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arXiv:2201.03021 · Journal of Physics A Mathematical and Theoretical

Bethe states on a quantum computer: success probability and correlation functions

Wen Li, Mert Okyay, Rafael I. Nepomechie

Published Jul 19, 2022• 20 citations
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Abstract

Abstract A probabilistic algorithm for preparing Bethe eigenstates of the spin-1/2 Heisenberg spin chain on a quantum computer has recently been found. We derive an exact formula for the success probability of this algorithm in terms of the Gaudin determinant, and we study its large-length limit. We demonstrate the feasibility of computing antiferromagnetic ground-state spin–spin correlation functions for short chains. However, the success probability decreases exponentially with the chain length, which precludes the computation of these correlation functions for chains of moderate length. Some conjectures for estimates of the Gaudin determinant are noted in an appendix.

DOI
10.1088/1751-8121/ac8255

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Abstract A probabilistic algorithm for preparing Bethe eigenstates of the spin-1/2 Heisenberg spin chain on a quantum computer has recently been found. We derive an exact formula for the success probability of this algorithm in terms of the Gaudin determinant, and we study its large-length limit. We demonstrate the feasibility of computing antiferromagnetic ground-state spin–spin correlation functions for short chains. However, the success probability decreases exponentially with the chain length, which precludes the computation of these correlation functions for chains of moderate length. Some conjectures for estimates of the Gaudin determinant are noted in an appendix.
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Several quantum many-body models in one dimension possess exact solutions via the Bethe ansatz method, which has been highly successful for understanding their behavior.Nevertheless, there remain physical properties of such models for which analytic results are unavailable and which are also not well described by approximate numerical methods.Preparing Bethe ansatz eigenstates directly on a quantum computer would allow straightforward extraction of these quantities via measurement.We present a quantum algorithm for preparing Bethe ansatz eigenstates of the spin-1/2 XXZ spin chain that correspond to real-valued solutions of the Bethe equations.The algorithm is polynomial in the number of T gates and the circuit depth, with modest constant prefactors.Although the algorithm is probabilistic, with a success rate that decreases with increasing eigenstate energy, we employ amplitude amplification to boost the success probability.The resource requirements for our approach are lower than for other state-of-the-art quantum simulation algorithms for small error-corrected devices and thus may offer an alternative and computationally less demanding demonstration of quantum advantage for physically relevant problems.
We communicate results on correlation functions for the spin-1/2 Heisenberg chain in two particularly important cases: (a) for the infinite chain at arbitrary finite temperature T, and (b) for finite chains of arbitrary length L in the ground state. In both cases we present explicit formulas expressing the short-range correlators in a range of up to seven lattice sites in terms of a single function ω encoding the dependence of the correlators on T (L). These formulas allow us to obtain accurate numerical values for the correlators and derived quantities like the entanglement entropy. By calculating the low T (large L) asymptotics of ω we show that the asymptotics of the static correlation functions at any finite distance are T(2) (1/L(2)) terms. We obtain exact and explicit formulas for the coefficients for up to eight sites.
We review some recent advances in the computation of exact correlation functions of the XXZ-½ Heisenberg chain. We first give a general introduction to our method which is based on the algebraic Bethe ansatz and the resolution of the quantum inverse scattering problem, leading in particular to multiple integral representations for the correlation functions. Then we describe recently obtained compact formulas for the spin-spin correlation functions of the XXZ-½ Heisenberg chain. We outline how this leads to several explicit results including the known two point functions in the limit of free fermions, the so-called emptiness formation probability at anisotropy Δ=1/2 and its large distance asymptotic behaviour in the massless phase of the model.