Complete Hilbert-Space Ergodicity in Quantum Dynamics of Generalized Fibonacci Drives

Ergodicity of quantum dynamics is often defined through statistical properties of energy eigenstates, as exemplified by Berry’s conjecture in single-particle quantum chaos and the eigenstate thermalization hypothesis in many-body settings. In this work, we investigate whether quantum systems can exhibit a stronger form of ergodicity, wherein any time-evolved state uniformly visits the entire Hilbert space over time. We call such a phenomenon complete Hilbert-space ergodicity (CHSE), which is more akin to the intuitive notion of ergodicity as an inherently dynamical concept. CHSE cannot hold for time-independent or even time-periodic Hamiltonian dynamics, owing to the existence of (quasi)energy eigenstates which precludes exploration of the full Hilbert space. However, we find that there exists a family of aperiodic, yet deterministic drives with minimal symbolic complexity—generated by the Fibonacci word and its generalizations—for which CHSE can be proven to occur. Our results provide a basis for understanding thermalization in general time-dependent quantum systems.

Figure 1

Main idea of complete Hilbert-space ergodicity, illustrated for the case of a single qubit: any initial state (orange disk) explores the Bloch sphere uniformly over time.

Figure 2

(a) Sequence of Fibonacci words constructed by the concatenation rule $W_{j+1}=W_j{W}_{j-1}$, for order $m=1$. The Fibonacci drive applies a pair of unitaries $A_0$, $A_1$ in the sequence prescribed by $W_{\infty }$. (b) Symbolic complexity $S_n$ counts the number of subwords of length $n$ that appear in an infinite word. It is linear in $n$ for Sturmian words such as $W_{\infty }$ and exponential in $n$ for random words. (c) Quasiperiodicity of the Fibonacci drive. Consider a point moving in a straight line (black) which wraps around the torus. Each time the line crosses the orange region, the unitary $A_0$ is applied, and when it crosses the blue region, $A_1$ is applied [22].

Figure 3

Numerical analysis of complete Hilbert-space ergodicity in the Fibonacci drive of order 1. (a),(b) Trace distance ${\mathrm{\Delta }}^{(k=2)}(T)$ for single qubit rotations $A_0=e^{-i{\theta }_{X}X}$ and $A_1=e^{-i{\theta }_{Z}Z}$, for three pairs of angles $({\theta }_X,{\theta }_Z)$: i $(0.13\pi ,0.39\pi )$ (orange), ii $(0.26\pi ,0.39\pi )$ (blue), and iii $(0.39\pi ,0.39\pi )$ (purple), and two initial states: $|0⟩$ (darker), $|+⟩$ (lighter). Lower bound $B(d=2)$ for time-independent evolution (black dashed). (c) Power law exponent $\gamma$ as a function of $({\theta }_X,{\theta }_Z)$ obtained by averaging ${\mathrm{\Delta }}^{(2)}$ over a pair of random initial states, for $T$ equal to the Fibonacci numbers $F_n$ up to $F_{3000}\approx {10}^{626}$. (d) ${\mathrm{\Delta }}^{(k)}$ for many-body Fibonacci drive over a spin-$1/2$ chain of length $L$, averaged over 10 random product states $|\psi {⟩}^{\otimes L}$ (solid lines). The bound $B(2^L)$ is shown with horizontal dashed lines. A power law decay $\sim T^{-1/2}$ (black dashed) is provided as reference.