Chaos and thermalization in Clifford-Floquet dynamics

TL;DR

This work addresses how Clifford QCAs driven in a Floquet setting thermalize in higher dimensions and with multiple qubits per site. By mapping Clifford QCAs to pseudo-unitary CA via Laurent-polynomial update rules, it classifies dynamics through diffusive regimes (strong vs weak) and proves that soliton-free QCAs weakly (and under stronger diffusion, strongly) thermalize broad classes of states, including short-range entangled states near the infinite-temperature state $\omega_\infty$ and states produced by local measurements. It also shows that some Pauli stabilizer states may evade thermalization due to invariant substructures, and provides numerical evidence that diffusion-driven thermalization extends beyond strictly proven cases. Overall, the results establish a robust link between operator growth (diffusivity) and deterministic quantum thermalization, applicable to systems with arbitrary $N$ qubits per site in $d$ dimensions and to qudits.

Abstract

We study the ergodic properties of a unitary Floquet dynamics arising from the repeated application of a translationally-invariant Clifford Quantum Cellular Automata to an infinite system of qubits in d dimensions. One expects that if the QCA does not exhibit any periodicity, a generic initial state of qubits will thermalize, that is, approach the infinite-temperature state. We show that this is true for many classes of states, both pure and mixed. In particular, this is true for all initial states that are short-range entangled and close to the equilibrium state. We also point out a subtle distinction between weak and strong thermalization.

Figures (6)

• Figure 1: Support size of $X(n)=\alpha^n(X_0)$ as a function of $n$ for the QCA (\ref{['eq:Mexample']}) with $t(u)=u+1+u^{-1}$.
• Figure 2: Support size of $Z(n)=\alpha^n(Z_0)$ as a function of $n$ for the QCA (\ref{['eq:Mexample']}) with $t(u)=u+1+u^{-1}$.
• Figure 3: Support size of $Y(n)=\alpha^n(Y_0)$ as a function of $n$ for the QCA (\ref{['eq:Mexample']}) with $t(u)=u+1+u^{-1}$.
• Figure 4: Absolute values of expectation values $\omega_0\circ\beta(\alpha^n(P_q))$ of single-site Pauli monomials as functions of $n$. The product state $\omega_0$ corresponds to $p=0$, $\theta=30^{\circ}$, $\phi=45^{\circ}.$ The non-Clifford QCA $\beta$ corresponds to (\ref{['eq:nonClbeta']}) with $r=1$.
• Figure 5: Absolute values of expectation values $\omega_0\circ\beta(\alpha^n(P_q))$ of single-site Pauli monomials as functions of $n$. The product state $\omega_0$ corresponds to $p=0.1$, $\theta=30^{\circ}$, $\phi=45^{\circ}.$ The non-Clifford QCA $\beta$ corresponds to (\ref{['eq:nonClbeta']}) with $r=1$.

Theorems & Definitions (20)

• Theorem 2.1
• Theorem 2.2
• Definition 3.1
• Definition 3.2
• Theorem 3.1
• Theorem 4.1
• proof
• Definition 4.1
• Proposition 4.1
• Theorem 4.2