1. A method for defining a quantum circuit u comprising a sequence of representations of quantum gates H and quantum gates T, wherein the quantum gates H and T are represented by [ i 2 i 2 i 2 - i 2 ] ⁢ ⁢ and ⁢ [ ⅇ - ⅈ ⁢ π 8 0 0 ⅇ ⅈ ⁢ π 8 ] , respectively, the method comprising: initially transforming, by a digital computing device, the quantum circuit u to a quantum circuit x comprising either a representation of a single Clifford gate or one or more representations of a single Clifford gate, a first quantum gate T and a second quantum gate T that form a T gate pair followed by a representation of a single Clifford gate, wherein the quantum circuit x and the quantum circuit u are equivalent; transforming, by the digital computing device, the quantum circuit x to a normal representation n comprising a sequence of representations of one or more quantum gates selected from among quantum gates TH, quantum gates SH, and Clifford gates, wherein the quantum circuit x and the normal representation n are equivalent, and wherein a quantum gate S is represented by [ ⅇ - ⅈ ⁢ π 4 0 0 ⅇ ⅈ ⁢ π 4 ] ; and implementing a quantum circuit based on the normal representation.