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arXiv:2609.29858

Avoided crossings in spin-boson systems: consequences for adiabatic state preparation

Joona Marjamäki, Robert van Leeuwen, Riku Tuovinen

quant-phcond-mat.mes-hallSubmitted Sep 24, 2026• Announced Sep 25, 2026• 0 citations
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Abstract

Avoided crossings in the correlated many-body spectrum play a central role in determining the conditions for adiabatic state preparation in interacting quantum systems. We investigate this connection in finite spin-boson systems by combining exact diagonalization of the many-body spectrum with time-dependent simulations of adiabatic switching protocols. By comparing the adiabatically prepared state with the exact ground state, we determine the switching times required for reliable preparation as a function of the spin-boson coupling strength and bosonic mode structure. We show that regions of small many-body level spacings, associated with avoided crossings, impose the strongest constraints on adiabatic preparation and explain the observed dependence of the required switching times on the interaction strength. Furthermore, while the ground-state energy follows the overall spectral trends, local observables such as Pauli matrix expectation values exhibit additional non-monotonic behavior arising from coherent oscillations in finite systems. These findings can provide concrete benchmarks for both the theoretical treatment of open quantum systems and practical implementations in quantum simulation and qubit control.

Comments
9 pages, 8 figures

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Avoided crossings in the correlated many-body spectrum play a central role in determining the conditions for adiabatic state preparation in interacting quantum systems. We investigate this connection in finite spin-boson systems by combining exact diagonalization of the many-body spectrum with time-dependent simulations of adiabatic switching protocols. By comparing the adiabatically prepared state with the exact ground state, we determine the switching times required for reliable preparation as a function of the spin-boson coupling strength and bosonic mode structure. We show that regions of small many-body level spacings, associated with avoided crossings, impose the strongest constraints on adiabatic preparation and explain the observed dependence of the required switching times on the interaction strength. Furthermore, while the ground-state energy follows the overall spectral trends, local observables such as Pauli matrix expectation values exhibit additional non-monotonic behavior arising from coherent oscillations in finite systems. These findings can provide concrete benchmarks for both the theoretical treatment of open quantum systems and practical implementations in quantum simulation and qubit control.
Adiabatic ground-state preparation is fundamentally limited by the spectral structure of the time-dependent Hamiltonian, particularly by gap reductions and degeneracies that induce nonadiabatic transitions. We examine this dependence in the anisotropic Heisenberg (XXZ) model on an eight-site ring by comparing three strategies: optimization of the initial Hamiltonian, addition of auxiliary terms, and considering approximate counterdiabatic driving. Owing to anisotropy-dependent level crossings among low-energy states, the XXZ model provides a stringent benchmark. We find that performance is mainly constrained by spectral degeneracies between the ground and excited states. Simple strategies such as initial-Hamiltonian optimization or site-dependent Zeeman fields, suppresses critical crossings and drastically enhance ground-state preparation. In contrast, counterdiabatic terms alone do not improve the protocol when the spectral structure remains level-crossings, becoming effective only after degeneracies are removed. These results identify spectral engineering as a prerequisite for efficient adiabatic ground-state preparation in interacting spin systems.
A well-known method to prepare ground states of fermionic many-body Hamiltonians is adiabatic state preparation, in which an easy-to-prepare state is time-evolved towards an approximate ground state under a specific time-dependent Hamiltonian. However, which path to take in the evolution is often unclear, and a direct linear interpolation, which is the most common method, may not be optimal. In this work, we explore other types of adiabatic paths based on the spectral decomposition of the two-body projection of the residual Hamiltonian (the difference between the final and initial Hamiltonian). The decomposition defines a set of Hamiltonian terms which may be adiabatically interpolated in a piecewise or combined fashion. We demonstrate the usefulness of partially piecewise interpolation through examples involving Fermi-Hubbard models where, due to symmetries, level crossings occur in direct (fully combined) interpolation. We show that this specific deviation from a direct path appropriately breaks the relevant symmetries, thus avoiding level crossings and enabling an adiabatic passage. On the other hand, we show that a fully piecewise scheme, which interpolates every Hamiltonian term separately, exhibits a worst-case complexity of $O({L}^{6}/{\mathrm{\ensuremath{\Delta}}}^{3})$ as compared to $O({L}^{4}/{\mathrm{\ensuremath{\Delta}}}^{3})$ for direct interpolation, in terms of the number of one-body modes LLL and the minimal gap $\mathrm{\ensuremath{\Delta}}$ along the path. This suboptimality result suggests that only those terms which break necessary symmetries should be taken into account for piecewise interpolation, while the rest is treated with direct interpolation.
