Random Quantum Circuits

• Digital quantum simulators motivate studying collective quantum phenomena generated by local unitary operations, measurements, and feedback.These operations also support progress toward fault-tolerant quantum computing.Source: §1 · p. 3
• Quantum circuits evolve lattice qubits through local gates and measurements in discrete time, without requiring infinitesimal steps or energy conservation.This structure resembles Trotterized Hamiltonian evolution, but local operations generally need not be close to the identity.Source: §1 · p. 3
• Entanglement entropy and quantum mutual information provide basis-independent probes of universal structure when minimally structured unitary circuits locally approach infinite-temperature disorder.These quantities remain sensitive to locality and unitarity and expose effects not captured by specially chosen local correlation functions.Source: §1 · p. 3
• Random circuit ensembles make universal out-of-equilibrium phenomena theoretically tractable by yielding classical descriptions of evolving entanglement structures.The resulting statistical ensembles of multi-copy Feynman trajectories retain universal properties reflecting quantum correlations.Source: §1 · p. 4
• Monitored circuits introduce new dynamical universality classes, including a measurement-driven transition from entangled to disentangled phases.Weak monitoring generates highly entangled wavefunctions, whereas sufficiently frequent measurement confines evolution near product states; these phases can map to classical statistical mechanics models.Source: §1 · p. 4
• The article develops unitary circuits with entanglement growth, correlations, conserved charges, and Floquet symmetry, then analyzes monitored circuits, their transitions, symmetries, experiments, and open questions.Its circuit framework connects universal dynamics to thermalization, chaos, quantum simulation, and classical descriptions.Source: §1 · p. 4

2.1. Circuit building blocks and quantum entanglement

• Quantum circuits evolve spatially local qubits or qudits through local unitary gates and, for monitored dynamics, stochastic local projective measurements obeying Born’s rule.Measurements produce different outcome sequences across repeated runs, even when the unitary gates are held fixed.Source: §2.1 · p. 5, §2.4 · p. 10
• Entanglement entropy and mutual information provide basis-independent measures of correlations, while low-entanglement states can be efficiently represented using truncated Schmidt decompositions and matrix product states.The entanglement entropy is defined from the reduced density matrix of a subsystem, and its Schmidt spectrum quantifies the required product-state decomposition.Source: §2.1 · pp. 5–6

2.2. Quantum gates and measurements – a two-qubit example

• A controlled-Z gate transforms two initially disentangled qubits into a maximally entangled state with single-qubit entropy ln 2, whereas measuring one qubit disentangles the pair along each recorded trajectory.Averaging over measurement outcomes instead produces a mixed two-qubit state with entropy ln 2 and maximally mixed reduced states.Source: §2.2 · pp. 7–8

2.3. Brickwork random unitary circuit

• The brickwork random unitary circuit applies alternating layers of local two-site gates in discrete time, with independently Haar-random gates providing a minimally structured model of chaotic evolution.Because the circuit has no conservation laws, arbitrary states locally equilibrate to the infinite-temperature state with entropy density s_eq = ln q.Source: §2.3 · pp. 9–10

2.4. Measurements and trajectories

• In monitored circuits, measurement records define pure-state quantum trajectories whose dynamics is a random walk through Hilbert space, with local disentangling competing against unitary correlation spreading and producing a phase transition.Measurements occur at a finite rate, such as probability p per spin per timestep, which tunes the monitoring strength.Source: §2.4 · pp. 10–11
• Measurements place circuits in a landscape of monitored many-body phases and phase transitions, while also revealing transitions in the classical complexity of simulating open quantum dynamics.The measurement process may additionally serve as a toy model for quantum-information processing or encoding (106, 56, 107, 108).Source: §2.4 · p. 11

2.5. What can we hope to do with these models?

• Without measurements, random circuits offer tractable models of thermalization, entanglement generation, scrambling, and quantum-information complexity through exact or typical ensemble averages.All-to-all circuits model black-hole scrambling, while randomness promotes solvability in the spirit of random matrix theory, coding theory, and Sachdev-Ye-Kitaev models (22, 89, 90, 91; 114–117).Source: §2.5 · p. 11
• Solvable random circuits expose long-timescale quantum-information spreading and can be extended with symmetries, conservation laws, dualities, or classically simulable gate restrictions.These extensions enable studies of quantum chaos, hydrodynamics, space-time duality, and classically simulable dynamics (94, 118; 92, 93; 119; 120).Source: §2.5 · pp. 11–12
• Random circuits connect many-body theory with quantum computing through benchmarking, algorithms, NISQ-inspired dynamics, and experimentally motivated studies of quantum supremacy.Sampling sufficiently deep random-circuit outputs is believed prohibitively difficult classically, motivating experimental demonstrations and complexity-theoretic interest (121–126).Source: §2.5 · p. 12

