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Many quantum algorithms have daunting resource requirements when compared to what is availabletoday. To address this discrepancy, a quantum-classical hybrid optimization scheme known as thequantum variational eigensolver was developed (Peruzzo et al 2014 Nat. Commun. 5 4213) with thephilosophy that even minimal quantum resources could be made useful when used in conjunctionwith classical routines. In this work we extend the general theory of this algorithm and suggestalgorithmic improvements for practical implementations. Specifically, we develop a variationaladiabatic ansatz and explore unitary coupled cluster where we establish a connection from secondorder unitary coupled cluster to universal gate sets through a relaxation of exponential operatorsplitting.Weintroduce the concept of quantum variational error suppression that allows some errorsto be suppressed naturally in this algorithm on a pre-threshold quantum device. Additionally, weanalyze truncation and correlated sampling in Hamiltonian averaging as ways to reduce the cost of thisprocedure. Finally, we show how the use of modern derivative free optimization techniques can offerdramatic computational savings of up to three orders of magnitude over previously used optimizationtechniques.