# Quantum Brain Networks: a Perspective

E. R. Miranda  Interdisciplinary Centre for Computer Music Research (ICCMR),
University of Plymouth, United Kingdom  S. Venkatesh  Interdisciplinary Centre
for Computer Music Research (ICCMR), University of Plymouth, United Kingdom
C. Hernani-Morales  IDAL, Electronic Engineering Department, University of
Valencia, Avgda. Universitat s/n, 46100 Burjassot, Valencia, Spain  L. Lamata
Departamento de Física Atómica, Molecular y Nuclear, Universidad de Sevilla,
41080 Sevilla, Spain  J. D. Martín-Guerrero  IDAL, Electronic Engineering
Department, University of Valencia, Avgda. Universitat s/n, 46100 Burjassot,
Valencia, Spain  E. Solano<EMAIL_ADDRESS>Department of Physical
Chemistry, University of the Basque Country UPV/EHU, Apartado 644, 48080
Bilbao, Spain  IKERBASQUE, Basque Foundation for Science, Plaza Euskadi 5,
48009 Bilbao, Spain  International Center of Quantum Artificial Intelligence
for Science and Technology (QuArtist)
and Physics Department, Shanghai University, 200444 Shanghai, China  Kipu
Quantum, Kurwenalstrasse 1, 80804 Munich, Germany

###### Abstract

We propose Quantum Brain Networks (QBraiNs) as a new interdisciplinary field
integrating knowledge and methods from neurotechnology, artificial
intelligence, and quantum computing. The objective is to develop an enhanced
connectivity between the human brain and quantum computers for a variety of
disruptive applications. We foresee the emergence of hybrid classical-quantum
networks of wetware and hardware nodes, mediated by machine learning
techniques and brain-machine interfaces. QBraiNs will harness and transform in
unprecedented ways arts, science, technologies, and entrepreneurship, in
particular activities related to medicine, Internet of humans, intelligent
devices, sensorial experience, gaming, Internet of things, crypto trading, and
business.

##  I Introduction

Since the origin of quantum physics, the role of human observers in the
disturbing collapse of the wave function occupies a central role. Challenges
to our classical intuition led to a series of proposed paradoxes, mostly due
to the extrapolation of microscopic quantum phenomena to our distinct
macroscopic human experience. Counterintuitive Gedankenexperiments, as the
celebrated cases of the Schrödinger’s Cat Schrodinger1935  and Wigner’s Friend
Wigner1967 , illustrate the historical difficulties in assuming the
consequences of the quantum theory FrauchigerRenner . Moreover, risky
conjectures have also been raised concerning possible quantum phenomena in
brain processes, in particular for making sense of human free will, mind
models, decision-making, and consciousness Penrose1994 ; Fisher2015 ;
Jedlicka2017 . In this sense, it would be scientifically and technologically
groundbreaking, from fundamentals of hardware and wetware science to cutting-
edge applications, the development of a closer connection between human brains
and quantum computers (QCs). However, our understanding of the brain, mind,
and whatever we may mean by consciousness, is still rudimentary. This makes it
difficult to interfacing the brain directly with external quantum devices or
quantum processors Feynman1982 ; Nielsen2010 . Nevertheless, artificial
intelligence (AI) may come at our rescue to achieve this otherwise impossible
task at this point of the 21st century.

In the last decades, we may find bottom-up approaches to consider the merge of
biological properties with quantum phenomena. In the case of quantum biology,
possible quantum features might explain efficiency of photosynthesis
QuantumBiology . Also, neuromorphic technologies are being studied to save
energy and enhance AI applications NeuromorphicComputing . More recently, bio-
inspired quantum artificial life has been proposed and implemented in a
quantum computer QAL , while neuromorphic quantum computing using quantum
memristors for quantum neural networks is getting first experimental tests
QuantumMemristors2016 ; AtomicQuantumMemristor2021 ;
PhotonicQuantumMemristor2021 .

Figure 1: Quantum Brain Networks

In this Perspective, we explore and argue in favor of the technical
feasibility of a novel interdisciplinary field we coined as Quantum Brain
Networks (QBraiNs), a top-down approach based on solid ground and innovative
merge of neurotechnology, artificial intelligence, and quantum computers.

Definition: a QBraiN is a multidirectional wired/wireless network of
local/distributed nodes with wetware/hardware classical/quantum processors.

