Eigenstate solutions of the Fermi-Hubbard model via symmetry-enhanced variational quantum eigensolver

TL;DR

The work demonstrates that incorporating particle-number and $S_z$ symmetry into both quantum circuits and loss functions in a symmetry-enhanced VQE framework dramatically improves ground- and excited-state solutions of Hubbard models on small lattices. By encoding fermionic operators via the Jordan–Wigner transformation and enforcing conserved quantities, the method reduces the search space and resolves degeneracies, achieving high-fidelity eigenstates with far fewer parameters than conventional hardware-efficient Ansätze. The authors showcase complete eigenstate solutions for a two-site 1D Hubbard model and partial solutions for a four-site system, highlighting substantial reductions in iteration counts and circuit complexity. This symmetry-aware approach holds promise for scaling to larger, more realistic Hubbard models in the NISQ era, with accessible data and code for reproducibility.

Abstract

The Variational Quantum Eigensolver (VQE), as a hybrid quantum-classical algorithm, is an important tool for effective quantum computing in the current noisy intermediate-scale quantum (NISQ) era. However, the traditional hardware-efficient ansatz without taking into account symmetries requires more computational resources to explore the unnecessary regions in the Hilbert space. The conventional Subspace-Search VQE (SSVQE) algorithm, which can calculate excited states, is also unable to effectively handle degenerate states since the loss function only contains the expectation value of the Hamiltonian. In this study, the energy eigenstates of the one-dimensional Fermi-Hubbard model with two lattice sites and the two-dimensional Hubbard model with four lattice sites are calculated. By incorporating symmetries into the quantum circuits and loss function, we find that both the ground state and excited state calculations are improved greatly compared to the case without symmetries. The enhancement in excited state calculations is particularly significant. This is because quantum circuits that conserve the particle number are used, and appropriate penalty terms are added to the loss function, enabling the optimization process to correctly identify degenerate states. The results are verified through repeated simulations.

Figures (5)

