Quantum Algorithms to Determine Spin-Resolved Exchange-Correlation Potential for Strongly Correlated Materials

Abstract

Accurate exchange-correlation (XC) potentials are essential for density functional theory, yet reliable approximations remain challenging for strongly correlated systems. In this work, we present a quantum algorithmic framework to determine spin-resolved XC potentials using a variational quantum eigensolver. Using the Hubbard model as a prototypical strongly correlated lattice system, we prepare ground states in fixed spin sectors through a Hamiltonian variational ansatz combined with a continuation strategy that gradually increases the interaction strength. From the resulting many-body ground states, we extract the XC energy and compute the corresponding spin-resolved XC potentials via finite differences. The accuracy of the approach is benchmarked against exact diagonalization for one- and two-dimensional Hubbard systems of various lattice sizes. We demonstrate that the variational ansatz reproduces the ground-state energies and densities with high fidelity, enabling accurate construction of both magnetic and non-magnetic XC potentials. We analyzed the dependence of the XC potentials on the interaction strength, charge, spin densities, and magnetization. We also present an empirical complexity scaling relation for the computational cost of the method at a fixed fidelity. These results illustrate how quantum simulations can be used to construct spin-resolved XC functionals for correlated lattice models, providing a potential pathway for improving density functional approximations in strongly correlated materials.

Figures (9)

• Figure 1: Qubit mappings for a one-dimensional lattice under the JW transformation. The JW ordering is depicted by the black line. Each spin orbital needs two qubits, so $L$ sites system are mapped onto $2L$ qubits. (a) Zig-zag ordering of lattice sites, with connection between spin up and down on same site. (b) Box (block) ordering, grouping sites into local blocks to reduce long Pauli strings.
• Figure 2: Quantum circuit layout for a 4-site Hubbard chain at half-filling, using a single variational layer. $G$: Givens rotations; $O$: onsite evolution gates; $H$: hopping evolution gates. The first four qubits encode the spin-up orbitals and the last four encode the spin-down orbitals, following the box-ordering convention. The onsite and hopping gates implement short-time evolutions under the corresponding onsite and kinetic terms of the Fermi–Hubbard Hamiltonian, and the green block is stacked to form multiple layers. All onsite gates in a layer share the same time parameter, and for a $1\times L_y$ system, all parallel hopping operations also use a common time parameter. The layers of $G$ gates (shaded gray) are used to prepare the non-interacting ground state first. Then we use the layer of onsite and hopping gates (shaded green) to evolve the state in $U \neq 0$ regime. The full circuit is executed repeatedly to estimate the energy. A similar quantum circuit layout has been used in Ref. Stanisic2022
• Figure 3: Convergence of the VQE solution as a function of the number of evolution layers $S$ for the $L=8$ Hubbard model at half filling $(N_\uparrow,N_\downarrow)=(4,4)$. Left: relative error in the ground-state energy. Right: state infidelity $1-\mathcal{F}$ between the VQE state and the exact ground state. Solid lines are guides to the eye only. Results are shown for interaction strengths $U=1,2,3,$ and $4$ going from bottom to top. The curves are the average of $4$ random initial starting points.
• Figure 4: Minimum number of evolution layers $S_{\min}$ required to reach fidelity $\mathcal{F}\ge0.99$ at $U=4$ as a function of lattice size $L$. Two representative cases are shown: the easiest sector (blue) and the hardest sector (orange) for each lattice size.
• Figure 5: XC functionals as a function of density for different interaction strengths of $L=9$. (a) XC energy per site $\varepsilon_{\rm XC} = E_{\mathrm{XC}}(n)/L$ and (b) corresponding XC potential density $v_{\mathrm{XC}}(n)=V_{\mathrm{XC}}(n)/L$ for the spin-balanced case. (c) (d) The same as in Figs. (a) and (b) for fixed $n_\downarrow = 4/L$. Data symbols denote values obtained from VQE ground-state energies, while solid curves correspond to exact results. The ansatz shows monotonic convergence in fidelity as the number of layers increases. The circuit depth increased layer-by-layer, and the smallest depth achieving $\mathcal{F} \geq 0.99$ was used for each spin sector.