2,386 research outputs found
Vertices cannot be hidden from quantum spatial search for almost all random graphs
In this paper we show that all nodes can be found optimally for almost all
random Erd\H{o}s-R\'enyi graphs using continuous-time
quantum spatial search procedure. This works for both adjacency and Laplacian
matrices, though under different conditions. The first one requires
, while the seconds requires , where . The proof was made by analyzing the convergence
of eigenvectors corresponding to outlying eigenvalues in the norm. At the same time for , the property does
not hold for any matrix, due to the connectivity issues. Hence, our derivation
concerning Laplacian matrix is tight.Comment: 18 pages, 3 figur
On the relationship between continuous- and discrete-time quantum walk
Quantum walk is one of the main tools for quantum algorithms. Defined by
analogy to classical random walk, a quantum walk is a time-homogeneous quantum
process on a graph. Both random and quantum walks can be defined either in
continuous or discrete time. But whereas a continuous-time random walk can be
obtained as the limit of a sequence of discrete-time random walks, the two
types of quantum walk appear fundamentally different, owing to the need for
extra degrees of freedom in the discrete-time case.
In this article, I describe a precise correspondence between continuous- and
discrete-time quantum walks on arbitrary graphs. Using this correspondence, I
show that continuous-time quantum walk can be obtained as an appropriate limit
of discrete-time quantum walks. The correspondence also leads to a new
technique for simulating Hamiltonian dynamics, giving efficient simulations
even in cases where the Hamiltonian is not sparse. The complexity of the
simulation is linear in the total evolution time, an improvement over
simulations based on high-order approximations of the Lie product formula. As
applications, I describe a continuous-time quantum walk algorithm for element
distinctness and show how to optimally simulate continuous-time query
algorithms of a certain form in the conventional quantum query model. Finally,
I discuss limitations of the method for simulating Hamiltonians with negative
matrix elements, and present two problems that motivate attempting to
circumvent these limitations.Comment: 22 pages. v2: improved presentation, new section on Hamiltonian
oracles; v3: published version, with improved analysis of phase estimatio
On analog quantum algorithms for the mixing of Markov chains
The problem of sampling from the stationary distribution of a Markov chain
finds widespread applications in a variety of fields. The time required for a
Markov chain to converge to its stationary distribution is known as the
classical mixing time. In this article, we deal with analog quantum algorithms
for mixing. First, we provide an analog quantum algorithm that given a Markov
chain, allows us to sample from its stationary distribution in a time that
scales as the sum of the square root of the classical mixing time and the
square root of the classical hitting time. Our algorithm makes use of the
framework of interpolated quantum walks and relies on Hamiltonian evolution in
conjunction with von Neumann measurements.
There also exists a different notion for quantum mixing: the problem of
sampling from the limiting distribution of quantum walks, defined in a
time-averaged sense. In this scenario, the quantum mixing time is defined as
the time required to sample from a distribution that is close to this limiting
distribution. Recently we provided an upper bound on the quantum mixing time
for Erd\"os-Renyi random graphs [Phys. Rev. Lett. 124, 050501 (2020)]. Here, we
also extend and expand upon our findings therein. Namely, we provide an
intuitive understanding of the state-of-the-art random matrix theory tools used
to derive our results. In particular, for our analysis we require information
about macroscopic, mesoscopic and microscopic statistics of eigenvalues of
random matrices which we highlight here. Furthermore, we provide numerical
simulations that corroborate our analytical findings and extend this notion of
mixing from simple graphs to any ergodic, reversible, Markov chain.Comment: The section concerning time-averaged mixing (Sec VIII) has been
updated: Now contains numerical plots and an intuitive discussion on the
random matrix theory results used to derive the results of arXiv:2001.0630
Almost uniform sampling via quantum walks
Many classical randomized algorithms (e.g., approximation algorithms for
#P-complete problems) utilize the following random walk algorithm for {\em
almost uniform sampling} from a state space of cardinality : run a
symmetric ergodic Markov chain on for long enough to obtain a random
state from within total variation distance of the uniform
distribution over . The running time of this algorithm, the so-called {\em
mixing time} of , is , where
is the spectral gap of .
