Asymptotically improved circuit for a <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>d</mml:mi></mml:math>-ary Grover's algorithm with advanced decomposition of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>n</mml:mi></mml:math>-qudit Toffoli gate
Amit Saha, Ritajit Majumdar, Debasri Saha, Amlan Chakrabarti, Susmita Sur‐Kolay
Abstract
The progress in building quantum computers to execute quantum algorithms has recently been remarkable. Grover's search algorithm in a binary quantum system provides a considerable speed-up over the classical paradigm. It can be extended to a $d$-ary (qudit) quantum system also for utilizing the advantage of larger state space, which helps to reduce the runtime 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 is 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 $d\ensuremath{\ge}2$, 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 with both gate count and circuit depth reduced as compared to that in state-of-the-art works.