A new method is proposed for determining the ground state wave function of a quantum many-body system on a quantum computer, without requiring an initial trial wave function that has good overlap with the true ground state. The technique of Spectral Combing involves entangling an arbitrary initial wave function with a set of auxiliary qubits governed by a time dependent Hamiltonian, resonantly transferring energy out of the initial state through a plethora of avoided level crossings into the auxiliary system. The number of avoided level crossings grows exponentially with the number of qubits required to represent the Hamiltonian, so that the efficiency of the algorithm does not rely on any particular energy gap being large. We give an explicit construction of the quantum gates required for the realization of this procedure and explore the results of classical simulations of the algorithm on a small quantum computer with up to 8 qubits. We show that for certain systems and comparable results, Spectral Combing requires fewer quantum gates to realize than the Quantum Adiabatic Algorithm.
It is known that arrays of trapped ions can be used to efficiently simulate a variety of many-body quantum systems. Here we show how it is possible to build a model representing a spin chain interacting with bosons that is exactly solvable. The exact spectrum of the model at zero temperature and the ground-state properties are studied. We show that a quantum phase transition occurs when the coupling between spins and bosons reaches a critical value, which corresponds to a level crossing in the energy spectrum. Once the critical point is reached, the number of bosonic excitations in the ground state, which can be assumed as an order parameter, starts to be different from zero. The population of the bosonic mode is accompanied by a macroscopic magnetization of the spins. This double effect could represent a useful resource for phase transition detection since a measure of the phonon can give information about the phase of the spin system. A finite-temperature phase diagram is also given in the adiabatic regime.
Background: Solving nuclear many-body problems with an ab initio approach is widely recognized as a computationally challenging problem. Quantum computers offer a promising path to address this challenge. There are urgent needs to develop quantum algorithms for this purpose. Objective: In this work, we explore the application of the quantum algorithm of adiabatic state preparation with quantum phase estimation in ab initio nuclear structure theory. We focus on solving the low-lying spectra (including both the ground and excited states) of simple nuclear systems. Ideas: The efficiency of this algorithm is hindered by the emergence of small energy gaps (level crossings) during the adiabatic evolution. In order to improve the efficiency, we introduce techniques to avoid level crossings: 1) by suitable design of the reference Hamiltonian; 2) by insertions of perturbation terms to modify the adiabatic path. Results: We illustrate this algorithm by solving the deuteron ground state energy and the spectrum of the deuteron bounded in a harmonic oscillator trap implementing the IBM Qiskit quantum simulator. The quantum results agree well the classical results obtained by matrix diagonalization. Outlook: With our improvements to the efficiency, this algorithm provides a promising tool for investigating the low-lying spectra of complex nuclei on future quantum computers.
In this paper, we analyze the circuit complexity for preparing ground states of quantum many-body systems. In particular, how this complexity grows as the ground state approaches a quantum phase transition. We discuss different definitions of complexity, namely the one following the Fubini-Study metric or the Nielsen complexity. We also explore different models: Ising, ZZXZ or Dicke. In addition, different forms of state preparation are investigated: analytic or exact diagonalization techniques, adiabatic algorithms (with and without shortcuts), and Quantum Variational Eigensolvers. We find that the divergence (or lack thereof) of the complexity near a phase transition depends on the non-local character of the operations used to reach the ground state. For Fubini-Study based complexity, we extract the universal properties and their critical exponents. In practical algorithms, we find that the complexity depends crucially on whether or not the system passes close to a quantum critical point when preparing the state. For both VQE and Adiabatic algorithms, we provide explicit expressions and bound the growth of complexity with respect to the system size and the execution time, respectively.