One-Dimensional Lazy Quantum walk in Ternary System

Quantum walks play an important role for developing quantum algorithms and quantum simulations. Here we present one dimensional three-state quantum walk(lazy quantum walk) and show its equivalence for circuit realization in ternary quantum logic for the first of its kind. Using an appropriate logical mapping of the position space on which a walker evolves onto the multi-qutrit states, we present efficient quantum circuits considering the nearest neighbour position space for the implementation of lazy quantum walks in one-dimensional position space in ternary quantum system. We also address scalability in terms of $n$-qutrit ternary system with example circuits for a three qutrit state space.Comment: 13 pages, 12 figures, and 10 table

Asymptotically Improved Grover's Algorithm in any Dimensional Quantum System with Novel Decomposed nn-qudit Toffoli Gate

As the development of Quantum computers becomes reality, the implementation of quantum algorithms is accelerating in a great pace. Grover's algorithm in a binary quantum system is one such quantum algorithm which solves search problems with numeric speed-ups than the conventional classical computers. Further, Grover's algorithm is extended to a $d$-ary quantum system for utilizing the advantage of larger state space. In qudit or $d$-ary quantum system n-qudit Toffoli gate plays a significant role in the accurate implementation of Grover's algorithm. In this paper, a generalized $n$-qudit Toffoli gate has been realized using qudits to attain a logarithmic depth decomposition without ancilla qudit. Further, the circuit for Grover's algorithm has been designed for any d-ary quantum system, where d >= 2, with the proposed $n$-qudit Toffoli gate so as to get optimized depth as compared to state-of-the-art approaches. This technique for decomposing an n-qudit Toffoli gate requires access to higher energy levels, making the design susceptible to leakage error. Therefore, the performance of this decomposition for the unitary and erasure models of leakage noise has been studied as well