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Random Quantum Circuits
Matthew P. A. Fisher, Vedika Khemani, Adam Nahum, Sagar Vijay
TL;DR
The paper asks what collective phenomena and dynamical phases can emerge from quantum-simulator operations. It introduces quantum-circuit models and shows that monitoring can produce new dynamical universality classes, including an entangled-to-disentangled phase transition.
Problem
The paper asks what collective quantum phenomena and dynamical phases can emerge from operations native to quantum simulators.
Method
The article reviews discrete-time quantum-circuit models built from local unitary gates and measurements, covering unitary and monitored dynamics.
Results
Monitored circuits exhibit new dynamical universality classes, including a monitoring-induced transition between entangled and disentangled phases.
Takeaways & Limitations
Quantum circuits provide a tractable setting for studying monitored many-body phases and associated transitions in open quantum dynamics.
Takeaways & Limitations
The minimal-cut description of entanglement entropy is exact only in the limit q = ∞, although aspects of its phenomenology persist at finite q.
Abstract
from arXiv · showhide
Quantum circuits -- built from local unitary gates and local measurements -- are a new playground for quantum many-body physics and a tractable setting to explore universal collective phenomena far-from-equilibrium. These models have shed light on longstanding questions about thermalization and chaos, and on the underlying universal dynamics of quantum information and entanglement. In addition, such models generate new sets of questions and give rise to phenomena with no traditional analog, such as new dynamical phases in quantum systems that are monitored by an external observer. Quantum circuit dynamics is also topical in view of experimental progress in building digital quantum simulators that allow control of precisely these ingredients. Randomness in the circuit elements allows a high level of theoretical control, with a key theme being mappings between real-time quantum dynamics and effective classical lattice models or dynamical processes. Many of the universal phenomena that can be identified in this tractable setting apply to much wider classes of more structured many-body dynamics.
1. Introduction
The introduction presents random quantum circuits as tractable models of far-from-equilibrium many-body dynamics, enabled by local gates and measurements and motivated by digital quantum simulators. It emphasizes universal entanglement and dynamical phenomena, including new universality classes in monitored circuits.
- Motivation: Digital quantum simulators evolve qubits through discrete-time unitary operations, measurements, and feedback, motivating quantum circuits as a new condensed-matter setting.These operations also advance the development of fault-tolerant quantum computers.
- Circuit models: Quantum circuits model lattices of spins or qubits evolving under local unitary gates and measurements, with finite time steps reminiscent of Hamiltonian Trotterization.The time step is not assumed to be infinitesimal.
- Universal probes: Minimally structured unitary circuits rapidly reach locally infinite-temperature steady states, shifting attention from conventional local-operator correlations toward abstract measures such as entanglement.Entanglement is central to understanding locally irreversible thermalization under unitary evolution.
- Randomness and universality: Random circuit ensembles provide theoretical control over universal out-of-equilibrium phenomena that can also arise in more structured quantum many-body dynamics.Randomness in local operations makes these models theoretically tractable.
- Circuit dynamics: The article examines both unitary circuits and monitored circuits, where repeated measurements by an external observer produce new dynamical universality classes.Unitary circuits address out-of-equilibrium dynamics and chaos, while monitored circuits involve nonunitary evolution.
2. Models and motivation
This section introduces spatially local quantum circuits of qubits or qudits, built from local gates and, for monitored dynamics, local projective measurements. It develops entanglement-based correlation measures and motivates random circuits as tractable models for nonequilibrium dynamics, scrambling, monitored phases, and mappings to classical statistical mechanics.
- Correlation measures: Entanglement entropy and bipartite mutual information provide basis-independent measures of quantum correlations used throughout the review.The entanglement entropy quantifies correlations in bipartitions, while mutual information measures correlations between subsystems whose union need not be the whole system.
- Circuit ingredients: Quantum circuits evolve spatially local qubits or qudits through local gates, with local projective measurements added for monitored dynamics.The review focuses on gates acting on a few nearby qubits and, in its second part, local projective measurements.