2.6. Quantum-classical mappings

• Random circuits map multi-layer quantum dynamics to effective classical ensembles, including Markov processes, lattice magnets, domain walls, and random walks.Averaging drastically reduces the surviving spacetime configurations; small-layer limits yield dephased operator dynamics, while general-layer mappings use pairing degrees of freedom and domain walls.Source: §2.6 · p. 12

• Local unitary dynamics are organized from minimally structured circuits toward circuits with hydrodynamic modes, time-translation symmetry, space-time rotation symmetry, or classical simulability.Source: §3 · p. 13
• Entanglement entropies and mutual informations quantify correlations using only locality, while also exposing structures that make these circuits analytically tractable.Source: §3 · p. 13
• Locality bounds information propagation, with geometric speed bounded by unity and the operator butterfly velocity vB typically lower (137–139).Source: §3 · p. 13

3.1.1. Membrane picture.

• Membrane minimization gives linear entanglement growth at early times and saturation to thermal entropy through a bulk-closing configuration for finite regions.The membrane is anchored to ∂A, with cost controlled by a slope-dependent tension E(v); saturation occurs at a time proportional to the region’s linear size.Source: §3.1.1 · pp. 13–14
• For a product-state quench, parity symmetry makes the optimal membrane straight, giving Sy(t) = seqvEt while distinguishing entanglement velocity vE from operator velocity vB.The line tension satisfies E(0) = vE, whereas causality gives E(vB) = vB and E′(vB) = 1.Source: §3.1.1 · p. 14

3.1.2. Minimal cut.

• At q →∞, entanglement is exactly the cost of a minimal directed cut, whose deterministic line tension controls growth and whose randomness produces KPZ fluctuations.The minimal-cut description is exact only at q = ∞; many of its phenomenological features survive at finite q.Source: §3.1.2 · pp. 15–16
• For a finite 1D region, the minimal cut predicts saturation at t⋆= ℓ/2vE with Sℓ(t > t⋆) = seqℓ, while the late-time membrane forms a V shape moving at ±vB.Before saturation, two endpoint curves contribute; afterward, a single curve connects the endpoints along the circuit lightcone.Source: §3.1.2 · p. 16

3.1.3. The pairing order parameter.

• Haar averaging pairs forward and backward Feynman trajectories, mapping the second-Rényi purity to an Ising-like model whose domain wall becomes the entanglement membrane.At q →∞ the energy-dominated domain wall reproduces the minimal cut, whereas finite q includes both energy and entropy contributions to E(v).Source: §3.1.3 · pp. 16–18
• Finite-q nonpaired fluctuations renormalize the membrane cost and generate a random local polymer potential, yielding directed-polymer universality for S2A at large scales.At large but finite q, ⊥ spins are dilute and renormalize the local domain-wall cost by order q−2.Source: §3.1.3 · pp. 18–19
• The pairing construction generalizes to permutation-valued order parameters for higher Rényi entropies, producing distinct line tensions and a Catalan-number Page correction, but von Neumann entropy requires difficult analytic continuation.Direct von Neumann calculations are currently available only in special circuit classes or alternative limits described in the passage.Source: §3.1.3 · p. 19
• Initially local operators spread ballistically under Heisenberg evolution, with their fronts broadening diffusively in (1+1) dimensions.This behavior occurs in minimally structured unitary quantum circuits.Source: §3.1.3 · p. 20
• Atypical operator Feynman trajectories can provide dominant contributions to time-ordered correlations and mean-squared correlators.The passage distinguishes two classes of atypical trajectories contributing to the mean-squared correlator.Source: §3.1.3 · p. 20
• In the appropriate scaling limit, holographic HRT entanglement prescriptions are consistent with the membrane picture.The comparison is formulated using surfaces in a higher-dimensional Anti-de Sitter space.Source: §3.1.3 · p. 20

Operator dynamics

• Scrambling delocalizes initially local information through an increasingly complex Pauli-string operator O(r,t), represented as a quantum-mechanical operator wavefunction.The string basis is orthonormal, operator normalization fixes the coefficient amplitudes, and each site takes one of four labels: identity, X, Y, or Z.Source: §3.2 · pp. 20–21
• Haar averaging maps operator-string evolution to a Markov process whose initially local seed grows as a classical cluster with biased-diffusing boundaries.In 1+1D, the cluster boundaries provide an effective stochastic description of operator growth.Source: §3.2 · p. 21

