Nonergodic Delocalized States for Efficient Population Transfer within a Narrow Band of the Energy Landscape

Abstract

We address the long-standing problem of the structure of the low-energy eigenstates and long-time coherent dynamics in quantum spin-glass models. Below the spin-glass freezing transition, the energy landscape of the spin system is characterized by a proliferation of local minima where classical dynamics gets trapped. A theoretical description of quantum dynamics in this regime is challenging due to the complex nature of the distribution of the tunneling matrix elements between the local minima of the energy landscape. We study the transverse-field-induced quantum dynamics of the following "impurity band" (IB) spin model: zero energy of all spin configurations except for a small fraction of spin configurations ("marked states") that form a narrow band at a large negative energy. At a zero transverse field, the IB model demonstrates the freezing transition at inverse temperature βf∼1 characterized by a nonzero value of the Edwards-Anderson order parameter. At a finite transverse field, the low-energy dynamics can be described by the effective down-folded Hamiltonian that acts in the Hilbert subspace involving only the marked states. We obtain in an explicit form the heavy-tailed probability distribution of the off-diagonal matrix elements of the down-folded Hamiltonian. This Hamiltonian is dense and belongs to the class of preferred basis Levy matrices. Analytically solving nonlinear cavity equations for the ensemble of down-folded Hamiltonians allows us to describe the statistical properties of the eigenstates. In a broad interval of transverse fields, they are nonergodic, albeit extended. It means that the band of marked states splits into a set of narrow minibands. Accordingly, the quantum evolution that starts from a particular marked state leads to a linear combination of the states belonging to a particular miniband. An analytical description of this qualitatively new type of quantum dynamics is a key result of our paper. Based on our analysis, we propose the population transfer (PT) algorithm: The quantum evolution under constant transverse field B? starts at a low-energy spin configuration and ends up in a superposition of ω spin configurations inside a narrow energy window. This algorithm crucially relies on the nonergodic nature of delocalized low-energy eigenstates. In the considered model, the run-time of the best classical algorithm (exhaustive search) is tcl=2n/ω. For n≫B≫1, the typical run-time of the quantum PT algorithm tclen/(2B2) scales with n and ω as that of Grover's quantum search, except for the small correction to the exponent. Unlike the Hamiltonians proposed for analog quantum unstructured search algorithms, the model we consider is nonintegrable and the transverse field delocalizes the marked states. As a result, our PT protocol does not require fine-tuning of the transverse field and may be initialized in a computational basis state. We find that the run-times of the PT algorithm are distributed according to the alpha-stable Levy law with tail index 1. We argue that our approach can be applied to study the PT protocol in other transverse-field spin-glass models, with a potential quantum advantage over classical algorithms.