Contextual Subspace Variational Quantum Eigensolver

Abstract

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.

The Variational Quantum Eigensolver: a review of methods and best practices

“…[223]. We also recommend that the resulting qubit Hamiltonian is further reduced using tapering off methods based on symmetries [242][243][244] (see Sec. 4.3).…”

“…It is worth noting that a similar approach is briefly outlined in [316], along with a proposed method to identify point-group symmetries. The Contextual Subspace VQE (CSVQE) [244], proposes to separate the expectation value of the Hamiltonian into two contributions: a contextual part, computed using VQE, and a non-contextual part, computed using a conventional computer. A set of Pauli strings observables is considered non-contextual if it is possible to measure and assign value to them simultaneously without contradiction [244,[317][318][319][320][321][322][323].…”

Qsun: an open-source platform towards practical quantum machine learning applications

“…where k is the scaling factor that transforms the output data from 4). The cost function is the mean squared error as described in equation (11).…”

“…For comparison, we next provide the analytical solution for minimizing the cost function (11) with (15). We vectorize:…”

“…The self-consistency is performed until convergence. VQAs have been extensively employed to tackle numerous tasks, including the variational quantum eigensolvers (VQEs) [10][11][12][13][14][15][16], quantum dynamics simulation [17][18][19][20][21][22], mathematical applications [23][24][25][26][27][28][29][30][31], quantum machine learning (QML) [32][33][34][35][36][37][38][39][40], and new frontiers in quantum foundations [8,[41][42][43][44][45][46][47].…”