25 research outputs found
Quantum walks: a comprehensive review
Quantum walks, the quantum mechanical counterpart of classical random walks,
is an advanced tool for building quantum algorithms that has been recently
shown to constitute a universal model of quantum computation. Quantum walks is
now a solid field of research of quantum computation full of exciting open
problems for physicists, computer scientists, mathematicians and engineers.
In this paper we review theoretical advances on the foundations of both
discrete- and continuous-time quantum walks, together with the role that
randomness plays in quantum walks, the connections between the mathematical
models of coined discrete quantum walks and continuous quantum walks, the
quantumness of quantum walks, a summary of papers published on discrete quantum
walks and entanglement as well as a succinct review of experimental proposals
and realizations of discrete-time quantum walks. Furthermore, we have reviewed
several algorithms based on both discrete- and continuous-time quantum walks as
well as a most important result: the computational universality of both
continuous- and discrete- time quantum walks.Comment: Paper accepted for publication in Quantum Information Processing
Journa
Group transference techniques for the estimation of the decoherence times and capacities of quantum Markov semigroups
Capacities of quantum channels and decoherence times both quantify the extent
to which quantum information can withstand degradation by interactions with its
environment. However, calculating capacities directly is known to be
intractable in general. Much recent work has focused on upper bounding certain
capacities in terms of more tractable quantities such as specific norms from
operator theory. In the meantime, there has also been substantial recent
progress on estimating decoherence times with techniques from analysis and
geometry, even though many hard questions remain open. In this article, we
introduce a class of continuous-time quantum channels that we called
transferred channels, which are built through representation theory from a
classical Markov kernel defined on a compact group. We study two subclasses of
such kernels: H\"ormander systems on compact Lie-groups and Markov chains on
finite groups. Examples of transferred channels include the depolarizing
channel, the dephasing channel, and collective decoherence channels acting on
qubits. Some of the estimates presented are new, such as those for channels
that randomly swap subsystems. We then extend tools developed in earlier work
by Gao, Junge and LaRacuente to transfer estimates of the classical Markov
kernel to the transferred channels and study in this way different
non-commutative functional inequalities. The main contribution of this article
is the application of this transference principle to the estimation of various
capacities as well as estimation of entanglement breaking times, defined as the
first time for which the channel becomes entanglement breaking. Moreover, our
estimates hold for non-ergodic channels such as the collective decoherence
channels, an important scenario that has been overlooked so far because of a
lack of techniques.Comment: 35 pages, 2 figures. Close to published versio
Evolution of entanglement structure in open quantum systems
The thesis presents research related to the dynamics of quantum systems, both isolated and in the presence of interactions with their environment. Generally, I employ matrix product state (MPS) techniques to explore quantum dynamics in open and closed systems. In the first part I present a study of quantum chaos and how mapping to a MPS variational manifold allow the use of techniques developed in the study of classical many-body systems. Using code developed for this project the Lyapunov spectrum is extracted to give an alternative perspective on eigenstate thermalization, pre-thermalization and integrability. In the second part, I present a novel combination of MPS methods with a Langevin description of the open system. I use this to show how coupling to the environment restricts the growth of entanglement. The consequences of this are relevant for simulations of open quantum systems and their use in in quantum technologies. Finally I discuss applications of these ideas to quantum search. I consider adiabatic and quantum walk algorithms for optimal scaling quantum search algorithms, and hybridizations between the two. The robustness of the different underlying physical mechanisms is investigated in a simple infinite-temperature model, and in a low-temperature limit using the MPS Langevin equation
Quantum information processing in continuous time
Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Physics, 2004.Includes bibliographical references (p. 127-138) and index.This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.Quantum mechanical computers can solve certain problems asymptotically faster than any classical computing device. Several fast quantum algorithms are known, but the nature of quantum speedup is not well understood, and inventing new quantum algorithms seems to be difficult. In this thesis, we explore two approaches to designing quantum algorithms based on continuous-time Hamiltonian dynamics. In quantum computation by adiabatic evolution, the computer is prepared in the known ground state of a simple Hamiltonian, which is slowly modified so that its ground state encodes the solution to a problem. We argue that this approach should be inherently robust against low-temperature thermal noise and certain control errors, and we support this claim using simulations. We then show that any adiabatic algorithm can be implemented in a different way, using only a sequence of measurements of the Hamiltonian. We illustrate how this approach can achieve quadratic speedup for the unstructured search problem. We also demonstrate two examples of quantum speedup by quantum walk, a quantum mechanical analog of random walk. First, we consider the problem of searching a region of space for a marked item. Whereas a classical algorithm for this problem requires time proportional to the number of items regardless of the geometry, we show that a simple quantum walk algorithm can find the marked item quadratically faster for a lattice of dimension greater than four, and almost quadratically faster for a four-dimensional lattice. We also show that by endowing the walk with spin degrees of freedom, the critical dimension can be lowered to two. Second, we construct an oracular problem that a quantum walk can solve exponentially faster than any classical algorithm.(cont.) This constitutes the only known example of exponential quantum speedup not based on the quantum Fourier transform. Finally, we consider bipartite Hamiltonians as a model of quantum channels and study their ability to process information given perfect local control. We show that any interaction can simulate any other at a nonzero rate, and that tensor product Hamiltonians can simulate each other reversibly. We also calculate the optimal asymptotic rate at which certain Hamiltonians can generate entanglement.by Andrew MacGregor Childs.Ph.D
Simulation of quantum walks and fast mixing with classical processes
International audienc
Quantum walk and Wigner function on a lattice
La informació quàntica és un camp relativament jove de la Física, que té com
a objectiu explorar les lleis de la mecànica quàntica per a la transmissió i el
processament de la informació. Com a exemple d’aplicacions es poden esmentar
les comunicacions segures, basades en la distribució de clau quàntica, i algoritmes
quàntics que superen als seus homàlegs clàssics per a un determinat nombre
de problemes. A més, les eines desenvolupades en el context de la informació
quàntica han demostrat ser de gran utilitat per aprofundir en la comprensió dels
sistemes quàntics, per exemple, en el context dels problemes de molts cossos
quàntics.
Una de les principals aplicacions de la potència de la mecànica quàntica en
tasques computacionals ́és la manipulació de sistemes quàntics al laboratori per
tal de realitzar simulacions quàntiques, i els diferents estudis experimentals s’estan
realitzant en l’actualitat cap aquest objectiu. Especialment prometedores són les
primeres simulacions quàntiques de sistemes atòmics ultrafreds atrapats en xarxes
optiques, on els resultats superen els càlculs clàssics.
Aquesta tesi aplica eines d’informació quàntica a la descripció i l’estudi de
diversos sistemes quàntics i a processos que succeeixen en un espai discret, és a
dir, en una xarxa. Fins i tot una sola partícula quàntica amb spin 1/2 pot donar
lloc a fenomens que difereixen de forma radical de qualsevol analogia clàssica. En
alguns casos, la nostra comprensió dels processos físics és més intuïtiva per al cas
continu, i per tant, el nostre estudi es connecta fins al límit continu adequat.
La tesi s’estructura en dues parts. La primera d’elles s’emmarca en l’estudi
i comprensió d’un algoritme quàntic en particular, el passeig quàntic. Per tal
d’explotar el passeig quàntic i aplicar-lo a la construcció d’algoritmes quàntics,
és important entendre i controlar el seu comportament tant com siga possible.
Una de les característiques analitzades en aquesta tesi és el passeig quàntic
discret en N dimensions des de la perspectiva de les relacions de dispersió. Fent ús de condicions inicials esteses en l’espai de posicions, s’obté una equació d’ona en el límit continu. Aquesta equació ens permet d’entendre algunes propietats
conegudes i dissenyar interessants comportaments. Apliquem l’estudi al passeig
quàntic en dos i tres dimensions per a la moneda de Grover, on la relació dedispersió presenta punts i interseccions particulars on la dinàmica és especialment
diferent.
D’altra banda, s’analitza el comportament del passeig quàntic com un procés
Markovià. Amb aquest objectiu, s’investiga l’evolució temporal de la matiu densitat reduïda per un passeig quàntic de temps discret en una xarxa unidimensional.
S’analitza la dinàmica de la matriu densitat reduïda en el cas estàndard, sense decoherència, i quan el sistema està exposat als efectes de decoherència. Analitzem
el comportament Markovià de l’evolució en el sentit definit en [1] examinant la
distància de traça per a possibles parells de estats inicials com una funció del
temps. Arribem a la conclusió que l’evolució de la matriu densitat reduïda en
el cas lliure és no Markoviana i, quan el nivell de soroll augmenta, la dinàmica
s’aproxima a un procés Markovià.
