Improving Ground State Accuracy of Variational Quantum Eigensolvers with Soft-coded Orthogonal Subspace Representations

We propose a new approach to improve the accuracy of ground state estimates in Variational Quantum Eigensolver (VQE) algorithms by employing subspace representations with soft-coded orthogonality constraints. As in other subspace-based VQE methods, such as the Subspace-Search VQE (SSVQE) and Multistate Contracted VQE (MCVQE), once the parameters are optimized to maximize the subspace overlap with the low-energy sector of the Hamiltonian, one diagonalizes the Hamiltonian restricted to the subspace. Unlike these methods, where hard-coded orthogonality constraints are enforced at the circuit level among the states spanning the subspace, we consider a subspace representation where orthogonality is soft-coded via penalty terms in the cost function. We show that this representation allows for shallower quantum circuits while maintaining high fidelity when compared to single-state (standard VQE) and multi-state (SSVQE or MCVQE) representations, on two benchmark cases: a 3times 3 transverse-field Ising model and random realizations of the Edwards--Anderson spin-glass model on a 4times 4 lattice.