One of the most promising suggested applications of quantum computing is solving classically intractable chemistry problems. This may help to answer unresolved questions about phenomena like: high temperature superconductivity, solid-state physics, transition metal catalysis, or certain biochemical reactions. In turn, this increased understanding may help us to refine, and perhaps even one day design, new compounds of scientific and industrial importance. However, building a sufficiently large quantum computer will be a difficult scientific challenge. As a result, developments that enable these problems to be tackled with fewer quantum resources should be considered very important. Driven by this potential utility, quantum computational chemistry is rapidly emerging as an interdisciplinary field requiring knowledge of both quantum computing and computational chemistry. This review provides a comprehensive introduction to both computational chemistry and quantum computing, bridging the current knowledge gap. We review the major developments in this area, with a particular focus on near-term quantum computation. Illustrations of key methods are provided, explicitly demonstrating how to map chemical problems onto a quantum computer, and solve them. We conclude with an outlook for this nascent field.

Imaginary time evolution is a powerful tool for studying quantum systems. While it is possible to simulate with a classical computer, the time and memory requirements generally scale exponentially with the system size. Conversely, quantum computers can efficiently simulate quantum systems, but not non-unitary imaginary time evolution. We propose a variational algorithm for simulating imaginary time evolution on a hybrid quantum computer. We use this algorithm to find the ground state energy of many-particle systems; specifically molecular Hydrogen and Lithium Hydride, finding the ground state with high probability. Our method can also be applied to general optimisation problems and quantum machine learning. As our algorithm is hybrid, suitable for error mitigation, and can exploit shallow quantum circuits, it can be implemented with current quantum computers.

Quantum simulation of chemistry and materials is predicted to be an important application for both near-term and fault-tolerant quantum devices. However, at present, developing and studying algorithms for these problems can be difficult due to the prohibitive amount of domain knowledge required in both the area of chemistry and quantum algorithms. To help bridge this gap and open the field to more researchers, we have developed the OpenFermion software package (www.openfermion.org). OpenFermion is an open-source software library written largely in Python under an Apache 2.0 license, aimed at enabling the simulation of fermionic and bosonic models and quantum chemistry problems on quantum hardware. Beginning with an interface to common electronic structure packages, it simplifies the translation between a molecular specification and a quantum circuit for solving or studying the electronic structure problem on a

Variational algorithms may enable classically intractable simulations on near-future quantum computers. However, their potential is limited by hardware errors. It is therefore crucial to develop efficient ways to mitigate these errors. Here, we propose a stabiliser-like method which enables the detection of up to 60 -80 % of depolarising errors. Our method is suitable for near-term quantum hardware. Simulations show that our method can significantly benefit calculations subject to both stochastic and correlated noise, especially when combined with existing error mitigation techniques.

Calculating the energy spectrum of a quantum system is an important task, for example to analyze reaction rates in drug discovery and catalysis. There has been significant progress in developing algorithms to calculate the ground state energy of molecules on near-term quantum computers. However, calculating excited state energies has attracted comparatively less attention, and it is currently unclear what the optimal method is. We introduce a low depth, variational quantum algorithm to sequentially calculate the excited states of general Hamiltonians. Incorporating a recently proposed technique [O. Higgott, D. Wang, and S. Brierley, arXiv:1805.08138], we employ the low depth swap test to energetically penalize the ground state, and transform excited states into ground states of modified Hamiltonians. We use variational imaginary time evolution as a subroutine, which deterministically propagates toward the target eigenstate. We discuss how symmetry measurements can mitigate errors in the swap test step. We numerically test our algorithm on Hamiltonians which encode 3SAT optimization problems of up to 18 qubits, and the electronic structure of the lithium hydride molecule. As our algorithm uses only low depth circuits and variational algorithms, it is suitable for use on near-term quantum hardware.