Control algebra for holonomic quantum computation with squeezed coherent states
Dennis Lucarelli
Department of Systems Science and Mathematics
St. Louis
Missouri 63130
A condition for determining the holonomy group associated to a principal bundle with connection is proven. Within the context of Holonomic Quantum Computation, this condition aids in determining the universality of the system. We apply the theory to the product bundle arising from the full two-qubit system of holonomic computation with squeezed coherent states and conclude that the control algebra generates the Lie algebra $u(4) ~.$
quant-ph0202055.tex
\documentclass[12pt]{article}
\usepackage{amsfonts}
\usepackage{amsmath}
\begin{document}
\title{Control algebra for holonomic quantum computation with 
squeezed coherent states}
\author{Dennis Lucarelli \\Department of Systems Science and Mathematics \\
Washington University, St. Louis, Missouri 63130}
\date{}
\maketitle
\begin{abstract}
A condition for determining the holonomy group associated to a 
principal bundle with connection is proven.  Within the context of Holonomic
Quantum Computation, this condition aids in determining the universality of
the system.  We apply the theory to the product bundle arising from the 
full two-qubit system of holonomic computation with squeezed coherent states 
and conclude that the control algebra generates the Lie algebra $u(4) ~.$
\end{abstract}

\section{Introduction}Geometric phases have long been a source of fascination and insight
into classical and quantum physical theories \cite{phase}.  In recent years, they have
proven to be useful in describing the dynamics and control of certain nonholonomic mechanical systems
with symmetry.  Inspired by the appearance of geometric phases in biology,
engineers have sought to create motion in machines via cyclic variations in {\it shape space}.
These endeavors \cite{kelly} and the characterization of optimal trajectories
\cite{koon, mont} remain active areas of research.

Most recently, the geometric phase has been realized as a way of
constructing logic gates in a quantum computer \cite{zan}.
Holonomic Quantum Computation (HQC) employs non-Abelian geometric
phases (holonomies) for the purpose of quantum information
processing. This paper is an elaboration of \cite{luc}. There it
was reported that the \emph{Optical Holonomic Computer}
\cite{pach1,pach2,fujii1,fujii2,fujii3} is universal by showing that the holonomy group
of the two-qubit interaction model contains $SU(2) \times SU(2)$.
These transformations act non-trivially on the standard
computational basis and support, in particular, SWAP gates.
Coupled with the ability to perform single qubit rotations, the
claim of universality follows by appealing to standard results
from quantum computation \cite{chuang}.

These notes and \cite{luc} completely solve the problem posed in
the recent work of K. Fujii \cite{fujii1,fujii2,fujii3}.  That
is, for the two-qubit system, we explicitly construct a set of
holonomic transformations that span the Lie algebra $u(4)$ and
thus attain universality of the system.  Note however that
we do not carry out the program proposed in \cite{fujii3}.  
In Pachos-Chountasis
\cite{pach1} and Pachos-Zanardi \cite{pach2}, 
the universality of
the model was shown by constructing a universal 
two-qubit logic
gate.  This $U(4)$ transformation was 
integrated directly
by choosing a loop in a 2-submanifold of parameter 
space along
which the connection components commute but with non-vanishing
curvature. For these restricted loops, the non-Abelian nature of
the path ordered integral may be avoided. However, that
transformation was not an element of the holonomy group as
calculated by the application of the Ambrose-Singer Theorem in
\cite{fujii2}.

In \cite{luc}, we stated simple conditions for determining the
holonomy group of a principal bundle with connection.  These
conditions are seen to be a corollary to Chow's theorem from
control theory.  The obvious relevance to HQC is that the holonomy
group represents the set of controlled transformations attainable
for manipulating quantum information.

In this section, we review some geometric constructions and
provide some details regarding the corollary stated in \cite{luc}.
This proof is not new.  A similar proof can be found in
\cite{mont} and,  while not explicitly stated, all the necessary
ingredients can be found in \cite{kn}.


