Algebraic Bethe Circuits

Abstract

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.

Accepted for publication in Quantum

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• Krylov complexity and Trotter transitions in unitary circuit dynamics
• Persistent currents in ultracold gases
• The Bethe Ansatz as a Quantum Circuit
• Deterministic Bethe state preparation
• The Floquet Baxterisation
• Integrable Quantum Circuits from the Star-Triangle Relation
• Large-scale simulations of Floquet physics on near-term quantum computers
• Qudit Dicke state preparation
• Symmetric quantum states: a review of recent progress
• Estimating Bethe roots with VQE
• Algebraic classification of Hietarinta's solutions of Yang-Baxter equations~:~invertible $4\times 4$ operators
• Bethe Ansatz, Quantum Circuits, and the F-basis
• Noise-aware variational eigensolvers: a dissipative route for lattice gauge theories
• From Heisenberg to Hubbard: An initial state for the shallow quantum simulation of correlated electrons
• Reducing Circuit Depth in Quantum State Preparation for Quantum Simulation Using Measurements and Feedforward
• Non-hermitian integrable systems from constant non-invertible solutions of the Yang-Baxter equation
• Entanglement at the interplay between single- and many-bodyness
• Majorana fermions solve the tetrahedron equations as well as higher simplex equations
• Efficient Eigenstate Preparation in an Integrable Model with Hilbert Space Fragmentation
• Fractal decompositions and tensor network representations of Bethe wavefunctions
• Double-bracket quantum algorithms for high-fidelity ground state preparation
• Effective Bethe Ansatz for Spin-1 Non-integrable Models
• Spin-$s$ $U(1)$-eigenstate preparation