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