Applications of variational methods for quantum computers

The primary subject of this dissertation is the analysis and improvement of variational methods that combine the use of classical and gate based quantum computers. The secondary subject is the development of matrix based error mitigation and benchmarking protocols fornoisy quantum computers. Variational methods run on quantum computer emulators are used to find the ground state energies of the Heisenberg and Hubbard models and selected molecules in chemistry. An algorithm is developed and deployed to automate the creation of variational circuits. The theory and overview of variational methods and gradient based optimisation algorithms are presented. We learn that while variational methods make it possible to use current generation quantum computers, guarantees of always finding the ground state energy are elusive.We introduce noise in our emulations and adapt the optimisation algorithms to withstand it. We observe the emergence of local minima and barren plateaus which hinder variational methods from finding the ground state energies. It is discerned that clever choices of initial states and parameters are necessary ingredients for success. We develop the technique of quasi-dynamical evolution inspired by quantum annealing. It overcomes the limitations of standard variational algorithms by systematically improving the ground state energy estimate. Our tests show that the heuristic improves the energy estimate even in facile settings. We introduce seven criteria for ideal error mitigation protocols. A new protocol is developedon its basis. Our tests on IBM Q quantum computers show noticeable error mitigation.The matrix generated during the execution of the protocol helps detect and visualise errorsand biases. We invent and use small depth quantum circuits for benchmarking quantumcomputers