Scrambling diagnostics

• vB approaches the circuit lightcone speed v = 1 as q →∞, while finite q gives vB < 1; the OTOC saturates inside |r| ≲vBt and is exponentially small outside.The OTOC diagnoses scrambling of locally accessible quantum information and has been used across chaotic, disordered, and integrable systems (165, 166, 167).Source: §3.2 · p. 21

Fluctuations

• In 1+1D, operator edges broaden as t1/2, while higher-dimensional stochastic cluster growth can fall in the KPZ class, with β = 1/3 for d = 2.The endpoint boundaries execute biased random walks in 1+1D; in d > 1, the cluster edge behaves as a fluctuating interface.Source: §3.2 · p. 21

Correlation decay

• Time-ordered correlators are governed by atypical compact operator trajectories, producing exponential relaxation G(vt,t)2 ∼exp(−r(v)t) as local degrees of freedom decohere on order-1 timescales.The remaining degrees of freedom act as an effective bath, consistent with the absence of slow modes.Source: §3.2 · pp. 21–22
• In 1+1D Haar circuits, endpoint trajectories unbind across vB, producing a non-analytic decay rate r(v); compact trajectory dominance also makes Heisenberg-picture numerics more efficient.The compact-trajectory structure extends to dual-unitary circuits (186) and improves numerical evaluation despite ballistic operator growth (187, 188).Source: §3.2 · p. 22

3.3.1. Circuits with continuous symmetries.

• U(1)-symmetric circuits produce diffusive conserved-charge dynamics, hydrodynamic power-law tails, and ballistic operator fronts with diffusive charge remnants (92, 93).The framework explicitly derives how reversible unitary dynamics generates dissipative diffusion, while conserved operators convert locally into nonconserved strings that spread at the butterfly speed.Source: §3.3.1 · pp. 22–24

3.3.2. Floquet Circuits.

• In the large-q limit, chaotic Floquet circuits recover the random-matrix spectral-form-factor ramp K(t) = t from t translation pairings, with a Thouless time tTh ∼ log L in modified models (95, 94).For t ≫ tTh, the effective Potts model orders and reproduces random-matrix behavior; dual-unitary circuits remove the long Thouless-time scale entirely.Source: §3.3.2 · pp. 25–26, §3.3.3 · p. 27

3.3.3. Dual-Unitary Circuits.

• Dual-unitary circuits exhibit maximal chaos while remaining exactly tractable, including random-matrix spectral correlations, entanglement dynamics, correlators, and eigenstate thermalization without approximation (209, 210, 217, 216, 218, 219, 220, 221, 222, 223).Their double causal structure restricts nonzero correlations to the light-cone rays |x| = |t|.Source: §3.3.3 · pp. 26–27

3.3.4. Classically Simulable Circuits.

• Clifford circuits are efficiently simulable because Pauli strings remain single strings, requiring O(L^2) stabilizer information even for highly entangled states.Their stabilizer Rényi entropies are n-independent, and random Clifford evolution nevertheless produces ballistic entanglement growth comparable to Haar circuits.Source: §3.3.4 · pp. 27–28
• Random Clifford ensembles exactly reproduce Haar averages for selected OTOCs and operator-endpoint densities as unitary 3-designs, but fail to capture U(1) circuits’ mixed diffusive-ballistic spreading (49, 50, 234, 235).With U(1) symmetry, Clifford dynamics shows diffusive spreading for all operators rather than the richer coexistence of conserved diffusion and ballistic operator growth.Source: §3.3.4 · p. 28
• QSSEP allows exact calculation of late-time nonequilibrium Green-function moments, with simple averages mapping to classical exclusion and higher moments diagnosing quantum coherences through nontrivial combinatorics (239, 240, 241).The crossover from the free-fermion limit to the strongly interacting regime remains an open direction.Source: §3.3.4 · pp. 28–29
• Automaton circuits implement reversible classical cellular automata as unitary permutations, yet can generate volume-law entanglement from product states outside the computational basis.Their computational-basis dynamics is classical, with U|n⟩ = e^iθ_n|π(n)⟩ for a permutation π of the 2^N basis states.Source: §3.3.4 · p. 29
• Automaton circuits make operator growth classically tractable and exhibit broadening operator fronts, subdiffusive hydrodynamics, kinetically constrained dynamics, measurement transitions, and integrability.Pauli operators evolve into sums of Pauli products, while the operator wavefunction follows the same automaton circuit in a rotated basis (242).Source: §3.3.4 · p. 29
• Rule 54 provides an integrable Floquet automaton with conserved left- and right-moving solitons, enabling solutions for nonequilibrium steady states, operator and entanglement spreading, and generalized hydrodynamics.It captures features of more complicated integrable systems while retaining tractable solutions (247, 254, 255, 248, 249).Source: §3.3.4 · p. 29

