Asymptotically Improved Circuit for $d$-ary Grover's Algorithm with Advanced Decomposition of $n$-qudit Toffoli Gate

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.