Systems and methods are provided for performing addition of qubit states using entangled quaternionic roots of single qubit gates. A method includes identifying a single qubit gate in a quantum circuit of a quantum computer, wherein the quantum circuit includes two summand qubits entangled with a third qubit that stores a measurable non-linear sum of the two summand qubits. The method includes mapping the single qubit gate to a representative gate that is phase-equivalent to the single qubit gate, mapping the representative gate to a unit quaternion, calculating an nth root of the unit quaternion, and mapping the nth root of the unit quaternion to a unitary matrix that represents a fractional single qubit gate. The method includes configuring the quantum circuit to include at least one fractional single qubit gate representing the unitary matrix in place of the single qubit gate for performing the addition.