The Bethe Ansatz as a Quantum Circuit

Roberto Ruiz1, Alejandro Sopena1, Max Hunter Gordon1,2, Germán Sierra1, and Esperanza López1

1Instituto de Física Teórica, UAM/CSIC, Universidad Autónoma de Madrid, Madrid, Spain
2Normal Computing Corporation, New York, New York, USA

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The Bethe ansatz represents an analytical method enabling the exact solution of numerous models in condensed matter physics and statistical mechanics. When a global symmetry is present, the trial wavefunctions of the Bethe ansatz consist of plane wave superpositions. Previously, it has been shown that the Bethe ansatz can be recast as a deterministic quantum circuit. An analytical derivation of the quantum gates that form the circuit was lacking however. Here we present a comprehensive study of the transformation that brings the Bethe ansatz into a quantum circuit, which leads us to determine the analytical expression of the circuit gates. As a crucial step of the derivation, we present a simple set of diagrammatic rules that define a novel Matrix Product State network building Bethe wavefunctions. Remarkably, this provides a new perspective on the equivalence between the coordinate and algebraic versions of the Bethe ansatz.

Presently, there is a considerable focus on the development of quantum circuits aimed at producing many-body states with distinct characteristics, such as serving as eigenstates of quantum lattice Hamiltonians. Among the proposed algorithms to address this challenge are the Quantum Approximate Optimization Algorithm (QAOA) and the Variational Quantum Eigensolver (VQE). In this paper, we introduce an alternative approach, which involves starting with the Bethe ansatz solution of the renowned spin-1/2 XXZ Hamiltonian and leveraging it to derive a quantum circuit capable of preparing the exact energy eigenstates. The method for constructing this circuit entails reframing the Bethe ansatz in the context of Matrix Product States (MPS). We subsequently reconstruct these MPS using a set of intuitive diagrammatic rules, which, in the process, yield the unitary gates for the circuit that prepares the desired state. As a result, we have successfully derived the analytical expressions for the gates required to generate not only the ground state but also the excited states of the XXZ Hamiltonian. This breakthrough paves the way for a deeper understanding of the practical preparation of such states.

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Cited by

[1] Lorenzo Piroli, Georgios Styliaris, and J. Ignacio Cirac, "Approximating many-body quantum states with quantum circuits and measurements", arXiv:2403.07604, (2024).

[2] David Raveh and Rafael I. Nepomechie, "Deterministic Bethe state preparation", arXiv:2403.03283, (2024).

[3] David Raveh and Rafael I. Nepomechie, "Estimating Bethe roots with VQE", arXiv:2404.18244, (2024).

[4] Roberto Ruiz, Alejandro Sopena, Max Hunter Gordon, Germán Sierra, and Esperanza López, "The Bethe Ansatz as a Quantum Circuit", Journal of Physics Conference Series 2667 1, 012022 (2023).

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