  Universal fault-tolerant quantum computers will require error-free execution
of long sequences of quantum gate operations, which is expected to involve
millions of physical qubits. Before the full power of such machines will be
available, near-term quantum devices will provide several hundred qubits and
limited error correction. Still, there is a realistic prospect to run useful
algorithms within the limited circuit depth of such devices. Particularly
promising are optimization algorithms that follow a hybrid approach: the aim is
to steer a highly entangled state on a quantum system to a target state that
minimizes a cost function via variation of some gate parameters. This
variational approach can be used both for classical optimization problems as
well as for problems in quantum chemistry. The challenge is to converge to the
target state given the limited coherence time and connectivity of the qubits.
In this context, the quantum volume as a metric to compare the power of
near-term quantum devices is discussed.
  With focus on chemistry applications, a general description of variational
algorithms is provided and the mapping from fermions to qubits is explained.
Coupled-cluster and heuristic trial wave-functions are considered for
efficiently finding molecular ground states. Furthermore, simple
error-mitigation schemes are introduced that could improve the accuracy of
determining ground-state energies. Advancing these techniques may lead to
near-term demonstrations of useful quantum computation with systems containing
several hundred qubits.