We take it for granted that such a definition of a QBraiN will need suitable
adaptations when implemented in different technological platforms and applied
to different use cases. We also assume several parts of the same human brain
may be considered as QBraiN nodes. It is noteworthy to mention that every step
in the development of QBraiN models or building prototypes may settle decisive
milestones in challenging domains. Among them, we could mention philosophy of
science, fundamentals of physics, brain science, mind and consciousness
models, augmented classical-quantum sensorial experience, enhanced classical-
quantum computing, and innovative wetware-hardware applications (Fig. 1).

##  II Neurotechnology

Neurotechnology enables us to understand the human brain and mind, while
improving and repairing their accessible functions. For the sake of
simplicity, we associate here brain to hardware and mind to software. Along
these lines, brain scanning technologies, such as EEG 111EEG stands for
electroencephalogram. It refers to the brain’s spontaneous electrical activity
over a period of time, as recorded from multiple electrodes placed on the
scalp or inside the skull. and fMRI 222fMRI stands for functional Magnetic
Resonance Imaging. It measures brain activity by detecting changes associated
with blood flow., allow clinicians and researchers to visualise the brain and
monitor its activity. Increasingly sophisticated brain scanning technologies
are also enabling the development of innovative brain-machine interfaces (BMIs
333Also referred to as BCIs or Brain-Computer Interfaces.). These devices
endow humans with the ability to control machines by thinking, detectable as
brain activity, without the physical constraints of the body IntroBCI . The
ultimate goal of BMI research is to network brains and machines to directly
communicate with each another. Despite tremendous research progress and
encouraging demonstrations of possible applications, from typewriting texts
SpellingBCI  to making music EduardoBook , BMI systems are not widely
available for general use yet. In fact, currently, brain-machine communication
takes place mostly from humans to machines, leaving still open bidirectional
and more integrated networks for further developments. In this sense, it is
noteworthy to mention that machines can only communicate with us through our
senses. The development of robust connectivity of our nervous system with
devices outside our body is hampered by the lack of adequate transducers, and
the inability to electronically compute humongous amount of data in real time.
Emerging ultra-sensitive quantum brain sensors QuantumBrainSensor  and quantum
computers with processing speeds beyond state-of-the-art classical
supercomputers are bound to impact significantly on BMI research
EduardoMasterpiece . The success of quantum technologies will enable the
development of new hybrid classical-quantum networks of wetware and hardware,
enhancing intelligent creativity and communication.

##  III Artificial Intelligence

Since the origin of humankind, the use of tools is ubiquitous in all human
activities, be for survival, leisure, arts, or technologies. These tools have
reached sophisticated levels of abstraction with the emergence of intelligent
devices. They cannot only perform a given task but also communicate with other
machines in order to solve challenging problems iot . The concept of AI as
machines that can perform smart tasks appears naturally in this framework.
These machines and intelligent devices are the source of an ever-increasing
amount of information, that oftentimes is given by means of computable data.
This implies the use of data that can be analysed by machines in order to
extract information leading to knowledge discovery, that is Machine Learning
(ML) alpaydin_ML .

AI, and particularly ML, has grown steadily in the last decade. Its success is
mostly due to the fact that Deep Learning (DL) DL_nature , a ML approach based
on models that may be characterized by millions of parameters, has shown its
ability to solve cognitive tasks that were computationally intractable until
very recently. Among them, we could mention image segmentation and
classification, natural language processing, sentiment analysis, generative
art and music 444Here we refer to computer-generated pieces of art and music
of equivalent quality of human-made ones., and autonomous vehicles. One may
wonder how the solutions to these problems may be harnessed further by means
of networking with quantum computing technologies.

The bottleneck of AI progress has been frequently associated with hardware
development. The situation is different at present, where ML innovations
strongly depend on the involved creativity to tackle a given problem.
Furthermore, an important enhancement to AI may happen if we are able to
involve features of neurotechnology and, even more, if we may borrow key
concepts from quantum computing.

