Approximate programmable quantum processors

A quantum processor is a programmable quantum circuit in which both the data and the program, which specifies the operation that is carried out on the data, are quantum states. We study the situation in which we want to use such a processor to approximate a set of unitary operators to a specified level of precision. We measure how well an operation is performed by the process fidelity between the desired operation and the operation produced by the processor. We show how to find the program for a given processor that produces the best approximation of a particular unitary operation. We also place bounds on the dimension of the program space that is necessary to approximate a set of unitary operators to a specified level of precision.