Search via Quantum Walk

Abstract

We propose a new method for designing quantum search algorithms for finding a "marked" element in the state space of a classical Markov chain. The algorithm is based on a quantum walk \'a la Szegedy (2004) that is defined in terms of the Markov chain. The main new idea is to apply quantum phase estimation to the quantum walk in order to implement an approximate reflection operator. This operator is then used in an amplitude amplification scheme. As a result we considerably expand the scope of the previous approaches of Ambainis (2004) and Szegedy (2004). Our algorithm combines the benefits of these approaches in terms of being able to find marked elements, incurring the smaller cost of the two, and being applicable to a larger class of Markov chains. In addition, it is conceptually simple and avoids some technical difficulties in the previous analyses of several algorithms based on quantum walk.Comment: 21 pages. Various modifications and improvements, especially in Section

Quantum Algorithms for the Subset-Sum Problem

“…This section summarizes Ambainis's unique-collision-finding algorithm [2] (from the edge-walk perspective of [23]); introduces a new way to streamline Ambainis's algorithm; and applies the streamlined algorithm to the subset-sum context, obtaining cost 2 n/3 in a different way from Section 2. This section's subset-sum algorithm uses collision finding as a black box, but the faster algorithms in Section 5 do not.…”

Time-Efficient Quantum Walks for 3-Distinctness

“…Superficially, our modification appears small. Indeed, the proof of the complexity of our framework is nearly the same as that of [12]. However, there are some subtle differences in the implementation of the walk.…”

“…Quantizing this algorithm leads to efficient quantum algorithms [12]. The quantization considers P as a walk on directed edges of G. We write (x, y) ∈ E when we consider an edge {x, y} ∈ E with orientation.…”

“…Using a quantum walk algorithm with costs S , U , C , ε , δ (as in [12]) as a checking subroutine straightforwardly gives complexity S + 1…”

“…Recently, [11] reproduced several known learning graph upper bounds via explicit algorithms in an extension of the quantum walk search framework of [12]. This work produced a new quantum algorithmic tool, quantum walks with nested checking.…”