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 $N$, 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 $N$ for all eigenstates, despite the models being interacting.

💡 Research Summary

The paper introduces a novel approach for preparing arbitrary eigenstates of integrable quantum many‑body models on a digital quantum computer using the adiabatic algorithm. While adiabatic evolution is traditionally employed to obtain ground states, its application to highly excited states has been limited because the energy gaps between excited levels shrink exponentially with system size. The authors overcome this obstacle by exploiting the extensive set of conserved charges that characterise integrable systems.

First, the authors revisit the standard adiabatic protocol for the XXZ Heisenberg chain, a paradigmatic interacting integrable model. By working in fixed magnetisation sectors, the adiabatic gap is governed only by states with the same total magnetisation. Using the thermodynamic Bethe ansatz, they show that for anisotropy Δ∈(0,1) the minimal gap scales as O(1/N), where N is the number of spins. Combined with a first‑order Trotter‑Suzuki decomposition, this yields a circuit depth that grows polynomially with N, dramatically improving on previous methods that required exponential resources.

The central contribution is a general construction that enables the preparation of any target eigenstate |v⟩. Given a complete set of commuting conserved operators {Q_k(g)} (k=1,…,N) that depend smoothly on a set of tunable parameters g, the authors define a “parent Hamiltonian”

  H_v(g) = Σ_k