Algebraic Bethe Circuits

Alejandro Sopena1, Max Hunter Gordon1,2, Diego García-Martín3,1, Germán Sierra1, and Esperanza López1

1Instituto de Física Teórica, UAM/CSIC, Universidad Autónoma de Madrid, Madrid, Spain
2Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA
3Barcelona Supercomputing Center, Barcelona, Spain

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The Algebraic Bethe Ansatz (ABA) is a highly successful analytical method used to exactly solve several physical models in both statistical mechanics and condensed-matter physics. Here we bring the ABA into unitary form, for its direct implementation on a quantum computer. This is achieved by distilling the non-unitary $R$ matrices that make up the ABA into unitaries using the QR decomposition. Our algorithm is deterministic and works for both real and complex roots of the Bethe equations. We illustrate our method on the spin-$\frac{1}{2}$ XX and XXZ models. We show that using this approach one can efficiently prepare eigenstates of the XX model on a quantum computer with quantum resources that match previous state-of-the-art approaches. We run small-scale error-mitigated implementations on the IBM quantum computers, including the preparation of the ground state for the XX and XXZ models on $4$ sites. Finally, we derive a new form of the Yang-Baxter equation using unitary matrices, and also verify it on a quantum computer.

The Bethe Ansatz is a tremendously successful analytical technique to solve a large class of many-body quantum systems. However, there remain properties of these systems that are inaccessible to classical methods. This motivates applying quantum computing to their study. Starting from the tensor-network structure provided by the Algebraic Bethe Ansatz (ABA), we present a quantum-classical algorithm to deterministically prepare ABA eigenstates on a quantum computer. We note that these eigenstates may also prove useful inputs to other quantum algorithms, and may be employed to benchmark quantum hardware.

The main issue encountered when translating the ABA into a quantum circuit is that the matrices that make up the tensor network are not unitary. Hence, we use classical resources to bring the ABA to unitary form by iteratively computing a QR decomposition. This leads to a set of unitary matrices that must be compiled down to quantum gates. We have thus transformed the ABA into a quantum circuit, implemented some of them in IBM quantum computers and applied advanced mitigation techniques obtaining precise results. We moreover find a novel representation of the Yang-Baxter equation in terms of unitary matrices, that may be of independent interest and have broad applicability.

Our work represents an important step in quantum simulation. We bring together techniques from statistical physics, condensed-matter physics and near-term quantum computing to provide an exact alternative approach to directly prepare quantum many-body eigenstates. This opens up a new promising area of research leveraging the power of the Bethe ansatz on quantum computers.

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