4.1. Measurement-induced entanglement transition

• At p < pc trajectories have volume-law entanglement, whereas measurements drive an area-law phase for p > pc; at criticality, entanglement is scale invariant.The transition occurs across microscopic models and dimensions, including all-to-all settings, and the classical minimal-cut limit captures its qualitative phases but not generic critical exponents.Source: §4.1 · pp. 30–31

4.1.1. Coherent trajectories versus dissipative information loss.

• The trajectory-averaged state becomes an infinite-temperature mixed state, even though individual conditioned trajectories retain nontrivial correlations and entanglement.Nonlinear trajectory observables require paired trajectories with identical outcomes, making post-selected measurements exponentially costly; the transition also separates efficiently simulable area-law states from potentially hard volume-law states.Source: §4.1.1 · pp. 31–32

4.1.2. Entanglement domain walls and the stability of the volume law phase.

• Entanglement domain walls map the volume-law phase to directed-polymer physics, yielding SA = s0|A| + b|A|^β with β = 1/3 and wandering exponent ζ = 2/3.Measurements reduce entanglement from local cuts, but the average reduction decays as x^-∆ with ∆≈1.25, making the volume-law phase stable; typical realizations decay as a stretched exponential.Source: §4.1.2 · pp. 32–33

4.1.3. Mixed state purification transition.

• Purification occurs at a system-size-independent rate for p > pc but takes exponentially long for p < pc, identifying the area- and volume-law phases respectively.The mixed phase retains an extensive encoded subspace with finite code rate and code distance scaling as L^β, although these stabilizer-like codes are not known to be practical; the code rate also diagnoses all-to-all transitions.Source: §4.1.3 · pp. 33–34

4.1.4. The critical point

• In 1+1 dimensions, hybrid Clifford simulations find pc ≈0.17, with dynamical exponent z = 1 and conformal symmetry, while Haar-random and Hamiltonian models show consistent continuous transitions.For finite q, the percolation fixed point is unstable and the critical theory remains analytically difficult because of replica limits and infinitely many relevant perturbations.Source: §4.1.4 · pp. 34–35

4.1.4. The critical point.

• In the q →∞ limit, all-to-all monitored circuits map to temporally local classical min-cut or time-slice percolation problems, while finite-q geometry suggests a tree-tensor-network transition.With postselected outcomes, the tree transition is exactly solvable and the entanglement order parameter vanishes as exp(−c/√(p−p_c)).Source: §4.1.4 · p. 36

4.2. Structured monitored circuits

• Symmetry-enriched monitored circuits support multiple area- and volume-law phases with symmetry-breaking or SPT orders, separated by transitions detectable only through nonlinear Edwards-Anderson observables.Their replicated statistical mechanics combines the circuit symmetry G with intrinsic dynamical symmetries, enabling orders absent from conventional static classifications.Source: §4.2 · pp. 36–37
• Measurement-only Ising-symmetric Clifford dynamics has an area-law transition at p_Z = 0.5 between spin-glass ordered cat states and paramagnetic product states, with no volume-law phase.At p_Z = 1, χ_SG > 0; at p_X = 1, χ_SG →0.Source: §4.2 · p. 37
• U(1)-symmetric monitored circuits are predicted to split their volume-law phase into charge-fuzzy and charge-sharp regimes, with charge sharpening and ancilla purification times scaling as O(L) versus O(1).A proposed field theory predicts an infinite-order Kosterlitz–Thouless transition between these phases.Source: §4.2 · p. 37
• Toric-code stabilizer measurements with weak Pauli monitoring produce topologically ordered area-law states with S_topo = 2 ln 2 and exponentially slow purification in system size.For a maximally mixed initial state, entropy saturates to S = S_topo in constant time while complete purification takes exponentially long in the linear dimension.Source: §4.2 · p. 38
• Free-fermion monitored dynamics cannot sustain volume-law entanglement at any nonzero monitoring rate, has purification time at most O(N^2), yet supports super-area-law phases and critical points.The absence of a volume-law phase is linked to continuous replica symmetry lowering entanglement-domain-wall costs.Source: §4.2 · p. 38

