A Stabilizer Framework for the Contextual Subspace Variational Quantum Eigensolver and the Noncontextual Projection Ansatz

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 utilize even the
modest NISQ machines available today. We discuss a method of ground
state energy estimation predicated on a partitioning of 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); however, 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 the 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
projection ansatz using a quantum simulator and demonstrate chemically
precise ground state energy calculations for a suite of small molecules
at a significant reduction in the required qubit count and circuit
depth.