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arXiv:2105.10187 · Physical Review A

Optimal parent Hamiltonians for time-dependent states

Davide Rattacaso, Gianluca Passarelli, Antonio Mezzacapo, Procolo Lucignano, Rosario Fazio

Published Aug 23, 2021• 7 citations
arXivHTMLPDF

Abstract

Given a generic time-dependent many-body quantum state, we determine the associated parent Hamiltonian. This procedure may require, in general, interactions of any sort. Enforcing the requirement of a fixed set of engineerable Hamiltonians, we find the optimal Hamiltonian once a set of realistic elementary interactions is defined. We provide three examples of this approach. We first apply the optimization protocol to the ground states of the one-dimensional Ising model and a ferromagnetic $p$-spin model but with time-dependent coefficients. We also consider a time-dependent state that interpolates between a product state and the ground state of a $p$-spin model. We determine the time-dependent optimal parent Hamiltonian for these states and analyze the capability of this Hamiltonian of generating the state evolution. Finally, we discuss the connections of our approach to shortcuts to adiabaticity.

DOI
10.1103/physreva.104.022611

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Given a generic time-dependent many-body quantum state, we determine the associated parent Hamiltonian. This procedure may require, in general, interactions of any sort. Enforcing the requirement of a fixed set of engineerable Hamiltonians, we find the optimal Hamiltonian once a set of realistic elementary interactions is defined. We provide three examples of this approach. We first apply the optimization protocol to the ground states of the one-dimensional Ising model and a ferromagnetic ppp-spin model but with time-dependent coefficients. We also consider a time-dependent state that interpolates between a product state and the ground state of a ppp-spin model. We determine the time-dependent optimal parent Hamiltonian for these states and analyze the capability of this Hamiltonian of generating the state evolution. Finally, we discuss the connections of our approach to shortcuts to adiabaticity.
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Tensor network states form a powerful framework for both the analytical and numerical study of strongly correlated phases. Vital to their analytical utility is that they appear as the exact ground states of associated parent Hamiltonians, where canonical proof techniques guarantee a controlled ground space structure. Yet, while those Hamiltonians are local by construction, the known techniques often yield complex Hamiltonians which act on a rather large number of spins. In this paper, we present an algorithm to systematically simplify parent Hamiltonians, breaking them down into any given basis of elementary interaction terms. The underlying optimization problem is a semidefinite program, and thus the optimal solution can be found efficiently. Our method exploits a degree of freedom in the construction of parent Hamiltonians---the excitation spectrum of the local terms---over which it optimizes such as to obtain the best possible approximation. We benchmark our method on the AKLT model and the toric code model, where we show that the canonical parent Hamiltonians (acting on 3 or 4 and 12 sites, respectively) can be broken down to the known optimal two-body and four-body terms. We then apply our method to the paradigmatic resonating valence bond (RVB) model on the kagome lattice. Here, the simplest previously known parent Hamiltonian acts on all the 12 spins on one kagome star. With our optimization algorithm, we obtain a vastly simpler Hamiltonian: we find that the RVB model is the ground state of a parent Hamiltonian whose terms are all products of at most four Heisenberg interactions, and whose range can be further constrained, providing a major improvement over the previously known 12-body Hamiltonian.
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We use local adiabatic evolution to experimentally create and determine the ground-state spin ordering of a fully connected Ising model with up to 14 spins. Local adiabatic evolution---in which the system evolution rate is a function of the instantaneous energy gap---is found to maximize the ground-state probability compared with other adiabatic methods while requiring knowledge only of the lowest $\ensuremath{\sim}$NNN of the 2N{2}^{N}2N Hamiltonian eigenvalues. We also demonstrate that the ground-state ordering can be experimentally identified as the most probable of all possible spin configurations, even when the evolution is highly nonadiabatic.
Abstract. Adiabatic quantum computation employs a slow change of a time-dependent control function (or functions) to interpolate between an initial and final Hamiltonian, which helps to keep the system in the instantaneous ground state. When the evolution time is finite, the degree of adiabaticity (quantified in this work as the average ground-state population during evolution) depends on the particulars of a dynamic trajectory associated with a given set of control functions. We use quantum optimal control theory with a composite objective functional to numerically search for controls that achieve the target final state with a high fidelity while simultaneously maximizing the degree of adiabaticity. Exploring properties of optimal adiabatic trajectories in model systems elucidates the dynamic mechanisms that suppress unwanted excitations from the ground state. Specifically, we discover that the use of multiple control functions makes it possible to access a rich set of dynamic trajectories, some of which attain a significantly improved performance (in terms of both fidelity and adiabaticity) through the increase of the energy gap during most of the evolution time. Submitted to: New J. Phys. † Author to whom any correspondence should be addressed. ar