La segona part d’aquesta tesi proposa una generalització de la coneguda funció
de Wigner per a una partícula que es mou en una xarxa infinita en una dimensió.
L’estudi de la mecànica quàntica en l’espai de fases a través de les distribucions
de quasi-probabilitat s’aplica en molts camps de la física i la funció de Wigner és
probablement la més utilitzada.
S’estudia la funció de Wigner per a un sistema quàntic en un espai d’Hilbert
discret, de dimensió infinita, tal com una partícula sense spin en moviment en una
xarxa infinita unidimensional. Es discuteixen les peculiaritats d’aquest escenari i la construcció de l’espai fàsic associat, i es proposa una definició significativa de la funció de Wigner en aquest cas, a més es caracteritza el conjunt d’estats purs per als quals la funció de Wigner és no negativa. També ampliem la definició proposada per incloure un grau intern de llibertat, com ara l’spin.
La dinàmica d’una partícula en una xarxa amb, i sense spin, en diferents casos, també s’analitza en termes de la funció de Wigner corresponent. Mostrem solucions explícites en el cas d’evolució hamiltoniana sota un potencial depenent de la posició que pot incloure un acoblament d’spin, i per a l’evolució governada per una equació mestra sota alguns simples models de decoherència.
Proposem una mesura de la no-classicitat dels estats en un sistema amb un
espai d’Hilbert discret i infinit que és consistent amb el límit continu. I, en
darrer lloc, discutim la possibilitat d’ampliar el concepte de negativitat de la
funció de Wigner al cas en el qual s’inclou el grau de llibertad d’spin.Quantum information is a relatively young field of Physics, that aims to exploit
the laws of quantum mechanics for the transmission and processing of information. As illustrative applications one can mention secure communications, based
on quantum key distribution, and quantum algorithms that outperform their classical counterparts for a number of problems. Furthermore, the tools developed
in the context of Quantum Information have proven extremely useful to deepen
the understanding of quantum systems, for instance in the context of quantum
many-body problems.
One of the main applications of the power of quantum mechanics to computational tasks is the manipulation of quantum systems in the lab in order to perform
quantum simulations, and different experimental approaches are currently being
pursued towards this goal. Especially promising are ultracold atomic systems
trapped in optical lattices, where the first quantum simulations that outperform
the feasible classical calculations have already been realized.
This thesis applies quantum information tools to the description and the study
of several quantum systems and processes that happen on a discrete space, i.e.
on a lattice. Even a single quantum particle with spin 1/2 hopping on a lattice
can give rise to phenomena that dramatically differ from any classical analogy.
In some cases, our understanding of the physical processes is more intuitive for
the continuous case, and hence we connect our study to the proper continuum
limit.
The thesis is structured in two parts. The first one is framed within the study
and understanding of a particular quantum algorithm, namely the quantum walk.
In order to exploit the quantum walk and apply it to the construction of quantum
algorithms, it is important to understand and control its behavior as much as
possible.
One of the features analyzed in this thesis is the discrete time quantum walk
in N dimensions from the perspective of its dispersion relations. Making use of
the spatially extended initial conditions, a wave equation in the continuum limit
is obtained. This equation allows us to understand some known properties, and
to design interesting behaviors. We apply the study to the two and three dimensional Grover quantum walk, where the dispersion relation presents particular
points and intersections where the dynamics is specially distinct.
On the other hand, we analyze the behavior of the quantum walk as a Markovian process. With this aim, we investigate the time evolution of the chirality
reduced density matrix for a discrete time quantum walk on a one-dimensional
lattice. We analyze the dynamics of the reduced density matrix in the standard
case, without decoherence, and when the system is exposed to the effects of decoherence. We analyze the Markovian behavior in the sense defined in [1] examining
the trace distance for possible pairs of initial states as a function of time which
gives us the distinguishability of two states and it is related with the Markovian
behavior of the system. We conclude that the evolution of the reduced density
matrix in the free case is non-Markovian and, as the level of noise increases, the
dynamics approaches a Markovian process.
The second part of this thesis proposes a generalization of the known Wigner
function for a particle moving on an infinite lattice in one dimension. The study
of the quantum mechanics in phase space through quasi-probability distributions
is applied in many fields of physics and the Wigner function is probably the most
commonly used one.
