  The progress in building quantum computers to execute quantum algorithms has
recently been remarkable. Grover's search algorithm in a binary quantum system
provides considerable speed-up over classical paradigm. Further, Grover's
algorithm can be extended to a $d$-ary (qudit) quantum system for utilizing the
advantage of larger state space, which helps to reduce the run-time of the
algorithm as compared to the traditional binary quantum systems. In a qudit
quantum system, an $n$-qudit Toffoli gate plays a significant role in the
accurate implementation of Grover's algorithm. In this article, a generalized
$n$-qudit Toffoli gate has been realized using higher dimensional qudits to
attain a logarithmic depth decomposition without ancilla qudit. The circuit for
Grover's algorithm has then been designed for any $d$-ary quantum system, where
$d \ge 2$, with the proposed $n$-qudit Toffoli gate to obtain optimized depth
compared to earlier approaches. The technique for decomposing an $n$-qudit
Toffoli gate requires access to two immediately higher energy levels, making
the design susceptible to errors. Nevertheless, we show that the percentage
decrease in the probability of error is significant as we have reduced both
gate count and circuit depth as compared to that in state-of-the-art works.
