectra
• Qubit recycling and the path counting problem
• The butterfly effect in a Sachdev-Ye-Kitaev quantum dot system
• Rare-event properties in a classical stochastic model describing the evolution of random unitary circuits
• Haar averaged moments of correlation functions and OTOCs in Floquet systems
• Entanglement dynamics from universal low-lying modes
• Attention to Quantum Complexity
• Moments of Quantum Channel Ensembles
• Noisy Monitored Quantum Circuits
• Measurement-and Feedback-Driven Non-Equilibrium Phase Transitions on a Quantum Processor
• Operator delocalization in disordered spin chains via exact MPO marginals
• Domain-wall melting in all-to-all QSSEP from random-matrix theory
• Random quantum circuits, chaos and quantum thermalization
• Engineering entanglement geometry via spacetime-modulated measurements
• Non-Commutative weak measurements: Entanglement, Symmetry Breaking, and the Role of Readout
• Thermalization dynamics of macroscopic weakly nonintegrable maps
• Scrambling dynamics across a thermalization-localization quantum phase transition
• Fast Scrambling in Classically Simulable Quantum Circuits
• Information propagation in a non-local model with emergent locality
• Statistics-encoded tensor network approach in disordered quantum many-body spin chains
• Roughening Transition in Quantum Circuits
• Tri-unitary quantum circuits
• Solvable random matrix ensemble with a logarithmic weakly confining potential
• Fast suppression of classification error in variational quantum circuits
• Non-trivial Lyapunov spectrum from fractal quantum cellular automata
• Symmetry resolved out-of-time-order correlators of Heisenberg spin chains using projected matrix product operators
• Recent developments in the holographic description of quantum chaos
• Entanglement structure for finite system under dual-unitary dynamics
• Time Glasses: Symmetry Broken Chaotic Phase with a Finite Gap
• Krylov complexity of many-body localization: Operator localization in Krylov basis
• Phase transitions from Heating to non-heating in SU(1, 1) quantum dynamics: applications to Bose-Einstein condensates and periodically driven coupled oscillators
• Weak-Memory Dynamics in Discrete Time
• Phase Transitions in the Classical Simulability of Open Quantum Systems
• Process Tensor Approaches to Non-Markovian Quantum Dynamics
• More global randomness from less-random local gates
• Entanglement Growth from Entangled States: A Unified Perspective on Entanglement Generation and Transport
• Anticoncentration is (almost) all you need
• Krylov Winding and Emergent Coherence in Operator Growth Dynamics