- Correlation measures: A two-qubit unitary interaction can create entanglement, while measuring one outcome leaves the two-qubit system in a pure state with disentangled constituent qubits.For the entangled state, each single-qubit reduced density matrix is maximally mixed and has entanglement entropy ln 2; after measurement, the qubits are disentangled.
- Motivation: Repeated measurements generate a new random-walk-like dynamics through Hilbert space, opening a landscape of monitored phases, phase transitions, and computational-complexity transitions.The monitored model is also presented as a toy model for broader questions in open or monitored quantum dynamics.
- Motivation: Random unitary circuits model nonequilibrium dynamics, thermalization, entanglement generation, and scrambling, while randomness promotes solvability.All-to-all random circuits have also been used as models for scrambling in black holes.
- Mappings: A central theme is mapping real-time quantum dynamics to effective classical statistical-mechanics models, with large-q limits exposing geometric properties such as min-cut and lightcone structures.These mappings provide insight into observables in simple limits, including large local Hilbert-space dimension q.
3. Unitary circuit dynamics
This section examines local unitary circuit dynamics, beginning with minimally structured models and extending to circuits with additional symmetries or classical simulability. It uses entanglement and locality-based bounds to characterize quantum-information spreading and entanglement growth.
- Scope: The section begins with minimally structured local unitary dynamics before treating circuits with hydrodynamic modes, time-translation symmetry, space-time rotation symmetry, or classical simulability.These extensions are organized as additional symmetry or invariance properties.
- Entanglement and correlations: Entanglement entropies and mutual informations quantify correlations without requiring structure beyond locality, illuminating quantum-information spreading over large lengthscales.The discussion then turns to conventional local correlation functions.
- Locality bounds: Local gates bound information-spreading speed by unity geometrically, while the butterfly velocity vB for operator spreading is typically lower.Circuit structure also constrains entanglement growth.
3.1. The entanglement membrane and the pairing order parameter
The entanglement membrane describes entropy as the minimum spacetime cost of a surface governed by a model-dependent tension E(v). In random circuits, pairing structures map Rényi entropies to domain-wall or directed-polymer problems whose large-scale behavior includes universal fluctuations.
- Unitary evolution typically drives weakly entangled product-like states into volume-law states, underlying pure-state thermalization and limiting tensor-network simulations.
- The entanglement membrane: The membrane picture computes SA by minimizing an effective entanglement cost over a d-dimensional surface in d + 1-dimensional spacetime anchored to ∂A.
- The entanglement membrane: The membrane tension E(v) depends on slope velocity v and is constrained by causality and unitarity, while E(0) = vE gives the product-state entanglement growth rate.For parity-symmetric systems, E(v) is convex with its minimum at v = 0; vE and vB are generally distinct.
- The entanglement membrane: At q →∞, entropy is exactly the cost of a minimal directed cut, which yields a deterministic large-scale line tension while retaining realization-dependent subleading fluctuations.The minimal-cut description is exact only at q = ∞, although aspects of its phenomenology survive at finite q.
- The pairing order parameter: Unitarity organizes multiple circuit paths into pairing degrees of freedom, mapping e−S2A to an Ising model with a single directed domain wall and, at large scales, to a directed polymer in a random medium.Distinct Rényi entropies generally have distinct line tensions En(v), and domain-wall counting produces the Page subleading correction.
3.2. Spreading and decay of correlations
Random quantum circuits describe scrambling through an operator-wavefunction and an effective classical cluster-growth process. In one dimension, operator fronts spread ballistically with diffusively broadening edges, while atypical compact trajectories govern time-ordered correlator decay and produce an unbinding transition in the decay rate.
- Operator dynamics: Scrambling is represented by a local Pauli operator whose increasing complexity and spatial extent store initially local information nonlocally.The operator expands into Pauli strings, with normalized coefficients interpreted as an operator wavefunction.
- Operator dynamics: Haar averaging maps operator-string evolution to a Markov process, and initially local operators to classical cluster growth with stochastically moving boundaries.In 1+1D, the boundaries undergo biased diffusion.