We present a natural quantum version of this algorithm based on repeated
measurements of the {\em quantum walk} . We show that it
samples almost uniformly from with logarithmic dependence on
just as the classical walk does; previously, no such
quantum walk algorithm was known. We then outline a framework for analyzing its
running time and formulate two plausible conjectures which together would imply
that it runs in time when is
the standard transition matrix of a constant-degree graph. We prove each
conjecture for a subclass of Cayley graphs.Comment: 13 pages; v2 added NSF grant info; v3 incorporated feedbac
Spatial search by continuous-time quantum walks on complex networks
Spatial search by continuous-time quantum walks on complex networks is focused
on using a quantum walk in continuous time in order to find a single or multiple
marked vertices within the complex network. The specific formalism used here is to
consider a coupling constant that shifts the state of the quantum walker from the
initial state to the target state, which is the marked vertex.
The thesis begins with establishing the mathematical framework of network theory,
quantum walks and numerical methods that will be used in the remainder of the
thesis. Then spatial search by continuous-time quantum walk is studied on regular
and semi-regular graphs, where most analytical results can be found. This will get
us acquainted with spatial search by quantum walk. The complex networks studied
are Barabasi-Albert graphs and the Internet network on the level of autonomous
systems. Different renormalized and pruned versions of the Internet network are
studied. The parameters of the quantum walk that are focused on are the optimal
values for the coupling constant, success probability, time and search time.Kvanttikulkujen spatiaalinen etsintä jatkuvassa ajassa kompleksisissa verkoissa
keskittyy yhden tai useamman merkityn solmukohdan löytämiseen kompleksisesta
verkosta käyttämällä kvanttikulkua jatkuvassa ajassa. Tässä työssä käytetty formalismi käsittelee kytkentävakiota, mikä siirtää kvanttikulkijan tilan alkutilasta
tavoitetilaan, eli merkittyyn solmukohtaan.
Tämä Pro Gradu alkaa matemaattisen viitekehyksen käsittelemisellä, jota tarvitaan
lopputyössä. Tämä viitekehys sisältää verkkoteorian, kvanttikulut ja käytetyt numeeriset menetelmät. Tämän jälkeen kvanttikulun spatiaalista etsintää jatkuvassa
ajassa tutkitaan säännöllisissä ja miltei säännöllisissä verkoissa, missä analyyttiset ratkaisut on löydettävissä. Tämän tarkoituksena on tutustua spatiaaliseen
etsintään kvanttikululla. Barabasi-Albert -graafit ja Internet-verkko autonomisten
järjestelmien tasolla ovat tässä työssä tutkittavat kompleksiset verkot. Tässä
tutkitaan eri renormalisoituja ja karsittuja versioita Internet-verkosta. Kvanttikulun parametrit, joihin keskitytään, ovat optimaaliset arvot kytkentävakiolle,
onnistumistodennäköisyydelle, ajalle ja etsintäajalle
Materials property prediction using symmetry-labeled graphs as atomic-position independent descriptors
Computational materials screening studies require fast calculation of the
properties of thousands of materials. The calculations are often performed with
Density Functional Theory (DFT), but the necessary computer time sets
limitations for the investigated material space. Therefore, the development of
machine learning models for prediction of DFT calculated properties are
currently of interest. A particular challenge for \emph{new} materials is that
the atomic positions are generally not known. We present a machine learning
model for the prediction of DFT-calculated formation energies based on Voronoi
quotient graphs and local symmetry classification without the need for detailed
information about atomic positions. The model is implemented as a message
passing neural network and tested on the Open Quantum Materials Database (OQMD)
and the Materials Project database. The test mean absolute error is 20 meV on
the OQMD database and 40 meV on Materials Project Database. The possibilities
for prediction in a realistic computational screening setting is investigated
on a dataset of 5976 ABSe selenides with very limited overlap with the OQMD
training set. Pretraining on OQMD and subsequent training on 100 selenides
result in a mean absolute error below 0.1 eV for the formation energy of the
selenides.Comment: 14 pages including references and 13 figure
Decoherence in quantum walks - a review
The development of quantum walks in the context of quantum computation, as
generalisations of random walk techniques, led rapidly to several new quantum
algorithms. These all follow unitary quantum evolution, apart from the final
measurement. Since logical qubits in a quantum computer must be protected from
decoherence by error correction, there is no need to consider decoherence at
the level of algorithms. Nonetheless, enlarging the range of quantum dynamics
to include non-unitary evolution provides a wider range of possibilities for
tuning the properties of quantum walks. For example, small amounts of
decoherence in a quantum walk on the line can produce more uniform spreading (a
top-hat distribution), without losing the quantum speed up. This paper reviews
the work on decoherence, and more generally on non-unitary evolution, in
quantum walks and suggests what future questions might prove interesting to
pursue in this area.Comment: 52 pages, invited review, v2 & v3 updates to include significant work
since first posted and corrections from comments received; some non-trivial
typos fixed. Comments now limited to changes that can be applied at proof
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