##  IV Quantum Computers

The experimental success of quantum physics, despite difficulties for an
intuitive interpretation, led to consider quantum computers as an innovative
paradigm for efficiently simulating and computing the distinct quantum aspects
of nature Feynman1982 . Two decades ago, quantum information led the
transition from single-particle tests towards multiqubit experiments, then
came the emergence of quantum computers in a variety of platforms, while last
years have brought ramping worldwide investments in quantum technologies.
Recently, the first claims have been made on quantum supremacy Martinis2019 ;
Pan2020 , that is quantum processors outperforming classical computers without
a request for useful applications yet. Nowadays, there is an agitated debate
on whether reaching quantum advantage, quantum processors outperforming
classical computers for industrial cases, might be possible in the near
future. If achieved, currently impossible solutions for applications such as
molecular design, financial modeling, optimization problems, and machine
learning may be at reach Nori2014 ; Biamonte2017 ; Aspuru-Guzik2020 .

Some of the quantum platforms that are currently being considered are
superconducting circuits, trapped ions, cold atoms, photonics, quantum dots,
spin impurities in solids, nitrogen-vacancy centers in diamond, as well as
nuclear magnetic resonance. Each of them has advantages and disadvantages
depending on their specific applications, and they have been evolving in
parallel at a different speed in the past three decades. In this sense, any
requirement for a possible control of a quantum computer with human brain
activity would end up in a specific choice of technology. Fortunately, there
are also intense activities at the educational, professional, and business
level, for developing web-based classical simulators of remote quantum
computers.

##  V Quantum Brain Networks

We consider the current availability and reachability of the theoretical
methods and technological requirements for designing, building, and using
QBraiNs for a variety of interdisciplinary applications. Accordingly, we
provide below three subsections containing our vision on QBraiNs, their unique
connection possibilities for a variety of situations, and potentially
disruptive use cases.

###  V.1 Vision on QBraiNs

We propose a list of logical statements with the aim of arguing in favor of
our final position on QBraiNs.

1) It is possible to read and harness brain data with BMIs

BMIs can be accessed commercially with an increasing number of channels, with
different kinds of technologies and levels of complexity. For instance, there
are commercially available portable EEG devices ranging from simple systems
for gaming to sophisticated designs for research and medical purposes. There
are invasive or non-invasive BMI systems, as well as research to develop
wireless devices WirelessBCI2021 .

2) Brain data can be processed by classical computers via ML

The communication between brains and machines, be digital or analog, is a
historic conundrum. Essentially, this problem is due to the lack of trustful
and minimally complex brain and mind models, as well as inadequate
transducers. AI brings pragmatic solutions to this mismatch, not by proposing
a structured language but through bypassing it. There is rudimentary BMI
technology available today to build simple brain-controlled systems using ML
running on classical computers, including laptops, tablets, and mobile phones
555An example: https://www.volvocars.com/uk/about/humanmade/discover-
volvo/music-of-the-mind.

3) Classical computers can control quantum computers

There is an increasing number of open-source software platforms provided by
quantum computing companies and startups to control quantum computer
simulators and real quantum computers via the World Wide Web (see, as
paradigmatic examples, Qiskit from IBM and Cirq from Google). The worldwide
success and spread of such web-based services has created a revolution in the
dissemination of possible quantum computing solutions for important
scientific, industrial, and societal applications.

4) Human brains can be connected to quantum computers 666Experimental work and
a proof-of-concept are in progress by the authors of this Perspective.

As a natural corollary to the previous items, we propose that human brains can
be directly connected to quantum computers with currently available concepts,
methods, and technologies. In this manner, QBraiNs are bound to become a
reality and a playground for novel arts, leisure, science, technology, and
entrepreneurial activities.

###  V.2 Connectivity of QBraiNs

Depending on specific goals, different possibilities to connect classical and
quantum nodes of a QBraiN may emerge.

1) One-to-one bidirectional brain-QC transducer

A basic QBraiN will involve a single human brain, possibly with several BMI
nodes, connected to a quantum computer. The suitable choice of technology may
be decided on the basis of latency, signal processing, and selected use case.

2) Multiple brain-QC nodes

Local and distributed QBraiN nodes will provide a wetware-hardware classical-
quantum processing network with unprecedented computational power and
capabilities.

3) Hybrid brain-SC-QC

A natural extension will be to add to the QBraiN one or several nodes
involving supercomputing and high-performance cloud-based facilities.

###  V.3 Applications of QBraiNs

Here we present a first tentative list of applications of QBraiNs, mostly
related to some key forefronts in science and technology.

