1. The problem statement, all variables and given/known data The goal is to 'decompose' common 2 qubit quantum gates such as the pi/8 gate into a sequence of CNOTS and single qubit rotations. I have the book by Nielsen and Chuang and the info is sortof in there (universality proof of CNOT), but I don't get how to apply it, i.e. how to calculate in a systematic way what rotations to add to the CNOT. I would be very grateful is someone could help me out here. I'm of course interested in the decomposition of a particular gate, but even more so in a consistent method of how to handle this kind of problem. Thanks in advance! For the relevant matrices see for instance Wikipedia In addition: I now that H N H (Hadamard NOT Hadamard) gives the S-gate (1 0 / 0 -1), which in turn is the square of the pi/8 gate, but I can't generalize this. My best guess would be to combine the Hadamards with additional pi-rotations, but I have no clue if this is correct and if so what axis to choose.