We study the Wigner function for a quantum system with a discrete, infinite
dimensional Hilbert space, such as a spinless particle moving on a one dimensional
infinite lattice. We discuss the peculiarities of this scenario and of the associated
phase space construction, propose a meaningful definition of the Wigner function
in this case, and characterize the set of pure states for which it is non-negative.
We also extended the proposed definition to include an internal degree of freedom,
such as the spin.
The dynamics of a particle on a lattice with and without spin in different cases
are also analyzed in terms of the corresponding Wigner function. We show explicit solutions for the case of Hamiltonian evolution under a position dependent
potential that may include a spin coupling, and for the evolution governed by a
master equation under some simple models of decoherence.
We propose a measure of non-classicality for states in the system with a discrete
infinite dimensional Hilbert space which is consistent with the continuum limit.
And we discuss the possibility of extending a negativity concept for the Wigner
function in the case in which the spin degree of freedom is included
Functional inequalities in quantum information theory
Functional inequalities constitute a very powerful toolkit in studying various problems arising in classical information theory, statistics and many-body systems. Extensions of these tools to the noncommutative setting have been introduced in the beginning of the 90's in order to study the asymptotic properties of certain quantum Markovian evolutions. In this thesis, we study various extensions and problems arising from the specific noncommutative nature of such processes.
The first logarithmic Sobolev inequality to be proved, due to Gross, was for the Ornstein Uhlenbeck semigroup, that is the Brownian motion with friction on the real line. The generalization of this result to the quantum Ornstein Uhlenbeck semigroup was found very recently by Carlen and Maas, and de Palma and Huber by means of different techniques. The latter proof consists of a quantum generalization of the so-called entropy power inequality. Here, we consider another possible version of the entropy power inequality and use it to derive asymptotic properties of the frictionless quantum Brownian motion.
The proof of Carlen and Maas discussed in the previous paragraph relies on their new quantum extension of the classical notion of displacement convexity. This is classically known to imply most of the usual functional inequalities such as the modified logarithmic Sobolev inequality and Poincaré's inequality. Here, we further study the framework introduced by Carlen and Maas. In particular, we show how displacement convexity implies quantum functional and transportation cost inequalities. The latter are then used to derive certain concentration inequalities of quantum states in the spirit of Bobkov and Goetze. These concentration inequalities are used in order to derive finite sample size bounds for the task of quantum parameter estimation.
The main advantage of classical logarithmic Sobolev inequalities over other methods resides in their tensorization property: the strong log-Sobolev constant of the product of independent Markovian evolutions is equal to the maximum over the set of strong log-Sobolev constants of the individual evolutions. However, this property is strongly believed to fail in the non-commutative case, due to the non-multiplicativity of noncommutative Lp to Lq norms. In this thesis, we show tensorization of the logarithmic Sobolev constants for the simplest quantum Markov semigroup, namely the generalized depolarizing semigroup. Moreover, we consider a new general method to overcome the issue of tensorization for general primitive quantum Markov semigroups by looking at their contractivity properties under the completely bounded Lp to Lq norms. This method was first investigated in the restricted case of unital semigroups by Beigi and King.
Noncommutative functional inequalities considered in the present literature only deal with primitive quantum Markovian semigroups which model memoryless irreversible dynamics converging to a specific faithful state. However, quantum Markov semigroups can in general display a much richer behavior referred to as decoherence: In particular, under some mild conditions, any such semigroup is known to converge to an algebra of observables which effectively evolve unitarily. Here, we introduce the concept of a decoherence-free logarithmic Sobolev inequality, and the related notion of hypercontractivity of the associated evolution, to study the decoherence rate of non-primitive quantum Markov semigroups. Moreover, we utilize the transference method recently introduced by Gao, Junge and LaRacuente, in order to find decoherence times associated to a class of decoherent Markovian evolutions of great importance in the field of quantum error protection, namely collective decoherence semigroups.
Finally, we develop the notion of quantum reverse hypercontractivity, first introduced by Cubitt, Kastoryano, Montanaro and Temme in the unital case, and apply it in conjunction with the tensorization of the modified logarithmic Sobolev inequality for the generalized depolarizing semigroup in order to find strong converse rates in quantum hypothesis testing and for the classical capacity of classical-quantum channels. Moreover, the transference method also allows us to find strong converse bounds on the various capacities of quantum Markovian evolutions