The current application is directed to methods and systems which transform a given single-qubit quantum circuit expressed in a first quantum-gate basis into a quantum-circuit expressed in a second, discrete, quantum-gate basis. The discrete quantum-gate basis comprises standard, implementable quantum gates. The given single-qubit quantum circuit is expressed as a normal representation. The normal representation is generally compressed, in length, with respect to equivalent non-normalized representations. The method and systems additionally provide a mapping from normal representations to canonical-form representations, which are generally further compressed, in length, with respect to normal representations. The normal and canonical-form representations can be used to implement methods and systems for search-based quantum-circuit design. Neither this section nor the sections which follow are intended to either limit the scope of the claims which follow or define the scope of those claims.