Adiabatic quantum state preparation in integrable models

We propose applying the adiabatic algorithm to prepare high-energy eigenstates of integrable models on a quantum computer. We first review the standard adiabatic algorithm to prepare ground states in each magnetization sector of the prototypical XXZ Heisenberg chain. Based on the thermodynamic Bethe ansatz, we show that the algorithm circuit depth is polynomial in the number of qubits <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> , outperforming previous methods explicitly relying on integrability. Next, we propose a protocol to prepare arbitrary eigenstates of integrable models that satisfy certain conditions. For a given target eigenstate, we construct a suitable parent Hamiltonian written in terms of a complete set of local conserved quantities. We propose using such Hamiltonian as an input for an adiabatic algorithm. After benchmarking this construction in the case of the non-interacting XY spin chain, where we can rigorously prove its efficiency, we apply it to prepare arbitrary eigenstates of the Richardson-Gaudin models. In this case, we provide numerical evidence that the circuit depth of our algorithm is polynomial in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> for all eigenstates, despite the models being interacting.