- Operator spreading: In 1+1 dimensions, operator-cluster boundaries spread ballistically at ±vB, with vB approaching the circuit light-cone speed v = 1 as q →∞ and remaining below 1 at finite q.The OTOC saturates inside |r| ≲vBt and is exponentially small well outside the front.
- Operator spreading: Endpoint fluctuations broaden the operator edge diffusively as t1/2 in 1+1 dimensions, rounding the OTOC plateau.The same endpoint dynamics is described by biased random walks.
- Correlation decay: Time-ordered correlators are governed by atypical compact operator trajectories rather than typical ballistic growth, yielding exponential relaxation G(vt, t)2 ∼exp(−r(v)t).The relaxation reflects decoherence of local degrees of freedom on an order-1 timescale by the remaining degrees of freedom.
- Correlation decay: The decay rate r(v) has an unbinding transition: endpoint trajectories dominate while bound for v > vB and unbound for v < vB, producing a non-analyticity in r(v).Correlators are generically dominated by trajectories more spatially compact than the operator support, enabling more efficient Heisenberg-picture evaluation.
3.3. Structured unitary circuits
Structured quantum circuits reveal how conservation laws generate hydrodynamic behavior, while special circuit constructions make operator dynamics, entanglement, and spectral chaos analytically tractable. These models also expose unusual phenomena including Hilbert-space shattering and spacetime-localized correlations.
- Conservation laws: U(1)-symmetric random circuits produce diffusive conserved-density modes and power-law late-time tails in time-ordered and out-of-time-ordered correlators.Coarse-graining yields a diffusion equation with diffusion constant D = 1/2.
- Conservation laws: Conserved operators slowly convert into nonconserved operators, leaving a ballistic operator front with power-law tails from lagging emitted fronts.The conserved weight decreases as a power law because conversion occurs at a rate proportional to the square of the local conserved current.
- Conservation laws: Dipole-conserving circuits exhibit ergodicity breaking through shattering into exponentially many dynamically disconnected Hilbert-space sectors.Some states with identical charge and dipole quantum numbers still have no dynamical path between them.
- Floquet circuits: Chaotic Floquet circuits reproduce random-matrix spectral statistics: after a Thouless time tTh ∼log L, an ordered Potts description yields the linear SFF ramp K(t) = t.The ramp reflects level repulsion across the eigenspectrum, while shorter times correspond to a disordered regime with domain walls.
- Dual-unitary circuits: Dual-unitary circuits impose two compatible arrows of time, restricting nonzero correlations to the light-cone rays |x| = |t|.Spacetime duality is generally nonunitary, but dual-unitary gates retain unitarity under the spacetime-flipped mapping.
- Classically tractable circuits: Automaton circuits make operator growth classically tractable, while higher-order Green’s-function moments retain nontrivial quantum-coherence structure beyond the symmetric exclusion process.The operator wavefunction follows the same automaton circuit in a rotated basis, and higher moments differ from the simplest classical mapping.
4. Monitored dynamics
Monitored quantum circuits exhibit measurement-induced transitions between volume-law and area-law entangled trajectory phases, with corresponding changes in classical simulability, purification, and encoded quantum information. Their entanglement properties admit statistical-mechanics descriptions with universal scaling and robustness signatures.
- Measurement-induced transition: Measurements break tensor-network bonds and reduce line tension, causing the volume-law coefficient to vanish critically at the transition.For sufficiently small p, enough unbroken bonds remain for large-scale circuit connectivity.
- Classical simulation: Area-law states above pc can be efficiently represented by matrix product states, while volume-law states below pc are a priori exponentially costly to store and simulate.Thus the measurement transition also marks an easy-to-hard crossover for classical simulation of quantum processes.
- Entanglement scaling: In the volume-law phase, entanglement maps to a directed polymer in a random environment and follows SA = s0|A| + b|A|^β with β = 1/3.The mapping treats subsystem entanglement as a free-energy cost associated with changing final-time boundary conditions.
