Efficient quantum circuit implementation of quantum walks

Abstract

Quantum walks, being the quantum analogue of classical random walks, are expected to provide a fruitful source of quantum algorithms. A few such algorithms have already been developed, including the 'glued trees' algorithm, which provides an exponential speedup over classical methods, relative to a particular quantum oracle. Here, we discuss the possibility of a quantum walk algorithm yielding such an exponential speedup over possible classical algorithms, without the use of an oracle. We provide examples of some highly symmetric graphs on which efficient quantum circuits implementing quantum walks can be constructed, and discuss potential applications to quantum search for marked vertices along these graphs.

Physical Implementation of Quantum Walks

“…(3.166) Douglas and Wang (2009) developed a set of exact and efficient quantum circuits to implement discrete-time quantum walks on several highly symmetric graphs, such as a 16-length cycle (Fig. 3.89), a 2D hyper cycle, a twisted toroidal lattice , as well as the complete 3 n graphs (Fig.…”

“…Loke and Wang (2011) extended this work to include non-regular graphs such as the star graphs of arbitrary degree and Cayley trees with an arbitrary number of layers. The examples considered in Douglas and Wang (2009) and Loke and Wang (2011) are quite simple, but more complex variations can still be efficiently implemented such as composites of highly symmetric graphs, symmetric graphs with a small number of 'imperfections', as well as graphs possessing a certain bounded level of complexity. Quantum walks have been used to search for marked vertices along highly symmetric graphs, including the hypercube, complete graphs and complete multipartite graphs.…”

“…The work carried out by Douglas and Wang (2009) and Loke and Wang (2011) can be used to efficiently implement such an oracle -using O(log(n)) elementary gates for a graph of order n -given a highly symmetric graph such as those considered in Hines and Stamp (2007), Reitzner et al (2009), Douglas and and Loke and Wang (2011). The work carried out by Douglas and Wang (2009) and Loke and Wang (2011) can be used to efficiently implement such an oracle -using O(log(n)) elementary gates for a graph of order n -given a highly symmetric graph such as those considered in Hines and Stamp (2007), Reitzner et al (2009), Douglas and and Loke and Wang (2011).…”

Quantum walks on graphene nanoribbons using quantum gates as coins

“…Discrete quantum walks have been proven to implement the same universal quantum gate set and thus are able to implement any quantum algorithm [16]. Quantum walks can also be encoded as quantum circuits, which is also a universal model for quantum computation [17,18].…”

Search via Quantum Walk

“…Generally coined quantum walk can make quantum walk faster than the continuous time quantum walk ( Ambainis (2005)). Quantum walk can also be simulated by quantum circuit ( Douglas & Wang (2009)), thus we can realized the quantum based search algorithm by the quantum computer in principle. This makes quantum walk not only a conceived tool for algorithm, but also useful for the computation theory to explore more efficient algorithms for intractable problems.…”