Random quantum circuits, chaos and quantum thermalization

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Overview

Research area: Theoretical quantum many-body physics — quantum chaos, quantum thermalization, and random quantum circuit models (quantum information dynamics / statistical physics).

Technical level: Advanced. The notes assume graduate-level familiarity with quantum many-body theory, random matrix theory, density matrices and tensor-network notation, though they are written as a pedagogical introduction.

Scope: A short set of lecture notes (delivered June 2025 at the summer school Fundamental Problems in Statistical Physics XVI) introducing random quantum circuits as minimally structured models of generic many-body quantum dynamics, and sketching ensemble-averaged calculations of operator spreading, entanglement growth and spectral correlations.

What This Paper Is About

Generic, interacting many-body quantum systems are hard to treat because their behaviour depends sensitively on microscopic details. The notes argue that one should instead study statistical ensembles of evolution operators, and that random quantum circuits — built from local gates drawn at random — capture the universal features of such systems while remaining analytically tractable. The goal is to show how three signatures of quantum thermalization (operator growth, entanglement spreading, and spectral correlations) can be computed by averaging over circuit ensembles.

Key Contributions

1. A pedagogical framing of random quantum circuits as the natural many-body analogue of random matrix theory, tracing the logic from Dyson's 1950s–1960s nuclear-level statistics to ensembles of unitary evolution operators described by the Haar distribution and the circular unitary ensemble (CUE).
2. A precise construction of the core models: brickwork random unitary circuits (RUCs) built from independently Haar-random $q^2 \times q^2$ two-site gates on a chain of $L$ sites with local Hilbert space dimension $q$ (with $q=2$ for spin-half), together with extensions to random Floquet circuits (RFCs), U(1)-symmetric circuits on an enlarged local space of dimension $2q$, and the random phase model (RPM).
3. A self-contained derivation of operator spreading in the Heisenberg picture, reducing the ensemble-averaged evolution of an initially local operator to a biased classical random walk of the operator string's endpoint, with explicit results for the butterfly velocity and the diffusion constant.
4. A treatment of entanglement dynamics from an initial product state, via two complementary routes: a min-cut bound on the rank of the reduced density matrix, and a direct ensemble-averaged calculation of the purity (with the caveat that the average of $e^{-S_2(t)}$ is computed rather than the average of $S_2(t)$, which would need replica methods beyond the level of the notes).

Main Findings

• Operator strings as a basis: A time-evolved operator $O(x,t) = W^\dagger(t) O(x) W(t)$ is expanded in a complete basis of operator strings $\mathcal{S}$ with real amplitudes $a_{\mathcal{S}}(t)$, normalised so that $\sum_{\mathcal{S}} [a_{\mathcal{S}}(t)]^2 = 1$ at all times.
• Endpoint probabilities: The strings are coarse-grained by the position $k$ of their right-most non-identity operator, giving a normalised distribution $p_k(t)$ with $\sum_k p_k(t) = 1$.
• Single-gate transition probability: From counting arguments on the $q^4 - 1$ two-site operators orthogonal to the two-site identity, the probability that the string end hops to the neighbouring site is $p = q^2(q^2-1)/(q^4-1) = q^2/(q^2+1)$.
• Biased random walk: Since $1/2 < p \le 1$ for all physical $q$, the string end drifts outward, with mean position $\langle k(t)\rangle \simeq \langle k(0)\rangle + vt$ and butterfly velocity $v = 2p - 1 = (q^2-1)/(q^2+1)$.
• Diffusive broadening: Fluctuations around the drifting endpoint grow as $\langle k(t)^2\rangle - \langle k(t)\rangle^2 \simeq 2Dt$ with $D = 2q^2/(q^2+1)^2$.
• Decay of autocorrelation: The autocorrelation $\langle O(x,t)O(x)\rangle \equiv q^{-L}\mathrm{Tr}[O(x,t)O(x)]$ decays in time because almost all weight transfers to longer strings.
• Out-of-time-order correlator: For traceless operators squaring to the identity, $\tfrac{1}{2}N^{-L}\mathrm{Tr}|[O(x),O(y,t)]|^2 = 1 - C(x,y;t)$ with $C(x,y;t) \equiv N^{-L}\mathrm{Tr}[O(y,t)O(x)O(y,t)O(x)]$; $C \approx 1$ at small $t$ and $C \to 0$ at large $t$.
• Exact link between OTOC and operator spreading: $C(x,y;t) = \sum_{k<x} p_k(t)$, i.e. the correlator equals the total probability that the string end has not yet passed the point $x$.
• Lieb–Robinson bound: $|[O(x),O(y,t)]| \le C e^{-a(|x-y| - v|t|)}$ with $C$, $a$, $v$ positive model-dependent constants, so the commutator vanishes along rays with fixed $|x-y|/|t| > v$.
• Min-cut entropy bound: Cutting the circuit along the shortest path separating subsystem $A$ from the rest bounds the rank of $\rho_A$ by $q^{\ell_{\mathrm{cut}}}$, hence $S_\alpha \le \ln[\text{rank of } \rho_A]$; at early times the shortest path runs down to the initial state and grows roughly linearly in time, while at late times it exits sideways through the spatial boundary and the bound becomes time-independent — the expected equilibration behaviour.
• Entropy definitions used: $e^{-(\alpha-1)S_\alpha(t)} = \mathrm{Tr}[\rho_A(t)^\alpha]$, with the purity $e^{-S_2(t)}$ and the von Neumann entropy as $\lim_{\alpha\to 1} S_\alpha(t)$; a pure state gives $S_\alpha = 0$, the maximally mixed state gives $S_\alpha = L_A \ln q$.
• Model extensions and their rationale: RFCs impose discrete time-translation invariance with $W(t) = W^t$ and $W = W_2 \cdot W_1$, but are harder to average and require large $q$; U(1) symmetric circuits enlarge the local space to $2q$ and use block-diagonal gates with two $q^2 \times q^2$ blocks and one $2q^2 \times 2q^2$ block; the RPM introduces a coupling-strength parameter $\varepsilon$ (uncoupled at $\varepsilon = 0$, strongly coupled at $\varepsilon \gtrsim 1$) to avoid the strong-coupling artefact of the large-$q$ limit with Haar gates.
• Absence of conservation laws: Because Haar-random gates break continuous time-translation symmetry and other symmetries, RUCs have no conserved local densities — not even energy density — which is itself a source of simplicity.
• Not reported: The paper contains no numerical simulations, benchmark comparisons, dataset sizes, or experimental results; it is an analytic and conceptual set of lecture notes.

