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arXiv:2407.09224 · Physical Review A

Phase diagram of the Schwinger model by adiabatic preparation of states on a quantum simulator

Oleg Kaikov, Theo Saporiti, Vasily Sazonov, Mohamed Tamaazousti

Published Jun 2, 2025• 3 citations
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Abstract

We argue the feasibility of studying the phase structure of a quantum physical system on quantum devices via adiabatic preparation of states. We introduce a method and successfully test it in application to the Schwinger model in the presence of a topological $\ensuremath{\theta}$ term. We explore the first-order-phase-transition and the no-transition regions of the corresponding phase diagram. The core idea of the method is to separately evolve the ground and the first excited states with a time-dependent Hamiltonian, the time dependence of which interpolates between different values of $\ensuremath{\theta}$. Despite our approach being a direct application of the adiabatic theorem, in some cases we are able to demonstrate its advantages in comparison to a different method from the literature that also employs adiabatic state preparation.

DOI
10.1103/physreva.111.062202

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We argue the feasibility of studying the phase structure of a quantum physical system on quantum devices via adiabatic preparation of states. We introduce a method and successfully test it in application to the Schwinger model in the presence of a topological \ensuremathθ\ensuremath{\theta}\ensuremathθ term. We explore the first-order-phase-transition and the no-transition regions of the corresponding phase diagram. The core idea of the method is to separately evolve the ground and the first excited states with a time-dependent Hamiltonian, the time dependence of which interpolates between different values of \ensuremathθ\ensuremath{\theta}\ensuremathθ. Despite our approach being a direct application of the adiabatic theorem, in some cases we are able to demonstrate its advantages in comparison to a different method from the literature that also employs adiabatic state preparation.
Many physically interesting models show a quantum phase transition when a single parameter is varied through a critical point, where the ground state and the first excited state become degenerate. When this parameter appears as a coupling constant, these models can be understood as straight-line interpolations between different Hamiltonians HI{H}_{\mathrm{I}}HI​ and HF{H}_{\mathrm{F}}HF​. For finite-size realizations however, there will usually be a finite energy gap between ground and first excited state. By slowly changing the coupling constant through the point with the minimum energy gap one thereby has an adiabatic algorithm that prepares the ground state of HF{H}_{\mathrm{F}}HF​ from the ground state of HI{H}_{\mathrm{I}}HI​. The adiabatic theorem implies that in order to obtain a good preparation fidelity the runtime $\ensuremath{\tau}$ should scale with the inverse energy gap and thereby also with the system size. In addition, for open quantum systems not only nonadiabatic but also thermal excitations are likely to occur. It is shown that---using only local Hamiltonians---for the one-dimensional quantum Ising model and the cluster model in a transverse field the conventional straight-line path can be replaced by a series of straight-line interpolations, along which the fundamental energy gap is always greater than a constant independent on the system size. The results are of interest for adiabatic quantum computation since strong similarities between adiabatic quantum algorithms and quantum phase transitions exist.
Quantum electrodynamics in 1 + 1 D (QED2) shares intriguing properties with quantum chromodynamics (QCD), including confinement, string breaking, and interesting phase diagram when the nontrivial topological θ -term is considered. Its lattice regularization is a commonly used toy model for quantum simulations of gauge theories on near-term quantum devices. In this work, we address algorithms for adiabatic state preparation in digital quantum simulations of QED2. We demonstrate that, for specific choices of parameters, the existing adiabatic procedure leads to level crossing between states of different charge sectors, preventing the correct preparation of the ground state. We further propose a new adiabatic Hamiltonian and verify its efficiency in targeting systems with a nonzero topological θ -term and in studying string breaking phenomena.
Quantum phase transitions materialize as level crossings in the ground-state energy when the parameters of the Hamiltonian are varied. The resulting ground-state phase diagrams are straightforward to determine by exact diagonalization on classical computers, but are challenging on quantum computers because of the accuracy needed and the near degeneracy of the competing states close to the level crossings. On the other hand, classical computers are limited to small system sizes, which quantum computers may help overcome. In this work, we use a local adiabatic ramp for state preparation to allow us to directly compute ground-state phase diagrams on a quantum computer via time evolution. This methodology is illustrated by examining the ground states of the XY model with a magnetic field in the z-direction in one dimension. We are able to calculate an accurate phase diagram on both two- and three-site systems using IBM quantum machines.
It is not possible, using standard lattice techniques in Euclidean space, to calculate the complete fermionic spectrum of a quantum field theory. Algorithms running on quantum computers have the potential to access the theory with real-time evolution, enabling a direct computation. As a testing ground we consider the 1+1-dimensional Schwinger model with the presence of a θ\thetaθ term using a staggered fermions discretization. We study the convergence properties of two different algorithms -adiabatic evolution and the Quantum Approximate Optimization Algorithm- with an emphasis on their cost in terms of CNOT gates. This is crucial to understand the feasibility of these algorithms, because calculations on near-term quantum devices depend on their rapid convergence. We also propose a blocked algorithm that has the first indications of a better scaling behavior with the dimensionality of the problem.
We explore the first-order phase transition in the lattice Schwinger model in the presence of a topological θ\thetaθ-term by means of the variational quantum eigensolver (VQE). Using two different fermion discretizations, Wilson and staggered fermions, we develop parametric ansatz circuits suitable for both discretizations, and compare their performance by simulating classically an ideal VQE optimization in the absence of noise. The states obtained by the classical simulation are then prepared on the IBM's superconducting quantum hardware. Applying state-of-the art error-mitigation methods, we show that the electric field density and particle number, observables which reveal the phase structure of the model, can be reliably obtained from the quantum hardware. To investigate the minimum system sizes required for a continuum extrapolation, we study the continuum limit using matrix product states, and compare our results to continuum mass perturbation theory. We demonstrate that taking the additive mass renormalization into account is vital for enhancing the precision that can be obtained with smaller system sizes. Furthermore, for the observables we investigate we observe universality, and both fermion discretizations produce the same continuum limit.
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.