  Quantum computation is one of the most promising new paradigms for the
simulation of physical systems composed of electrons and atomic nuclei, with
applications in chemistry, solid-state physics, materials science, or molecular
biology. This requires a truncated representation of the electronic structure
Hamiltonian using a finite number of orbitals. While it is, in principle,
obvious how to improve on the representation by including more orbitals, this
is usually unfeasible in practice (e.g., because of the limited number of
qubits available) and severely compromises the accuracy of the obtained
results. Here, we propose a quantum algorithm that improves on the
representation of the physical problem by virtue of second-order perturbation
theory. In particular, our quantum algorithm evaluates the second-order energy
correction through a series of time-evolution steps under the unperturbed
Hamiltonian. An important application is to go beyond the active-space
approximation, allowing to include corrections of virtual orbitals, known as
multireference perturbation theory. Here, we exploit that the unperturbed
Hamiltonian is diagonal for virtual orbitals and show that the number of qubits
is independent of the number of virtual orbitals. This gives rise to more
accurate energy estimates without increasing the number of qubits. Moreover, we
demonstrate numerically for realistic chemical systems that the total runtime
has highly favourable scaling in the number of virtual orbitals compared to
previous work. Numerical calculations confirm the necessity of the
multireference perturbation theory energy corrections to reach accurate ground
state energy estimates. Our perturbation theory quantum algorithm can also be
applied to Symmetry-Adapted Perturbation Theory. As such, we demonstrate that
perturbation theory can help to reduce the quantum hardware requirements for
quantum chemistry.
