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|Title: ||Numerical treatment of the Liouville-von Neumann equation for quantum spin dynamics|
|Authors: ||Mazzi, Giacomo|
|Supervisor(s): ||Leimkuhler, Ben J.|
|Issue Date: ||2010|
|Publisher: ||The University of Edinburgh|
|Abstract: ||This thesis is concerned with the design of numerical methods for quantum simulation
and the development of improved models for quantum relaxation. Analysis is
presented for the treatment of quantum systems using the density matrix formalism.
This approach has been developed from the early days of quantum mechanics as a
tool to describe from a statistical point of view a large number of identical quantum
Traditional methods are well established and reliable, but they perform poorly for
practical simulation as the system size is scaled up. Ad hoc schemes for nuclear spin
dynamics appearing in the literature can be shown to fail in certain situations. The
challenge is therefore to identify efficient reduction methods for the quantum system
which are also based on a rigorous foundation. The method presented in the thesis, for
the time–independent Hamiltonian case, combines a quantum density matrix formalism
with a procedure based on Chebyshev polynomials; application of the method to
Nuclear Magnetic Resonance (NMR) spectroscopy is considered, and it is shown that
the new technique outperforms existing alternatives in term of computational costs.
The case of a time–dependent Hamiltonian in NMR simulation is studied as well
and some splitting methods are presented. To the author’s knowledge this is the first
time such methods have been applied within the NMR framework, and the numerical
results show a better error–to–cost rate than traditional methods.
In a separate strand of research, formulations for open quantum systems are studied
and new dynamical systems approaches are considered for this problem.
This thesis work is mainly focused on nuclear spin dynamics. Nuclear spin dynamics
constitutes the basis for NMR, which is a very powerful spectroscopy technique that
exploits the interaction between nuclear spins and magnetic fields. The same technique
is used to reveal the presence of hydrogen atoms in the blood for Magnetic Resonance
Imaging (MRI). Within this framework the role of simulations is extremely important,
as it provides a benchmark for studies of new materials, and the development of new
magnetic fields. The main computational issue is that with current software for NMR
simulation it is extremely expensive to deal with systems made of more than few (7–10)
spins. There is therefore a strong need to develop new algorithms capable of simulating
In recent years NMR simulations have been found to be one of the most favorable
candidates for quantum computing. There are two reasons for this: nuclear quantum
states maintain extremely long coherences, and it is possible to attain a very strong
control on the quantum state via the application of sequences of pulses. In order to
develop a proper quantum computer it is fundamental to understand how the entangled
states lose coherence and relax back to equilibrium by means of external interactions.
This process is described as relaxation in an open quantum system. The theory for such systems has been available for 50 years but there are still substantial limitations
in the two main approaches. There are also relatively few numerical approaches for
the simulation of such systems, for this reason it is important to develop numerical
alternatives for the description of open quantum systems.
The thesis is organized as follow: the first two chapters provide background material
to familiarize the reader with fundamental concepts of both quantum mechanics and
nuclear spin dynamics; in this part of the thesis no new results are presented.
The first chapter introduces the concept of quantum systems and the mathematical
environment with which we describe those systems. We also present the main equations
we need to solve to determine the dynamics of a quantum system in a statistical
In the second chapter we introduce the nuclear spin system, that is the physical
system that has been the main reference frame in this work, for both tests and practical
applications of the new algorithms. We describe how nuclear spin systems are at the
basis of very important applications like NMR spectroscopy and MRI. We present in
some detail the physical features of the NMR technique and the equations we need
to solve to describe the dynamics of a spin system; we also focus on the relevance of
numerical simulations for these systems, and consequently which must be the interest
in developing new algorithms, and the major obstacles which must be overcome.
In the third chapter we investigate the numerical challenges that arise in simulation
of quantum systems, we describe some of the methods that have been developed in the
literature, focusing on the performances and the computational costs of them, setting
the new developments of this thesis in the proper research frame. We discuss one of
the major issues: the evaluation of the matrix exponential.
We also present the analysis we have done of a recent method called Zero Track
Elimination (ZTE) that has been developed specifically for NMR simulations. This
analysis shows the limitations of this method but also gives a mathematical explanation
of why–and in which cases–it works.
In the fourth chapter we present the main result of the thesis, the development of
a new method that directly evaluates the expectation values for a quantum simulation
via a different application of the well known Chebyshev expansion. We have proved
that this new method can provide an excellent boost in terms of performance, with
computational costs that can be reduced by a factor ten in common cases. (The results
of this chapter and the new method have been presented in international conferences
and recently they have been submitted for publication).
We also present some attempts we have made in the application of splitting methods
for the evolution of the system in a time dependent environment. To our knowledge this
is the first time splitting methods have been used for NMR simulations. The results of this approach are as follows: for a particular splitting technique combined with a
Lanczos iteration method it is possible to speed up the calculation by a third if compared
with a Lanczos type method whilst keeping the error below a critical threshold. This
last approach is still a work in progress especially in terms of developing clever ways to
split the Hamiltonian.
The last chapter of this thesis deals with simulation of quantum systems interacting
with an external environment. After presenting the main theoretical approaches for the
description of such systems we then survey several the techniques that are currently used
for the numerical implementation of such theories. As a work in progress we present a
considerably different new approach we have been developing aiming to overcome some
of the issues that arise when treating this kind of system within usual frameworks. This
is somewhat speculative work that gives rise to some new directions in the development
of a numerical description for open quantum systems. We also present some numerical
results. (The main core of this chapter has been presented in international conferences).|
|Keywords: ||quantum dynamics|
Nuclear Magnetic Resonance
|Appears in Collections:||Mathematics thesis and dissertation collection|
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