- Entanglement scaling: A local measurement a distance x from a region boundary reduces entanglement as ⟨δS(x)⟩∼x−∆, with ∆≈1.25 observed in hybrid Clifford dynamics and related circuit behavior.The average is dominated by rare events, while typical realizations are more robust and exhibit stretched-exponential decay.
- Purification and quantum information: The same transition is a purification transition: p > pc purifies a maximally mixed state at a system-size-independent rate, whereas p < pc has exponentially diverging purification time.The low-measurement phase retains a residual entropy density sQ, representing quantum information propagated from initial to final time.
5. Experiments
Experiments have established programmable quantum circuits as useful many-body-physics platforms, including a 53-qubit random-circuit sampling demonstration and measurements of dynamical properties. Monitored circuits remain experimentally challenging because measurement randomness creates a fundamental postselection barrier.
- Experimental platforms: Programmable platforms now isolate coherent qubits, implement controlled local unitaries, and perform high-fidelity local measurements across superconducting-junction and trapped-ion architectures.These capabilities support readout, control, and feedback, although engineering challenges remain.
- Experimental platforms: Google’s 53-qubit “quantum supremacy” experiment sampled the output distribution of a random quantum circuit, marking the emergence of genuinely many-body coupled systems.The demonstration shifted experimental emphasis from individual circuit elements toward systems with vast Hilbert spaces.
- Experimental platforms: Experiments use quantum circuits as accessible platforms for studying non-equilibrium dynamics rather than only as prospective universal simulators of equilibrium many-body problems.The present operational mode of these devices directly opens regimes of dynamical many-body physics.
- Measurement protocols: Two-copy partial-SWAP measurements and ancilla-assisted protocols provide alternatives to tomography for measuring Rényi-type quantities and correlation functions.Partial-SWAP methods require preparing multiple copies, while Hadamard-test protocols access new correlation functions.
- Experimental studies: Experiments have probed information scrambling in chaotic quantum circuits, extending circuit platforms to fundamental questions about quantum dynamics and thermalization.These studies complement milestone experiments on analog simulator platforms.
- Monitored circuits: Monitored circuits face a prohibitive postselection barrier because preparing multiple copies of a trajectory with specified measurement outcomes is fundamentally difficult.The obstacle arises from randomness in the measurement process, not merely from experimental engineering limitations.
6. Outlook
The outlook identifies open questions about feedback, monitored entanglement transitions, effective-model structure, and extending random-circuit insights to conventional systems. It also asks whether monitored dynamics can support practical benchmarking and quantum error correction.
- Feedback and error correction: Feedback-conditioned operations remain poorly understood, including whether they can stabilize ordered quantum phases and be implemented in digital quantum simulators.Using feedback to stabilize a many-body phase is less stringent than recovering a particular encoded state with high fidelity.
- Monitored entanglement: Monitored entanglement transitions remain difficult to observe and fully characterize because pure-state entanglement measurements generally require extensive post-selection.Open directions include avoiding this barrier beyond Clifford or dual-unitary dynamics and understanding the transition through stabilizer evolution, continuous-symmetry statistical mechanics, and coarse-graining.
- Effective models: The effective-lattice-model formalism still lacks a complete understanding of its structure, including combinatorial features of replica-like limits and applications to broader observables.The discussion emphasizes both unitary and nonunitary dynamics.
- Beyond random circuits: Extending random-circuit results to non-random systems with fixed Hamiltonians remains an important unfinished direction.The target systems may be defined on lattices or in the continuum.
- Benchmarking and codes: Monitored dynamics may yield benchmarking tasks and quantum error-correcting codes, but the usefulness of alternatives to time-periodic constructions and practical decoding remains unresolved.Time-periodic monitored dynamics have already been used to construct dynamically evolving, fault-tolerant codes.
DISCLOSURE STATEMENT · LITERATURE CITED
The disclosure statement reports that the authors are unaware of affiliations, memberships, funding, or financial holdings that could be perceived as affecting this review’s objectivity.
- DISCLOSURE STATEMENT: The authors report no known conflicts affecting the review’s objectivity.The statement covers affiliations, memberships, funding, and financial holdings.