Quantum ergodicity and scrambling in quantum annealers

TL;DR

This paper investigates ergodicity and scrambling in quantum annealers beyond the ground-state regime by analyzing a chaotic interpolation between an integrable mixer and a problem Hamiltonian. It shows that the full adiabatic unitary is typically quantum chaotic, driving forward ramps to generate volume-law entanglement and Page-like eigenstate entanglement, while cyclic ramps reveal a structured, energy-dependent deviation from chaos due to adiabaticity in the bulk. The authors connect operator growth to out-of-time-ordered correlators, demonstrating scrambling for local and global operators, and show that cyclic driving can induce partial reversal of scrambling for select operators, highlighting a nontrivial balance between chaos and adiabaticity. These insights have practical implications for benchmarking quantum devices and contribute to understanding nonequilibrium dynamics in driven many-body quantum systems, including connections to quantum many-body scars and energy-sector-specific ergodicity.

Abstract

Quantum annealers play a major role in the ongoing development of quantum information processing and in the advent of quantum technologies. Their functioning is underpinned by the many-body adiabatic evolution connecting the ground state of a simple system to that of an interacting classical Hamiltonian which encodes the solution to an optimization problem. Here we explore more general properties of the dynamics of quantum annealers, going beyond the low-energy regime. We show that the unitary evolution operator describing the complete dynamics is typically highly quantum chaotic. As a result, the annealing dynamics naturally leads to volume-law entangled random-like states when the initial configuration is rotated away from the low-energy subspace. Furthermore, we observe that the Heisenberg dynamics of a quantum annealer leads to extensive operator spreading, a hallmark of quantum information scrambling. In contrast, we find that when the annealing schedule is returned to the initial configuration (i.e. via a cyclic ramp), a subtle interplay between chaos and adiabaticity emerges, and the dynamics shows strong deviations from full ergodicity.

Figures (8)

• Figure 1: (a) Schematic of a quantum annealing protocol. Two integrable Hamiltonians $H_{\rm M}$ and $H_{\rm P}$ are interpolated by chaotic Hamiltonian $H(s)$. Slowly varying the parameter $s$ from $0$ to $1$ will connect the ground state of $H_{\rm M}$ to that the ground state of $H_{\rm P}$. (b) The forward ramp from $s(t=0)=0$ to $s(t=T)=1$ leads to the unitary map $U(T)\equiv U_{\rm FW}$. Ramping back from $s=1$ to $s=0$ generates the backwards unitary $U_{\rm BW}$ such that the cyclic process is given by $U_{\rm cycle}=U_{\rm BW}U_{\rm FW}$. (c) Half-chain entanglement entropy of the state $\left|\psi(T)\right>$ at the end of a forward ramp under annealing dynamics, for the Ising model described in Sec. \ref{['sec:model']}. The initial state is the spin coherent state $\left|\psi(0)\right> = \left|\theta=0,\varphi\right>$, where all qubits are pointing along the direction $(\cos(\varphi),\sin(\varphi),0)$ on the Bloch sphere. For $\varphi\rightarrow 0,\:\pi$, the dynamics connects the ground state of $\pm H_{\rm M}$ to that of $\pm H_{\rm P}$ and generate little entanglement. For intermediate values of $\varphi$, the final states acquire volume-law entanglement entropy, similar to that of random states.
• Figure 2: Spectral properties of Hamiltonian $H(s)$ in Eq. (\ref{['eq:hami_s']}). (a) Mean level spacing ratio $\overline{r}$, defined from Eq. (\ref{['eq:ratios_def']}), as a function of the interpolation parameter $s$. Average is taken over the bulk of the spectrum, here defined by not considering the top 5% lowest and highest energy levels. Dotted (dash-dotted) lines indicate the values of $\overline{r}$ for chaotic (GOE) and integrable (Poisson) models. (b) and (c) Spectral gaps as a function of $s$ for system sizes $N\in [8,12]$ (light to dark tones). Figure (b) shows the ground state gap $\Delta_0(s)$, while (c) shows the average gap $\sum_j \Delta_j/(\overline{d}-1)$ with $\overline{d}$ the relevant Hilbert space dimension. (d) Scaling of the minimum ground state gap and average gap with system size $N$; minimum is taken over $s\in[0,1]$. All calculations are performed in the positive parity sector of the Hamiltonian $H(s)$, whose dimension is $d=2^{N-1}$.
• Figure 3: Properties of the final state, $|\psi(2T)\rangle$, for a cyclic ramp initialized with a spin coherent state pointing along the direction given by angles $(\theta, \varphi) = (\pi/2, \varphi)$ (see main text for details), for systems $N=8,10,12,14$. (a) Half-chain entanglement entropy $S_{A}$ of $|\psi(2T)\rangle$ as a function of the mean-energy density of the initial state which is determined by its direction. The dashed lines indicate the Page value for the respective system size. The dotted line indicates $S_{A} = \log(2)$. (b) Fidelity between initial and final states of the cyclic ramp as function of the initial state mean-energy density. A perfect cyclic evolution is characterized by a unit fidelity. Only states with a small mean-energy density neighborhood of either $|+\rangle^{\otimes N}$ and $|-\rangle^{\otimes N}$ can be successfully returned to their initial configuration. Parameters are: $T=600$, $\Delta t = 0.05$.
• Figure 4: Spectral properties of the adiabatic unitary operator for both forward and cyclic ramps. (a) Mean level spacing ratio (MLSR) as a function of time for $0\leq t \leq 2 T$ covering both forward and backward ramps. The green star indicates the first time instant $t^\ast$ at which the unitary becomes fully chaotic ($s\simeq0.14$ for $N=14$), and the inset shows the behavior of this quantity with system size. The dash-dot line shows the MLSR for an integrable system and the dotted line that of a chaotic system. The bottom panels shows $s(t)$, with the forward ramp stopping at $t/T = 1$ and the cyclic ramp defined over the full range of $t/T$. (b) Half-chain entanglement entropy of all the eigenstates of $U(T)$ as a function of their mean energy with respect to the problem Hamiltonian $H_{\rm P}$. (c) Half-chain entanglement entropy of all the eigenstates of $U(2T)$ as a function of their mean energy with respect to mixer Hamiltonian $H_{\rm M}$. Both in (b) and (c) the dots are colored based on the local density, with black indicating very low density and light orange indicating very high density, the dashed lines (bottom to top), indicate $\mathcal{S} = 0, \log(2), 0.5\log(N) - 0.5$, respectively (see App. \ref{['app:evol_dicke']} for further information about the relevance of these values). all panels the parameters are: $N=14$, $T=600$, $dt = 0.5$.
• Figure 5: (a) Half-chain entanglement entropy of all the eigenstates of $U(2T)$ as function of their mean energy with respect to $H_{\rm P}$. Darker color indicates low point density, lighter color indicate high point density. Purples, greens, blues, black, correspond to system sizes of $N = 8,10,12,14$, respectively. The bottom two dashed lines indicate $\mathcal{S} = 0, \log(2)$, the top four dashed lines indicate the Page value of the entanglement entropy for the respective system size. (b) Scaling exponent with system size of the locally averaged half-chain entanglement entropy. A rapid convergence to the Page value of $\log(2)/2$ indicates that most of the eigenstates considered in (a) exhibit volume-law entanglement entropy and thus are ergodic states.