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arXiv:1907.11368

On the relationships between Z-, C-, and H-local unitaries

Cook, Jeremy

Published Jul 9, 2019• 0 citations
arXivHTMLPDF

Abstract

Quantum walk algorithms can speed up search of physical regions of space in both the discrete-time [arXiv:quant-ph/0402107] and continuous-time setting [arXiv:quant-ph/0306054], where the physical region of space being searched is modeled as a connected graph. In such a model, Aaronson and Ambainis [arXiv:quant-ph/0303041] provide three different criteria for a unitary matrix to act locally with respect to a graph, called $Z$-local, $C$-local, and $H$-local unitaries, and left the open question of relating these three locality criteria. Using a correspondence between continuous- and discrete-time quantum walks by Childs [arXiv:0810.0312], we provide a way to approximate $N\times N$ $H$-local unitaries with error $δ$ using $O(1/\sqrtδ,\sqrt{N})$ $C$-local unitaries, where the comma denotes the maximum of the two terms.

DOI
10.48550/arxiv.1907.11368

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In this Letter we make progress on a long-standing open problem of Aaronson and Ambainis [Theory Comput. 1, 47 (2005)]: we show that if U is a sparse unitary operator with a gap Delta in its spectrum, then there exists an approximate logarithm H of U which is also sparse. The sparsity pattern of H gets more dense as 1/Delta increases. This result can be interpreted as a way to convert between local continuous-time and local discrete-time quantum processes. As an example we show that the discrete-time coined quantum walk can be realized stroboscopically from an approximately local continuous-time quantum walk.
Quantum walks can be defined in two quite distinct ways: discrete-time and continuous-time quantum walks (DTQWs and CTQWs). For classical random walks, there is a natural sense in which continuous-time walks are a limit of discrete-time walks. Quantum mechanically, in the discrete-time case, an additional ``coin space'' must be appended for the walk to have nontrivial time evolution. Continuous-time quantum walks, however, have no such constraints. This means that there is no completely straightforward way to treat a CTQW as a limit of a DTQW, as can be done in the classical case. Various approaches to this problem have been taken in the past. We give a construction for walks on ddd-regular, ddd-colorable graphs when the coin flip operator is Hermitian: from a standard DTQW we construct a family of discrete-time walks with a well-defined continuous-time limit on a related graph. One can think of this limit as a ``coined'' continuous-time walk. We show that these CTQWs share some properties with coined DTQWs. In particular, we look at a spatial search by a DTQW over the two-dimensional (2D) torus (a grid with periodic boundary conditions) of size $\sqrt{N}\ifmmode\times\else\texttimes\fi{}\sqrt{N}$, where it was shown that a coined DTQW can search in time O(NlogN)O(\sqrt{N}logN)O(N​logN), but a standard CTQW takes $\mathrm{\ensuremath{\Omega}}(N)$ time to search for a marked element. The continuous limit of the DTQW search over the 2D torus exhibits the O(NlogN)O(\sqrt{N}logN)O(N​logN) scaling, like the coined walk it is derived from. We also look at the effects of graph symmetry on the limiting walk, and show that the properties are similar to those of the DTQW as shown in Krovi and Brun, Phys. Rev. A 75, 062332 (2007).
The interest in quantum walks has been steadily increasing during the last two decades. It is still worth to present new forms of quantum walks that might find practical applications and new physical behaviors. In this work, we define discrete-time quantum walks on arbitrary oriented graphs by partitioning a graph into tessellations, which is a collection of disjoint cliques that cover the vertex set. By using the adjacency matrices associated with the tessellations, we define local unitary operators, whose product is the evolution operator of our quantum walk model. We introduce a parameter, called alpha, that quantifies the amount of orientation. We show that the parameter alpha can be tuned in order to increase the amount of quantum walk-based transport on oriented graphs.
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We propose a class of continuous-time quantum walk models on graphs induced by a certain class of discrete-time quantum walk models with a parameter [Formula: see text]. Here the graph treated in this paper can be applied to both finite and infinite cases. The induced continuous-time quantum walk is an extended version of the (free) discrete-Schrödinger equation driven by the normalized Laplacian: the element of the weighted Hermitian takes not only a scalar value but also a matrix value depending on the underlying discrete-time quantum walk. We show that each discrete-time quantum walk with an appropriate setting of the parameter [Formula: see text] in the long time limit identifies with its induced continuous-time quantum walk and gives the running time for the discrete-time to approximate the induced continuous-time quantum walk with a small error [Formula: see text]. We also investigate the detailed spectral information on the induced continuous-time quantum walk.