Methods and systems 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 can map normal representations to canonical-form representations, which are generally further compressed, in length, with respect to normal representations.