1) Fundamental brain-QC tests

QBraiNs will allow interdisciplinary researchers and engineers to realize
fundamental tests related to quantum physics, quantum information, psychology
and behavioural models, quantum measurement and human perception in general
BCIbehavior2021 .

2) Brains controlling hybrid classical-quantum computers

Distributed nodes of wetware-hardware classical-quantum processors will allow
QBraiN developers and practitioners to explore enhanced computational power
and communication in Internet of humans, Internet of things, and the crypto
space.

3) Quantum-enhanced brains

QBraiNs will allow humans to enter into a new era of interconnectivity with
machines. We envision a wide class of possibilities for enhancing analysis and
creativity, as well as intuitive and rational control of intelligent
classical-quantum systems.

4) Quantum Arts

QBraiNs will allow artists and spectators a unique two-level sensorial quantum
experience. A first quantum level will be related to the signal sequence:
classical input, quantum filtering, classical output. And a a second quantum
level will involve quantum feedback of looped emotions and thoughts.

##  VI Concluding Remarks

We have discussed the conceptual and technological readiness for a promising
interdisciplinary field, Quantum Brain Networks (QBraiNs), involving the
hybrid classical-quantum communication between human brains, intelligent
machines, and quantum computers. Beyond the necessary upgrades in upcoming
hardware and wetware developments, including transducers and control systems,
we believe QBraiNs will have a wide variety of disruptive applications in
association with versatile connectivity options. In this sense, we have
provided feasible use cases where wetware and hardware interconnectivity, at
the classical and quantum levels, would be transformational for current
processing and communication technologies. This Perspective should open wide
discussions in yet unexplored directions of the long-term relation between
humans and machines. Finally, we understand that QBraiNs may open another path
of ethical issues to consider, an important topic that goes beyond the scope
of this Perspective.

##  VII Acknowledgments

We acknowledge support from QMiCS (820505) and OpenSuperQ (820363) projects of
the EU Flagship on Quantum Technologies, National Natural Science Foundation
of China (NSFC) (12075145), STCSM (2019SHZDZX01-ZX04, 18010500400 and
18ZR1415500), Spanish Government PGC2018-095113-B-I00, PID2019-104002GB-C21
and PID2019-104002GB-C22 (MCIU/AEI/FEDER, UE), Basque Government IT986-16, EU
FET Open Grant Quromorphic and EPIQUS. We thank the ICCMR, University of
Plymouth, UK, for support and access to BMI resources.

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[a]Yannick Meurice

# Scalar QED with Rydberg atoms

James Corona  Sergio Cantu  Fangli Liu  Shengtao Wang  Kenny Heitritter  Steve
Mrenna  Jin Zhang  Shan-Wen Tsai

###### Abstract

We review recent suggestions to quantum simulate scalar electrodynamics (the
lattice Abelian Higgs model) in $1+1$ dimensions with rectangular arrays of
Rydberg atoms. We show that platforms made publicly available recently allow
empirical explorations of the critical behavior of quantum simulators. We
discuss recent progress regarding the phase diagram of two-leg ladders,
effective Hamiltonian approaches and the construction of hybrid quantum
algorithms targeting hadronization in collider physics event generators.
FERMILAB-CONF-23-0831-CSAID

##  1 Introduction

In these Proceedings we discuss the possibility of hybrid quantum/classical
computing for event generators such as PYTHIA [1, 2]. Our long-term goal is to
replace the current hadronization part which turns quarks and gluons into
hadrons and is currently implemented with the phenomenological Lund model, by
an ab-initio lattice calculation performed by a quantum computing device. More
technical details and references can be found in [3].

Recently there has been a lot of interest for quantum simulation for gauge
theories [4, 5, 6, 7, 8, 9] and in particular for the real-time dynamics. In
the following, we consider the possibility where the hadronization part of
event generators could be described by a simplified lattice model: the compact
Abelian Higgs Model (Scalar QED) in $1+1$ dimensions. We explain that is
(literally!) possible to attempt model building for a quantum simulator with
arrays of Rydberg atoms [10, 11, 12]: publicly available interfaces [13] allow
users to engage in quantum simulations in a rather straightforward way. The
matching between the target model and the simulator is non-trivial. However,
we show that for some region of the parameters of the simulator, it is
possible to construct an effective theory for the simulator which has only one
(important) term differing from the target model. More details about this
question can be found in a recent preprint [28]. We also briefly discuss
practical examples of current work with QuEra.

