Contextual Subspace Variational Quantum Eigensolver

We describe the contextual subspace variational quantum eigensolver (CS-VQE), a hybrid quantum-classical algorithm for approximating the ground state energy of a Hamiltonian. The approximation to the ground state energy is obtained as the sum of two contributions. The first contribution comes from a noncontextual approximation to the Hamiltonian, and is computed classically. The second contribution is obtained by using the variational quantum eigensolver (VQE) technique to compute a contextual correction on a quantum processor. In general the VQE computation of the contextual correction uses fewer qubits and measurements than the VQE computation of the original problem. Varying the number of qubits used for the contextual correction adjusts the quality of the approximation. We simulate CS-VQE on tapered Hamiltonians for small molecules, and find that the number of qubits required to reach chemical accuracy can be reduced by more than a factor of two. The number of terms required to compute the contextual correction can be reduced by more than a factor of ten, without the use of other measurement reduction schemes. This indicates that CS-VQE is a promising approach for eigenvalue computations on noisy intermediate-scale quantum devices.

Efficient estimation of hydraulic conductivity heterogeneity with non-redundant measurement information

Solving the inverse problem of identifying groundwater model parameters with measurements is a computationally intensive task. Although model reduction methods provide computational relief, the performance of many inversion methods depends on the amount of often highly correlated measurements. We propose a measurement reduction method that only incorporates essential measurement information in the inversion process. The method decomposes the covariance matrix of the model output and projects both measurements and model response on the eigenvector space corresponding to the largest eigenvalues. We combine this measurement reduction technique with two inversion methods, the Iterated Extended Kalman Filter (IEKF) and the Sequential Monte Carlo (SMC) methods. The IEKF method linearizes the relationship between measurements and parameters, and the cost of the required gradient calculation increases with increase of the number of measurements. SMC is a Bayesian updating approach that samples the posterior distribution through sequentially sampling a set of intermediate measures and the number of sampling steps increases with increase of the information content. We propose modified versions of both algorithms that identify the underlying eigenspace and incorporate the reduced information content in the inversion process. The performance of the modified IEKF and SMC methods with measurement reduction is tested on a numerical example that illustrates the computational benefit of the proposed approach as compared to the standard IEKF and SMC methods with full measurement sets.