Saturation and recurrence of quantum complexity in random local quantum dynamics

TL;DR

This paper proves rigorous saturation and recurrence of quantum circuit complexity for chaotic time evolutions modeled by random local quantum circuits and stochastic local Hamiltonians, confirming long-standing conjectures about late-time complexity. It introduces the approximate equidistribution property as a quantitative bridge between spectral gaps, Haar randomness, and high-degree unitary designs, enabling universal results that apply to both discrete and continuous local dynamics on graphs with Hamiltonian paths. The authors derive explicit time scales: saturation occurs at t_sat ≈ exp(Θ(n))·log(1/ε) and recurrences occur at t_rec ≈ exp(exp(Θ(n))) with typical recurrence durations exponential in system size; they also establish a linear-in-depth lower bound on complexity for exponentially deep circuits. The work links the physics of black hole interiors and wormhole growth to the mathematical structure of complexity growth in chaotic quantum dynamics, offering a framework that could illuminate holographic complexity conjectures. Overall, the results demonstrate that saturation and recurrences of quantum complexity are universal features of broad classes of chaotic quantum evolutions and are underpinned by deep connections to unitary designs and equidistribution.

Abstract

Quantum complexity is a measure of the minimal number of elementary operations required to approximately prepare a given state or unitary channel. Recently, this concept has found applications beyond quantum computing -- in studying the dynamics of quantum many-body systems and the long-time properties of AdS black holes. In this context Brown and Susskind \cite{BrownSusskind17} conjectured that the complexity of a chaotic quantum system grows linearly in time up to times exponential in the system size, saturating at a maximal value, and remaining maximally complex until undergoing recurrences at doubly-exponential times. In this work we prove the saturation and recurrence of complexity in two models of chaotic time evolutions based on (i) random local quantum circuits and (ii) stochastic local Hamiltonian evolution. Our results advance an understanding of the long-time behaviour of chaotic quantum systems and could shed light on the physics of black hole interiors. From a technical perspective our results are based on establishing new quantitative connections between the Haar measure and high-degree approximate designs, as well as the fact that random quantum circuits of sufficiently high depth converge to approximate designs.

Figures (7)

• Figure 1: The conjectured BrownSusskind17 time evolution of the circuit complexity in an $n$ qubit system. The complexity exhibits a (linear) growth for an exponentially-long time until it saturates at its maximal value in time $t_{sat}=\exp(\Theta(n))$. Afterwards the complexity remains maximal until it undergoes a recurrence at doubly exponential times $t_{rec}= \exp(\exp(\Theta(n)))$.
• Figure 2: Diagram depicting our results for the circuit complexity $C_\varepsilon$ of depth-$t$ local random quantum circuits and unitaries generated by the SLH model after time $t$ acting on $n$ qudits. For typical realizations of random quantum circuits, complexity grows until an exponential time, at which point it saturates to its maximal value at time $t_{sat}\approx \exp(\Theta(n))\log(1/\varepsilon)$. Thereafter, the complexity remains maximal until it undergoes a recurrence at doubly exponential times $t_{rec}\approx (1/\varepsilon)^{\exp(\Theta(n))}$. The recurrence process typically takes exponential amount of time.
• Figure 3: Diagram depicting approximate equidistribution. For every ball $B(\boldsymbol{V},R)$ of radius $R$ in the space of unitaries, we have that the measure on that ball is upper and lower bounded by a comparable Haar volume.
• Figure 4: A plot depicting the conjectured random circuit complexity growth (upper curve in red), a long-time linear growth saturating when the circuit depth is $d^2$, alongside the best established lower bounds (lower curve in purple), a long-time sublinear growth (previously known bounds, see \ref{['sec:lineargrowth']}), followed by a linear regime and saturation at circuit depth $d^4$.
• Figure 5: Structure of the main results of the paper. Arrows are labelled by one of the informal results presented above together with the relevant rigorous statement.

Theorems & Definitions (101)

• Definition 1: Unitary complexity
• Definition 2: State complexity
• Definition 3: Approximate equidistribution on unitary channels and pure states
• Definition 4: Spectral gap
• Definition 5: Approximate unitary designs/expanders
• Definition 6: Local random quantum circuits
• Definition 7: $G$-local random quantum circuits
• Definition 8: Stochastic Local Hamiltonian (SLH) model
• Proposition 1: Spectral gap for quantum evolutions
• Proposition 2