• Figure 1: (a) A one-dimensional Hubbard model with two lattice sites, in which the circles represent the lattice sites, the numbers 0 and 1 in the center are the site indices, and the arrows indicate the electron hopping between the lattice sites. (b) Hardware efficient ansatz single-layer circuits acting on four qubits. During the computation, $n$ layers of such circuits will be applied. In a certain range, increasing the number of layers can enhance the circuit's ability to represent the state. (c) Quantum circuits that can conserve the particle number for states such as $\vert01\rangle$ or $\vert10\rangle$. (d) The single-layer circuit corresponding to the quantum gate in plot (c). (e) Quantum circuits that can conserve the particle number for arbitrary two-qubit state. (f) The single-layer circuit corresponding to the quantum gate in plot (e).
• Figure 2: The solved ground states under three different ansatzes. All data represent the average values from 200 independent computational runs, during which the initial parameters of the quantum circuit are randomly selected. The error bands are displayed in blue shade. The standard error is small and not visually prominent in several plots. To make the standard error more apparent, error bars are provided at every 30th iteration. Plots (a)-(d) show the energy expectation value $\langle\psi_{0}\left(\boldsymbol{\theta}\right)\vert\hat{H}\vert\psi_{0}\left(\boldsymbol{\theta}\right)\rangle$, fidelity $\left|\langle\psi_{0}\left(\boldsymbol{\theta}\right)\vert\alpha_{2,0}\rangle\right|^{2}$, particle number expectation value $\langle\psi_{0}\left(\boldsymbol{\theta}\right)\vert\hat{N}\vert\psi_{0}\left(\boldsymbol{\theta}\right)\rangle$, and total spin operator $z$-component $\langle\psi_{0}\left(\boldsymbol{\theta}\right)\vert\hat{S}{z}\vert\psi_{0}\left(\boldsymbol{\theta}\right)\rangle$ as a function of iteration number using hardware efficient ansatz to solve for the ground state. Plot (c) shows that certain optimization process is required to make the particle number approach the value 2, which suggests that resources are being spent on searching for unnecessary states. Plots (e)-(h) show the results corresponding to the first type of symmetry-preserving ansatz. Plot (g) indicates that this ansatz is able to preserve the particle number. Plots (i)-(l) show the results obtained by using the second type of symmetry-preserving ansatz. Plots (k)-(l) indicate that the circuit can simultaneously preserve both the particle number and the $z$-component of spin when the initial state has a $z$-component of spin equal to zero. This eliminates the requirement to search for states that do not satisfy the target energy eigenstate, particle number, or $z$-component of spin.
• Figure 3: The solutions of all eigenstates obtained by using three different methods. The expectation values of various quantities during the computation process, with $\vert\psi_{i}\rangle$ as the initial state, are presented. The eigenstates corresponding to $\vert\psi_{i}\rangle$, which need to be solved, are listed in Table \ref{['tab:eigenstates_initial_states']}. All data represent the averages of 200 computations with random initial parameters. (a)-(d) show the changes in energy, fidelity, particle number, and $z$-component of spin with respect to the number of iterations when using hardware efficient ansatz without adding penalty terms to the loss function. The fidelity plot in (b) demonstrates the inefficacy of this method in calculating degenerate states. Although the expectation value of the Hamiltonian in the loss function allows the identification of states with the correct energy, it fails to select one of the degenerate states. As a result, the energy in (a) converges well, but the particle number and $z$-component of spin in (c)-(d) fail to converge to the correct values. (e)-(h) show the changes of various quantities with respect to the number of iterations when using the hardware efficient ansatz with penalty terms. This method allows us to select which degenerate state to be optimized, resulting in a high fidelity. (i)-(p) show the results using the symmetry-preserving ansatz with penalty terms. This ansatz preserves the particle number, allowing faster convergence to the target energy eigenstate. (i)-(j), (k)-(l), (m)-(n) and (o)-(p) share the same legend, respectively. The fidelity in (n) starts at 1 because the initial state which we chose to satisfy the conditions is exactly the corresponding energy eigenstate. Similarly, the energy $\langle\psi_{15}\left(\boldsymbol{\theta}\right)\vert\hat{H}\vert\psi_{15}\left(\boldsymbol{\theta}\right)\rangle$ and the fidelity $\left|\langle\psi_{15}\left(\boldsymbol{\theta}\right)\vert\theta_{4,0}\rangle\right|^{2}$ corresponding to the state $\vert\psi_{15}\rangle$ remain consistently 4 and 1, respectively. For simplicity, they are not included in the figure.
• Figure 4: (a) The one-dimensional Hubbard model with four lattice sites incorporates periodic boundary conditions, where circles represent lattice points, the numbers in the middle are the point indices, and arrows indicate that electrons only hop between nearest-neighbor lattice sites. (b) The particle number preserving circuit for the four-site Hubbard model, applying 5 layers for the ground state calculation and 7 layers for the second excited state calculation. (c)-(d) The change of the average fidelity of hardware efficient ansatz and symmetry-preserving ansatz with the number of iterations when solving the ground state. The black vertical dashed line indicates the iteration number when the average fidelity first reaches $\geq0.95$. All data are the average of 5 independent computations. (e)-(f) Comparison between the two ansatzes when solving the second excited state.
• Figure 5: Partial optimization processes of two different ansatz. (a) Variation of the loss function with the number of iterations for a hardware efficient ansatz without penalty terms. (b) Variation of the loss function with the number of iterations for a symmetry-preserving ansatz with penalty terms, where the initial state's particle number $\langle\hat{N}\rangle$ is 2. (c) Particle number of the state during the optimization process with the symmetry-preserving ansatz, showing that the circuit maintains a particle number of 2. (d) $z$-component of spin $\langle\hat{S}_{z}\rangle$ of the state during the optimization process with the symmetry-preserving ansatz, demonstrating that the circuit maintains the $z$-component of spin when the initial state's $z$-component of spin is 0.