%Nr: DS-1999-04
%Title: Quantum Algorithms and Quantum Entanglement
%Author: Barbara M. Terhal
Quantum Information Processing is an interdisciplinary field of research at the
crossroads of physics and computer science. The computer science aspect is
expressed by the fact that the main application of the research lies in
computers, information processing and communication. The physics aspect relies
on the use of quantum mechanics and quantum mechanical phenomena to enhance
this computational and information processing power.
The physical basis of the successful digital technology that has emerged since
the 1960s has undergone few changes. The developments in semiconductor
technology have enabled us to build smaller and faster CPU and memory chips.
This trend is accompanied by similar progress in the technology of magnetic
storage and retrieval. When extrapolating the current trend of miniaturization
of silicon chips to the future, we may expect to reach an atomic level around
the year 2020. At this scale quantum mechanics is an indispensable tool in
describing the phenomena of nature. Scaling of the current computer
architecture to this regime will be severely limited by the problem of heat
generation. With these considerations in mind, the notion of intrinsically
reversible quantum computing arises naturally. Essential properties of quantum
mechanical systems such as the superposition principle, interference and
quantum entanglement lie at the basis at this proposal for a new technology.
Over the past 15 years researchers in this field have found that the use of
quantum bits, as opposed to 'classical' bits, can present many advantages and
that quantum information is unlike classical information in many respects. In
terms of computational power the most striking achievement is the factoring
quantum algorithm developed by Peter Shor (see Section 2.1.1). A quantum
computer can factor a number N in time polynomially related to the size of the
input, i.e. O(log N ), whereas the best classical algorithm takes exponential
time. Another example is the quantum key distribution protocol invented by
Charlies Bennett and Gilles Brassard in 1982. The protocol enables two parties
to construct a shared secret random (classical) key by transmission of quantum
states and classical states over a channel on which potential eavesdroppers may
be listening.
The physical realizability of a quantum information processor is an issue which
presents a major challenge to physicists. On the theoretical side, the field
has seen the development of quantum error correcting codes and faulttolerant
computation which are sine qua nons for the ultimate feasibility of a quantum
computer. On the experimental side, we are at the stage of creating and
manipulating 1-4 qubits in a variety of physical systems such a liquid state
NMR, ions in a trap, cavity QED and solid state quantum dots.
Thesis
------
The two main topics of this thesis are Quantum Algorithms and Quantum
Entanglement.
What problems does a quantum computer solve faster than a classical machine? In
Chapter 2, we consider three types of problems. The first problem is the
determination of the mean value of a given function over a sample set, where
the function is presented as an oracle. We show in Section 2.2.1 that a
square-root speed up, compared to a classical setting, can be obtained by
applying the method of generalized counting which is based from Grover's
quantum search algorithm. The second set of problems (Section 2.3) considers
information retrieval from a database. We show that classical information
theoretic bounds are invalid in a quantum setting and we show how to apply a
single query quantum algorithm to the problem of coin weighing. Our last
problem, iterated function application, is an oracle problem for which we prove
that a quantum computer does not provide a speed-up (Section 2.4).
In Chapters 3 and 4 we investigate how to perform physical simulations of
quantum systems on a quantum computer. In Chapter 3, we consider how to (space)
efficiently implement an arbitary superoperator on a qubit system. We prove
that, contrary to a prior conjecture, there are superoperators which need at
least an additional two qubit system to be implemented. Our proof uses the
computer algebra technique of Gr"obner bases to determine that a system of
nonlinear equations does not have a solution.
In Chapter 4, we study the problem of computing correlations functions and
preparing a thermal equilibrium state of a physical system on a quantum
computer. In Section 4.1 we determine a quantum algorithm which, given a local
Hamiltonian and a temperature, provably converges to the thermal equilibrium
state of the corresponding Hamiltonian. The algorithm constitutes the first
example of a quantum algorithm based on a quantum Markov chain. In Section
4.2.3 we prove several general properties of quantum Markov chains which lie at
the basis of a theory of quantum Markov chains for quantum computational
purposes. In Sections 4.2.7 and 4.2.8 we study the properties of the quantum
algorithm by a numerical simulation.
In Section 4.3 an alternative quantum algorithm for thermal equilibration is
presented which uses Kitaev's eigenvalue-estimation routine. In Section 4.4 we
show that given the ability to create an approximation of the thermal
equilibrium state, the estimation of time-dependent correlation functions can
be efficiently (polynomial time) carried out on a quantum computer.
In Chapter 5 we report on several results in studying fundamental properties of
quantum mechanical systems. One of the crucial properties of quantum systems is
their capacity to be entangled. In Section 5.2 we summarize several features of
the emerging theory of entanglement, in particular the relation with positive
linear maps.
Quantum entanglement can be a source of non-locality as expressed by (a
violation of) a Bell inequality. The violation of a Bell inequality indicates
that the statistics of outcomes of local measurements on, for example, a
maximally entangled state, cannot be described by a local hidden variable
theory. The question of interest is whether all entangled states violate a Bell
inequality, and are therefore 'non-local' in this way. In Section 5.3 we show
that the problem of deciding whether a state violates a Bell inequality and the
problem of deciding whether a state is entangled are very similar problems in
convex geometry. The problems are however not identical. We prove under what
restrictions of the local hidden variable theory the two criteria do coincide.
Quantum entanglement is a resource in quantum information theory. Sharing of
quantum entanglement enables parties to perform (quantum) information
processing tasks (quantum teleportation, transmission of classical data etc.)
which are impossible or can only be carried out with less efficieny without the
use of entanglement. It is therefore of great importance to develop a
classification of different types of quantum entanglement in terms of its
ability to enhance quantum information processing. This entanglement
classification is derived from considering the properties of the set of quantum
operations that do not enhance entanglement; this is the set of local quantum
operations and classical communication.
In Section 5.4 a class of entangled states is constructed which are not
convertible (by local quantum operations and classical communication) to a set
of maximally entangled states. The entanglement of these states is called
bound, since the entanglement of these states is locked away and cannot be
fruitfully used for information processing. Few examples of this type of
entanglement were previously known; we develop a construction method that gives
rise to many examples of bound entanglement. The method introduces new concepts
such as unextendible and uncompletable product bases, for which we find a
useful representation by means of a (orthogonality) graph. The interest in
these basis is twofold. On the one hand they give rise to the aforementioned
bound entangled states (Section 5.4.3). On the other hand we prove that the
states in an uncompletable product basis have the feature of not being locally
distinguishable, even though these are mutually orthogonal product states
(Section 5.4.5). In this way, they form a new illustration of the phenomenon of
nonlocality without entanglement.
In Section 5.5 we show how an unextendible product basis can be used to
construct indecomposable positive linear maps. The problem of finding
(indecomposable) positive linear maps has been notoriously hard; the method
developed in this thesis offers a construction which generates a whole new
family of positive linear maps and is amenable to further generalizations.