##  2 The compact Abelian Higgs Model (CAHM)

The lattice compact Abelian Higgs model is a non-perturbative regularized
formulation of scalar quantum electrodynamics (scalar electrons-positrons +
photons with compact fields). The partition function reads:

$Z_{CAHM}=\prod_{x}\int_{-\pi}^{\pi}\frac{d\varphi_{x}}{2\pi}\prod_{x,\mu}\int_{-\pi}^{\pi}\frac{dA_{x,\mu}}{2\pi}e^{-S_{gauge}-S_{matter}},$

with

$S_{gauge}=\beta_{plaquette}\sum_{x,\mu<\nu}(1-\cos(A_{x,\mu}+A_{x+\hat{\mu},\nu}-A_{x+\hat{\nu},\mu}-A_{x,\nu})),$
$S_{matter}=\beta_{link}\sum\limits_{x,\mu}(1-\cos(\varphi_{x+\hat{\mu}}-\varphi_{x}+A_{x,\mu})).$

It has a local invariance: $\varphi_{x}^{\prime}=\varphi_{x}+\alpha_{x}$ and
where local changes in $S_{matter}$ are compensated by
$A_{x,\mu}^{\prime}=A_{x,\mu}-(\alpha_{x+\hat{\mu}}-\alpha_{x})$. $\varphi$ is
the Nambu-Goldstone mode of the original model. The Brout-Englert-Higgs mode
is decoupled (heavy). Using methods of Tensor Lattice Field Theory (TLFT)
reviewed in [27], one obtains an Hamiltonian and Hilbert space in $1+1$
dimension in the continuous-time limit:

$H=\frac{U}{2}\sum_{i=1}^{N_{s}}\left(L^{z}_{i}\right)^{2}+\frac{Y}{2}{\sum_{i}}(L^{z}_{i+1}-L^{z}_{i})^{2}-X\sum_{i=1}^{N_{s}}U^{x}_{i}\
$

with $U^{x}\equiv\frac{1}{2}(U^{+}+U^{-})$ and $L^{z}|m\rangle=m|m\rangle\ \
{\rm and}\ U^{\pm}|m\rangle=|m\pm 1\rangle.$ $m$ is a discrete electric field
quantum number ($-\infty<m<+\infty$). In practice, we need to apply
truncations: For a spin-$m_{max}$ truncation we have $U^{\pm}|\pm
m_{max}\rangle=0.$ We focus on the spin-1 truncation ($m=\pm 1,0$ and
$U^{x}=L^{x}/\sqrt{2}$). The $U$-term represents the electric field energy,
the $Y$-term, the matter charges (determined by Gauss’s law) and the $X$-term:
currents inducing temporal changes in the electric field. Using this
Hamiltonian, we have evolved an intial state with a bit of electric field in
the middle. By Gauss’s law, this is equivalent to a pair of bosonic particle-
antiparticle separated by one lattice spacing. An example is shown in Fig. 1.
and various stages can be reinterpreted in terms of patterns such as string
breaking, discussed in Refs. [1, 2].

Figure 1: Electric field $E_{j}$ versus position $j$ for the evolution of a
particle-antiparticle pair for the Abelian Higgs model in $1+1$ dimensions for
$Y/X=1$ and $U/X=0.1$ and 50 time steps with $\Delta t$ such that $X\Delta
t=0.1$.

##  3 The simulator

In [17], we adapted the optical lattice construction [28] using arrays of
${}^{87}Rb$ atoms [10, 11, 12, 13] separated by controllable (but not too
small) distances, coupled to the excited Rydberg state $|r\rangle$ with a
detuning $\Delta$. The ground state is denoted $|g\rangle$ and the two
possible states $|g\rangle$ and $|r\rangle$ can be seen as a qubit. We have
the occupations $n|g\rangle=0,\ n|r\rangle=|r\rangle$. The Hamiltonian reads

$H=\frac{\Omega}{2}\sum_{i}(|g_{i}\rangle\langle r_{i}|+|r_{i}\rangle\langle
g_{i}|)-\Delta\sum_{i}n_{i}+\sum_{i<j}V_{ij}n_{i}n_{j},$

with

$V_{ij}=\Omega R_{b}^{6}/r_{ij}^{6},$

for a distance $r_{ij}$ between the atoms labelled as $i$ and $j$. Note that
when $r=R_{b}$, the Rydberg radius, $V=\Omega$. This repulsive interaction
prevents two atoms close enough to each other to be both in the $|r\rangle$
state. This is the so-called blockade mechanism which can be used to produce
an effective spin-1 local Hilbert space. In order to shortcut the discussion
of the staggered interpretation of the electric field, we have plotted its
square $E^{2}_{j}=(n_{j}^{up}-n_{j}^{down})^{2}$ (see [17] for a discussion)
in Fig. 2 which also exhibit interesting behavior with a similar
interpretation as for Fig. 1.