4.2.1. Hybrid dynamics from spacetime-duality.

• Spacetime rotation converts unitary gates into forced or weak postselected measurements, and teleportation protocols shift postselection from the full spacetime volume to final-time Bell measurements.This construction improves experimental cost while retaining an analytic correspondence between monitored and unitary dynamics.Source: §4.2.1 · pp. 38–39
• Hybrid monitored circuits map spatial entanglement scaling to temporal entanglement growth, producing volume-law or logarithmic-to-fractal phases and universal boundary-dissipation corrections.Chaotic unitary circuits with S(t) ∼v_Et yield steady-state entropy density v_E; boundary dissipation adds t^1/3 and 3/2 log(t) temporal corrections, dualizing to |A|^1/3 spatial corrections.Source: §4.2.1 · p. 39

Experimental platforms and benchmarks

• Current platforms support non-equilibrium many-body-physics experiments despite universal programmable quantum computing remaining distant at present noise rates and system sizes.Superconducting-junction and trapped-ion platforms provide controlled unitaries and locally resolved measurements, while serving as experimental systems for many-body physics (38).Source: §5 · pp. 39–40
• Google’s 53-qubit random-circuit sampling demonstration marked a transition to genuinely many-body quantum systems and benchmarked near-term quantum devices.The experiment sampled the output distribution of a state evolved under a random quantum circuit (125, 277, 124, 278, 279).Source: §5 · p. 40

Information-theoretic observables

• Tomography, partial-SWAP measurements, and ancilla-assisted protocols enable experiments on entanglement, purity, and correlation functions beyond conventional local observables.Tomography reconstructs density matrices but is limited to small systems; partial-SWAP protocols measure purity using multiple copies, and Hadamard-test methods access new correlators (281, 282, 283, 284).Source: §5 · p. 40

Circuit experiments

• Recent circuit experiments measured scrambling, topological entanglement entropy, anyon braiding, and spectrum-averaged space-time order in a time-crystal circuit.These studies used ancilla-assisted out-of-time-ordered commutators and unequal space-time correlators, alongside toric-code state preparation and Floquet-circuit simulations (291, 292, 293).Source: §5 · p. 40

Monitored circuits

• Monitored-circuit experiments face a fundamental postselection barrier: an 8-qubit trapped-ion study used Clifford feed-forward, whereas larger non-Clifford evolutions require new approaches to reduce trajectory-reproduction overhead.IBM digital simulators reproduced individual trajectories by brute force, but scalable experiments may need structures such as space-time duality (294, 295).Source: §5 · p. 41

Feedback and open-system dynamics

• Feedback remains an open direction, including its use for stabilizing ordered phases, generating entanglement, and enabling interactive quantum dynamics alongside open-system effects.Quantum error correction illustrates feedback’s power to recover an unknown encoded state, while monitored open-system dynamics can produce mixed-state nonequilibrium phases.Source: §6 · p. 41

Experimental access to monitored entanglement

• Post-selection requirements hinder experimental observation of the measurement-driven entanglement phase transition, motivating alternative monitored dynamics and spacetime-rotation protocols.The outlook asks whether classes beyond Clifford and dual-unitary dynamics can avoid this barrier and prepare other useful quantum states.Source: §6 · p. 41

Theory of monitored transitions

• The monitored entanglement transition remains incompletely characterized, requiring insight from stabilizer evolution, continuous-symmetry statistical mechanics, generic-unitary coarse-graining, and rigorous phase-diagram results.A specific target is a rigorous proof of stability for the volume-law phase.Source: §6 · pp. 41–42

Effective-model formalism

• The effective lattice-model formalism still has unresolved combinatorial structures, replica-like limits, and applications to observables beyond the limited set discussed.These questions concern both unitary and nonunitary dynamics.Source: §6 · p. 42

Beyond random circuits

• Extending random-circuit insights to conventional many-body systems with fixed Hamiltonians, including lattice and continuum settings, remains an important unfinished task.The paper notes existing extensions to non-random systems but emphasizes that substantial work remains.Source: §6 · p. 42

Applications and quantum codes

• Monitored dynamics may provide near-term quantum-computing benchmarks and quantum-error-correction insights, but the usefulness of broader monitored evolutions for practical decoding remains unresolved.Time-periodic monitored dynamics have already produced dynamically evolving fault-tolerant codes (298, 299).Source: §6 · p. 42

• The authors disclose no known affiliations, memberships, funding, or financial holdings that might affect the review’s objectivity.Source: DISCLOSURE STATEMENT · p. 42