  Numerous scientific developments in this NISQ-era (Noisy Intermediate Scale
Quantum) have raised the importance for quantum algorithms relative to their
conventional counterparts due to its asymptotic advantage. For resource
estimates in several quantum algorithms, arithmetic operations are crucial.
With resources reported as a number of Toffoli gates or T gates with/without
ancilla, several efficient implementations of arithmetic operations, such as
addition/subtraction, multiplication/division, square root, etc., have been
accomplished in binary quantum systems. More recently, it has been shown that
intermediate qutrits may be employed in the ancilla-free frontier zone,
enabling us to function effectively there. In order to achieve efficient
implementation of all the above-mentioned quantum arithmetic operations with
regard to gate count and circuit-depth without T gate and ancilla, we have
included an intermediate qutrit method in this paper. Future research aiming at
reducing costs while taking into account arithmetic operations for computing
tasks might be guided by our resource estimations using intermediate qutrits.
Therefore, the enhancements are examined in relation to the fundamental
arithmetic circuits. The intermediate qutrit approach necessitates access to
higher energy levels, making the design susceptible to errors. We nevertheless
demonstrate that the percentage decrease in the probability of error is
significant due to the fact that we achieve circuit efficiency by reducing
circuit-depth in comparison to qubit-only works.