Methodology in Plain English

The strategy is to replace a single, microscopically complicated system with a random ensemble of systems, then average physical quantities over that ensemble — an idea imported from random matrix theory, where individual energy levels are unpredictable but their statistics are universal.

Concretely, the notes build a circuit on a one-dimensional chain of sites, each carrying a $q$-dimensional local space. Time advances in discrete steps; in each step, neighbouring pairs of sites are coupled by random two-site gates arranged in a brickwork pattern, with the pairing offset between successive steps so that every part of the chain eventually interacts. Each gate is drawn independently from the uniform (Haar) distribution over unitaries. Evolution over $t$ steps is the ordered product $W(t) = W_t W_{t-1}\cdots W_1$, built from tensor products of the individual gates.

For operator spreading, the time-evolved operator is expanded in a basis of "operator strings" (products of local operators, including the identity at each site). Because Haar-averaged gates treat all non-identity outcomes on a pair of sites democratically, the only thing that matters is counting how many final configurations move the string's endpoint. This turns the ensemble-averaged quantum evolution into a classical master equation — a biased random walk — whose drift and diffusion constants follow directly.

For entanglement, the notes use two complementary tactics. The first is geometric: draw the shortest path through the circuit separating subsystem $A$ from its complement; the number of legs this path cuts bounds the rank of the reduced density matrix and therefore the Rényi entropies. The second is algebraic: express the purity of the reduced density matrix as a tensor-network diagram, then average each gate using Haar moments, which is tractable at low order in the number of replicas.

Why This Matters

Random circuits isolate the ingredients that survive in generic quantum dynamics — locality and randomness — while discarding model-specific detail. The results are therefore candidate universal statements: a butterfly velocity that depends only on the local dimension $q$, and an entanglement growth pattern that follows from circuit geometry alone.

• Quantum computing hardware: Random circuits are the standard tool for benchmarking and certifying near-term quantum processors, where random gate sequences are used to probe noise and to demonstrate "quantum supremacy"-style computational advantage.
• Black-hole and holographic physics: Operator spreading and the OTOC are the discrete-circuit analogue of scrambling, informing toy models of information propagation in gravity.
• Thermalization and statistical mechanics: The circuit picture clarifies how reversible unitary evolution nevertheless produces apparent equilibrium and entanglement entropy growth.
• Scrambling and information protection: Understanding how quickly local information spreads bears on quantum error correction and on the prospects for recovering information from chaotic dynamics.

Industry relevance: The models are directly relevant to quantum-computing companies using random-circuit sampling for processor validation, and to the broader effort to characterise noise and information loss in real devices. The notes themselves explicitly do not cover experimental platforms, instead directing readers to a companion set of lectures.

Future Directions

• Beyond the averaged purity: Computing the ensemble average of $S_2(t)$ rather than of $e^{-S_2(t)}$ requires replica methods that lie outside these notes; carrying this through remains an open technical step for the entanglement discussion.
• Floquet circuits at finite $q$: Because RFC averages are harder than RUC averages and currently require large $q$, developing tractable treatments at physically relevant local dimension is an important gap.
• Weak-coupling regimes: The random phase model is introduced precisely because the large-$q$ Haar brickwork limit gives strong inter-site coupling, obscuring phenomena seen numerically at finite $q$; systematic study of the $\varepsilon$-controlled crossover is a natural continuation.
• Symmetry-resolved dynamics: Reintroducing U(1) conservation (and other symmetries) raises questions about how hydrodynamic modes, which behave classically at long times, coexist with the quantum-information diagnostics emphasised here.

Target Audience

Graduate students and researchers in condensed matter theory, quantum information and statistical mechanics who want a compact, self-contained entry point to random quantum circuits. It suits readers who already know basic quantum many-body formalism and want the calculational logic — Haar averaging, operator strings, min-cut bounds — rather than a literature survey; readers seeking experimental context or extensive citations are pointed instead to the review articles and companion lectures referenced in the notes.

Authors’ abstract

These notes accompany lectures given in June 2025 at the summer school \emph{Fundamental Problems in Statistical Physics XVI}. They offer a short introduction to random quantum circuits as simple models for generic many-body quantum systems. They give an outline of the motivation for introducing these models, starting from ideas of random matrix theory. They also provide a sketch of calculations of some of the quantities of most physical interest, based on an average over an ensemble of systems. These quantities give insights into operator spreading, entanglement dynamics and spectral correlations.