State preparation and symmetries

TL;DR

The paper demonstrates that enforcing exact and approximate symmetries during VQE state preparation dramatically enhances convergence and accuracy for two SU(2)-symmetric spin models: a neutrino-inspired all-to-all Hamiltonian and a 2D Heisenberg lattice with translational and mirror symmetries. By first projecting onto symmetry sectors (e.g., $J=0$, $J_z=0$, and lattice symmetries) and then applying a symmetry-preserving swap-based variational Ansatz, the authors achieve near-exact ground states with substantially larger spectral gaps and high fidelities (up to $>98\%$) compared to unconstrained VQE. The results also show drastic Hilbert-space reductions (from $2^{12}=4096$ to as few as 9 symmetry-compatible states) and improved energy gaps, highlighting the practical viability of symmetry-aware VQE on near-term hardware. The study implies broad applicability to larger lattices and fermionic problems, and suggests combining projection techniques with variational loops and time-projection methods for further gains.

Abstract

We demonstrate the importance of symmetries in Variational Quantum Eigensolver (VQE) algorithms to prepare the ground or specific low-lying states of quantum Hamiltonians. We examine two spin problems, one with random all-to-all couplings inspired by neutrino flavor evolution in supernovae, and the standard Heisenberg spin Hamiltonian on a $4 \times 3$ lattice. The neutrino Hamiltonian has the total spin $J$ and third component $J_{\rm{z}}$ as its only symmetries. The Heisenberg model has these symmetries plus translational invariance and reflection symmetry. We demonstrate that the convergence of variational methods is dramatically improved by keeping all symmetries. In both cases a nearly exact solution can be obtained in cases where standard unconstrained variational algorithms fail. Since variational algorithms can use standard Trotter steps as part of the optimization, allowing additional correlations that obey all the symmetries of the Hamiltonian will speed convergence of variational algorithms. This will lead to faster convergence than standard projection algorithms.

Figures (4)

• Figure 1: Energy difference ($E - E_{\rm{exact}}$) of the projected state at each step of the projection algorithm applied to the initial state of the Neutrino model, and the Heisenberg model. The unprojected initial state is shown as the first point at step $0$, while the subsequent points connected by solid lines represent the energy after each of the $11$ projection steps. The red line indicates the exact ground state energy.
• Figure 2: Swap convergence plots using the VQE algorithm for the Neutrino model (a), and the Heisenberg model (b) starting from the projected state obtained after $11$ steps of the projection algorithm represented by the orange dashed line. The red line is the exact ground state energy. The plots show the energy difference $E - E_{\rm{exact}}$. In both cases, the dark green curve shows the VQE energy as a function of iteration that preserve the symmetry of the model. In the Heisenberg case (b) additional strategies are shown: the green curve uses nearest-neighbor connectivity, and the olive-green curve uses all-to-all connectivity. Final translational/reflection symmetry projected state from VQE is shown as a blue dashed line.
• Figure 3: Fidelity of the prepared state with respect to the $4096$ classically computed eigenstates during optimization for the Neutrino model (a), and the Heisenberg model (b). The fidelity is computed as $|\braket{\psi | \phi}|^2$, where $\ket{\psi}$ is the prepared state and $\ket{\phi}$ is the reference state.
• Figure 4: Fidelity of the prepared state obtained using the VQE algorithm with all-to-all connectivity, with respect to the reference ground state of the Neutrino model. The fidelity is shown on a logarithmis scale, where each total spin subspace $S$ is indicated by a different color.