QUANTUM COMPUTATION

Quantum Computation: Theory and Implementation






Quantum computation is a new field bridging many disciplines, including theoretical physics, 
functional analysis and group theory, electrical engineering, algorithmic computer science, and 
quantum chemistry. The goal of this thesis is to explore the fundamental knowledge base of 
quantum computation, to demonstrate how to extract the necessary design criteria for a 
functioning tabletop quantum computer, and to report on the construction of a portable NMR 
spectrometer which can characterize materials and hopefully someday perform quantum 
computation. This thesis concentrates on the digital and system-level issues. Preliminary data 
and relevant code, schematics, and technology are discussed. 

Why build a quantum computer? Because it’s not there, for one thing, and the theory of quantum 
computation, which far outstrips the degree of implementation as of today, suggests that a 
quantum computer would be a incredible machine to have around. It could factor numbers 
exponentially faster than any known algorithm, and it could extract information from unordered 
databases in square-root the number of instructions required of a classical computer. As 
computation is a nebulous field, the power of quantum computing to solve problems of 
traditional complexity theory is essentially unknown. A quantum computer would also have 
profound applications for pure physics. By their very nature, quantum computers would take 
exponentially less space and time than classical computers to simulate real quantum systems, and 
proposals have been made for efficiently simulating many-body systems using a quantum 
computer. A more exotic application is the creation of new states which do not appear in nature: 
for example, a highly entangled state experimentally unrealizable through other means, the 3-
qubit Greenberger-Horne-Zeilinger state  
000 + 111 , 
has been prepared on an NMR quantum computer with only 3 operations, a Y(π/2) pulse 
followed by 2 CNOTs, applied to the spins of trichloroethylene (Laf97). Computational 
approaches to NMR are interesting for their own sake: Warren has coupled spins as far as 
millimeters apart, taking advantage of the properties of a phenomenon known as zero-quantum 
coherence to make very detailed MRI images (War98). Performing fixed-point analyses on 
iterated schemes for population inversion can result in very broadband spin-flipping operators – 
in such a scheme, the Liouville space of operators is contracted towards a fixed point, so that 
operators that are imperfect (due to irregularities in the magnetic field, end effects of the 
stimulating coil, impurities in the sample, etc.) nevertheless result in near-perfect operations 
(Tyc85) (Fre79). These simple uses of nonlinear dynamics and the properties of solution have 
shed light on certain computational aspects of nuclear magnetic resonance. 
But performing a quantum computation as simple as factoring 15 will involve the creation and 
processing of dozens of coherences over extended periods of time. This presents an incredible 
engineering challenge, and may open up new understandings on both computation and quantum 
mechanics. 
Undoubtedly we will someday have quantum computers, if only to maintain progress in 
computation. It has been widely speculated that within a few decades, the incredible growth 
which characterizes today’s silicon age will come to a gradual halt, simply because there will be 
no more room left at the bottom: memory bits and circuit traces will essentially be single atoms 
on chips, the cost of a fabrication plant will be “the GNP of the planet,” and there will probably 
be lots of interesting problems to solve which require still more computing power. This power 
will have to come from elsewhere – that is, by stepping outside of the classical computation 
paradigm. 
  8 
Other fields related to quantum computing, which might be said to fall into 

‘quantum control,’ include quantum teleportation, quantum manipulation, and quantum 
cryptography. Quantum teleportation is an appealing way to send quantum states over long 
distances, something no amount of classical information can ever do (probably), and has been 
realized for distances exceeding 100 kilometers (Bra96). Quantum means of cryptography are 
unbreakable according to the laws of physics, and using quantum states to send messages 
guarantees that eavesdroppers can be detected (Lo99). 
  
Furthermore quantum computing joins two of the most abstruse and surprisingly nonintuitive 
subjects known to humans: computation and quantum mechanics. Both have the power to shock 
the human mind, the former with its strange powers and weaknesses, and the latter with its 
nonintuitive picture of nature. Perhaps the conjunction of these two areas will yield new insights 
into each. For the novice in this field, past overview papers on quantum computing include 
(Bar96) (Llo95) (Ste97). 
There are powerful alternatives to quantum computing which may have more immediate 
significance for the computing community. Incremental improvements in silicon fabrication 
(such as MEMS-derived insights, low-power and reversible design principles, copper wiring 
strategies, intergate trench isolation, and the use of novel dielectrics), reconfigurable computing 
and field programmable gate arrays, single-electron transistors, superconducting rapid singleflux quantum logic, stochastic and defect-tolerant computers, ballistically-transporting 
conductors, printable and wearable computers, dynamical systems capable of computation, 
memory devices consisting of quantum dots and wells, and many other technologies are vying 
for the forefront. Any of them, if lucky and well-engineered, could jump to the forefront and 
perhaps render the remainder obsolete, much like the alternatives to silicon in the heyday of 
Bardeen and Intel. It is more likely that these diverse technologies will find a variety of niches in 
the world, given the amazing number of problems that people want to solve nowadays. 
This paper begins with a comprehensive critique of much of the knowledge in the quantum 
computation community, since that is perhaps fundamental to understanding why and how 
quantum computers should be built. Part I.1. covers elementary quantum mechanics, information 
theory, and computation theory, which is a prerequisite for what follows. Much of this was 
written over the past year to teach the author how all of these pictures can contribute to a 
coherent picture of the physics of computation. Part I.1. can also be thought of an ensemble of 
collected material, in preparation for a book which delves into the theories and experiments 
behind the physics of computation; it is the opinion of the author that the definitive book on the 
physics of computation has not yet been written, and that the world needs one. It is noted that 
very little synthesis has been performed on the information here presented; most of the 
paragraphs are summaries of various different sources, assembled for easy reference and as an 
aid to thinking. Some of these notes were presented at (Bab98).