T-count and Qubit Optimized Quantum Circuit Design of the Non-Restoring Square Root Algorithm

Quantum circuits for basic mathematical functions such as the square root are required to implement scientific computing algorithms on quantum computers. Quantum circuits that are based on Clifford+T gates can easily be made fault tolerant but the T gate is very costly to implement. As a result, reducing T-count has become an important optimization goal. Further, quantum circuits with many qubits are difficult to realize, making designs that save qubits and produce no garbage outputs desirable. In this work, we present a T-count optimized quantum square root circuit with only $2 \cdot n +1$ qubits and no garbage output. To make a fair comparison against existing work, the Bennett's garbage removal scheme is used to remove garbage output from existing works. We determined that our proposed design achieves an average T-count savings of $43.44 \%$, $98.95 \%$, $41.06 \%$ and $20.28 \%$ as well as qubit savings of $85.46 \%$, $95.16 \%$, $90.59 \%$ and $86.77 \%$ compared to existing works.