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arXiv:2404.18244 · Journal of Physics A Mathematical and Theoretical

Estimating Bethe roots with VQE

David Raveh, Rafael I. Nepomechie

Published Aug 9, 2024• 6 citations
arXivHTMLPDF

Abstract

Abstract Bethe equations, whose solutions determine exact eigenvalues and eigenstates of corresponding integrable Hamiltonians, are generally hard to solve. We implement a Variational Quantum Eigensolver approach to estimating Bethe roots of the spin-1/2 XXZ quantum spin chain, by using Bethe states as trial states, and treating Bethe roots as variational parameters. In numerical simulations of systems of size up to 6, we obtain estimates for Bethe roots corresponding to both ground states and excited states with up to 5 down-spins, for both the closed and open XXZ chains. This approach is not limited to real Bethe roots.

DOI
10.1088/1751-8121/ad6db2

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Abstract Bethe equations, whose solutions determine exact eigenvalues and eigenstates of corresponding integrable Hamiltonians, are generally hard to solve. We implement a Variational Quantum Eigensolver approach to estimating Bethe roots of the spin-1/2 XXZ quantum spin chain, by using Bethe states as trial states, and treating Bethe roots as variational parameters. In numerical simulations of systems of size up to 6, we obtain estimates for Bethe roots corresponding to both ground states and excited states with up to 5 down-spins, for both the closed and open XXZ chains. This approach is not limited to real Bethe roots.
Abstract The open spin-1/2 XXZ spin chain with diagonal boundary magnetic fields is the paradigmatic example of a quantum integrable model with open boundary conditions. We formulate a quantum algorithm for preparing Bethe states of this model, corresponding to real solutions of the Bethe equations. The algorithm is probabilistic, with a success probability that decreases with the number of down spins. For a Bethe state of L spins with M down spins, which contains a total of L M 2 M M ! terms, the algorithm requires L + M 2 + 2 M qubits.
We review recent results on the Bethe Ansatz solutions for the eigenvalues of the transfer matrix of an integrable open XXZ quantum spin chain using functional relations which the transfer matrix obeys at roots of unity. First, we consider a case where at most two of the boundary parameters {{alpha−\\alpha_-alpha−​,alpha+\\alpha_+alpha+​,beta−\\beta_-beta−​,beta+\\beta_+beta+​}} are nonzero. A generalization of the Baxter T−QT-QT−Q equation that involves more than one independent QQQ is described. We use this solution to compute the boundary energy of the chain in the thermodynamic limit. We conclude the paper with a review of some results for the general integrable boundary terms, where all six boundary parameters are arbitrary.
We consider the open spin- s XXZ quantum spin chain with N sites and general integrable boundary terms for generic values of the bulk anisotropy parameter, and for values of the boundary parameters which satisfy a certain constraint. We derive two sets of Bethe ansatz equations, and find numerical evidence that together they give the complete set of (2 s +1) N eigenvalues of the transfer matrix. For the case of s = 1, we explicitly determine the Hamiltonian, and find an expression for its eigenvalues in terms of Bethe roots.
We investigate special solutions to the Bethe Ansatz equations (BAE) for open integrable XXZ Heisenberg spin chains containing phantom (infinite) Bethe roots. The phantom Bethe roots do not contribute to the energy of the Bethe state, so the energy is determined exclusively by the remaining regular excitations. We derive the phantom Bethe roots criterion and focus on BAE solutions for mixtures of phantom roots and regular (finite) Bethe roots. We prove that in the presence of phantom Bethe roots, all eigenstates are split between two invariant subspaces, spanned by chiral shock states. Bethe eigenstates are described by two complementary sets of Bethe ansatz equations for regular roots, one for each invariant subspace. The respective ``semiphantom'' Bethe vectors are states of chiral nature, with chirality properties getting less pronounced when more regular Bethe roots are added. For the easy plane case, ``semiphantom'' Bethe states carry nonzero magnetic current and are characterized by quasiperiodic modulation of the magnetization profile, the most prominent example being the spin helix states (SHS). We illustrate our results investigating ``semiphantom'' Bethe states generated by one regular Bethe root (the other Bethe roots being phantom), with a simple structure of the invariant subspace, in all details. We obtain the explicit expressions for Bethe vectors and calculate the simplest correlation functions, including the spin-current for all the states in the single-particle multiplet.
Integrable quantum spin chains are exactly solvable quantum mechanical models of N quantum spins, of which the Heisenberg model solved by Bethe is the prototype. In the antiferromagnetic regime, such spin chains can be regarded as lattice versions of corresponding integrable relativistic quantum field theories. For integrable spin chains, quantities of physical interest (spectrum, S matrix, etc.) can be calculated exactly by direct means, starting from the microscopic Hamiltonian, while for the corresponding field theories, such exact information has been primarily obtained by indirect means, such as the “ bootstrap ” approach and semiclassical approximations. We have been focused basically on the Heisenberg XXX and XXZ chains and their generalizations. Basic aims of this work were the study of the symmetries of these models and the computation, by means of Bethe ansatz equations, of the bulk and boundary S matrices for the closed and open chains respectively...