Figure 2: Time evolution for $E^{2}_{j}=(n_{j}^{up}-n_{j}^{down})^{2}$ for a
5-rung ladder (ten ${}^{87}Rb$ atoms). At initial time, all the atoms are in
the ground state except for the top atom in the middle rung.
$\Omega=2\pi\times 2*10^{6}$ MHz, $\Delta=2\Omega$, $a_{x}=R_{b}$
$a_{y}=0.5R_{b}$.

##  4 The effective Hamiltonian and phase diagram

As reported in a recent preprint [28], we constructed a translation-invariant
effective Hamiltonian by integrating over the simulator high-energy states
produced by the blockade mechanism. Remarkably, for all the simulators
considered (ladders, prisms, and others), the effective Hamiltonians have the
three types of terms present for the CAHM (Electric field, matter charge and
currents energies) but, in addition, terms quartic in the electric field. For
positive detuning, the new terms create degenerate vacua resulting in a very
interesting phase diagram.

The phase diagram of the ladder simulator with rung size twice the lattice
spacing has been investigated experimentally. Evidence for an incommensurate
phase between crystalline commensurate phases will be reported in an upcoming
preprint. It is relatively straightforward for theoretical users to perform
analog simulations with Rydberg arrays using publicly available interfaces
[13]. We hope that this exploration will improve our understanding of
inhomogeneous phases and the Lifshitz regime of lattice quantum chromodynamics
[29, 30], as well as the examination of “chiral spiral” condensation [31]. One
should also compare the manipulation of the three states associated with a
rung in our approach with qutrit simulations [32] and decide if the quartic
term could be found in the context of Symanzik improvement [33, 34].

##  5 Conclusions

We have considered ladder-shaped Rydberg arrays with two atom per rung as
simulators for the compact Abelian Higgs model. Ultimately the matching
between simulator and target model should be understood in the continuum limit
(universal behavior). Effective Hamiltonians for the simulator were found with
same three types of terms as the target model plus an extra quartic term. The
two-leg ladder has a very rich phase diagram. Explorations of the phase
diagram with AWS/QuEra are ongoing as well as the possibility of an hybrid
interface with PYTHIA are pursued.

## Acknowledgemnts

Y.M. and K.H. were supported in part by the Dept. of Energy under Award Number
DE-SC0019139. Y. M. thanks the Amazon Web Services and S. Hassinger for
facilitating remote access to QuEra through the Amazon Braket. K.H.
acknowledges support from the URA Visiting Scholars Program. J.Z. is supported
by NSFC under Grants No. 12304172 and No. 12347101, Chongqing Natural Science
Foundation under Grant No. CSTB2023NSCQ-MSX0048, and Fundamental Research
Funds for the Central Universities under Projects No. 2023CDJXY-048 and No.
2020CDJQY-Z003. This work was also supported in part by the National Science
Foundation (NSF) RAISE-TAQS under Award Number 1839153 (SWT). Computations
were performed using the computer clusters and data storage resources of the
HPCC, which were funded by grants from NSF (MRI-1429826) and NIH
(1S10OD016290-01A1). S.M. is supported in part by the U.S. Department of
Energy, Office of Science, Office of High Energy Physics QuantISED program
under the grants “HEP Machine Learning and Optimization Go Quantum”, Award
Number 0000240323, and “DOE QuantiSED Consortium QCCFP-QMLQCF”, Award Number
DE-SC0019219. This manuscript has been authored by Fermi Research Alliance,
LLC under Contract No. DEAC02-07CH11359 with the U.S. Department of Energy,
Office of Science, Office of High Energy Physics. This research was supported
in part through computational resources provided by The University of Iowa,
Iowa City, Iowa.

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