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Quantum chemistry is a promising application for noisy intermediate-scale quantum (NISQ) devices.However, quantum computers have thus far not succeeded in providing solutions to problems of real scientific significance, with algorithmic advances being necessary to fully utilise even the modest NISQ machines available today. We discuss a method of ground state energy estimation predicated on a partitioning the molecular Hamiltonian into two parts: one that is noncontextual and can be solved classically, supplemented by a contextual component that yields quantum corrections obtained via a Variational Quantum Eigensolver (VQE) routine. This approach has been termed Contextual Subspace VQE (CS-VQE), but there are obstacles to overcome before it can be deployed on NISQ devices. The problem we address here is that of the ansatz -a parametrized quantum state over which we optimize during VQE. It is not initially clear how a splitting of the Hamiltonian should be reflected in our CS-VQE ansätze. We propose a 'noncontextual projection' approach that is illuminated by a reformulation of CS-VQE in the stabilizer formalism. This defines an ansatz restriction from the full electronic structure problem to the contextual subspace and facilitates an implementation of CS-VQE that may be deployed on NISQ devices. We validate the noncontextual Tim J.

The contextual subspace variational quantum eigensolver (CS-VQE) is a hybrid quantum-classical algorithm that approximates the ground state energy of a given qubit Hamiltonian. It achieves this by separating the Hamiltonian into contextual and noncontextual parts. The ground state energy is approximated by classically solving the noncontextual problem, followed by solving the contextual problem using VQE, constrained by the noncontexual solution. In general, computation of the contextual correction needs fewer qubits and measurements compared to solving the full Hamiltonian via traditional VQE. We simulate CS-VQE on different tapered molecular Hamiltonians and apply the unitary partitioning measurement reduction strategy to further reduce the number of measurements required to obtain the contextual correction. Our results indicate that CS-VQE combined with measurement reduction is a promising approach to allow feasible eigenvalue computations on noisy intermediate-scale quantum devices. We also provide a modification to the CS-VQE algorithm, that previously could cause an exponential increase in Hamiltonian terms, that now at worst will scale quadratically.