\subsection{Principal and associated bundles}

Let $Q$ be a principal $G-$bundle defined by the free (right) Lie
group action $ \Phi: Q \times G \xrightarrow{} Q$ and the
canonical projection $Q \xrightarrow{\pi} Q/G \equiv  M .$ If $U$
is a neighborhood of M, then $Q$ is locally diffeomorphic to the
product $ U \times G .$ A smooth map $\sigma : U \xrightarrow{}
Q$, such that $ \pi \circ \sigma  = \rm{id}_M$ is called a local
section over $U.$ 

Given any principal bundle $ Q \xrightarrow{\pi}
Q/G$ and an action of the Lie group $G$ on a vector space ${\cal V}$, we
may define
\begin{eqnarray}
\rho : & Q  \times {\cal V} & \xrightarrow{} \quad {\cal V}    \notag  \\
& (q,v) &\mapsto \quad (qg, \rho^{-1}(g)  v )
\end{eqnarray}
where $q \in Q \, , \, v \in {\cal V}  \, , \, g \in G \, $ and  $\rho$ is a representation of
the group $G.$
The {\it associated vector bundle} $E_\rho$ with typical fiber ${\cal V}$ is the set of equivalence
classes under this action
\begin{equation}
E_\rho = Q \times_\rho  {\cal V} = (Q \times {\cal V})  / G ,
\end{equation}
where points $(q,v)$ and  $ (qg,  \rho^{-1}(g)  v )$ are identified.  Let $ [ (q,v) ]$
denote a representative of the equivalence class.

In the mathematical theory of HQC, there are {\it two} associated
bundles of interest.  First, the quantum code is realized as the
${\bf C}^n$ vector bundle associated to the pullback of the
universal bundle by the classifying map \cite{fujii1,fujii2}.
Second, the {\it adjoint bundle} defined by taking ${\cal V}= \mathfrak
g$ (the Lie algebra of $G$) and $\rho$  as the adjoint action of a
Lie group on its Lie algebra is important in deriving conditions
for universal computation as we prove in Section 1.3.

We now state an important correspondence concerning the associated adjoint bundle.
First, recall that a  Lie algebra valued differential form $\omega$ is {\emph Ad-equivariant} if
\begin{equation}
\Phi_g^* \cdot \omega = Ad(g^{-1}) \cdot \omega \quad g \in G .
\end{equation}
If $\omega$ is a Lie algebra valued {\it function} $f$  on Q, then Ad-equivariance
is written as $ f(q g ) = Ad(g^{-1}) f(q), \quad g \in G .$  The sections $s$ of the adjoint bundle
associated to a principal bundle $Q$
are in one-to-one correspondence with Ad-equivariant functions on  $Q.$
To any Ad-equivariant function $f$ we may define a local section $s_f$ by
$ s_f(p) = [(q,f(q))],$ where $q$ is any point in the fiber over $p.$
This definition is independent of the choice of $q$ because of Ad-equivariance.
Conversely, given a local section $s: U \xrightarrow{} Q \times_{\rm Ad} \mathfrak{g}$,
we define a map $f_s: \pi^{-1}(U) \xrightarrow{} \mathfrak{g}$ such that $s(p) = [(p,f_s (p))]$
\cite{found}.

\subsection{Connections and covariant derivatives}
Recall that for any $\xi \in \mathfrak{g}$, the {\bf R}-action $\Phi_{ \rm{exp}(t\xi) } \, q $
defines a curve though $q \in Q$.  The {\it infinitesimal generator} of the action corresponding
to $\xi$ is defined as the vector field
\begin{equation}
\xi_q \equiv \frac{d}{dt} \bigg{|}_{t=0}   \Phi_{ \rm{exp}(t\xi) } \, q .
\end{equation}
The vertical subspace $V_qQ \subset T_q Q$ is defined to be the subspace of $T_qQ$ that is tangent
to the fiber over  $ p = \pi(q) \, , \, G_p$.  By the previous definition, we have the
identification $ V_qQ \cong \mathfrak{g}$.

A connection on a principal $G$-bundle ${\cal A}$ on $Q$ is an Ad-equivariant
Lie algebra-valued one-form ${\cal A}: TQ \to \mathfrak{g}$ such that $A(\xi_q) = \xi$.
The horizontal space $H_qQ$ is the linear space $H_qQ \equiv \{ X \in T_qQ | {\cal A}_q( X ) = 0 \}$.
These definitions provide the splitting
\begin{equation}
T_qQ = V_qQ \oplus H_qQ
\end{equation}
of the tangent vectors into horizontal and vertical components.
Note that
\begin{equation}
V_q Q = {\rm Ker}T_q \pi, \quad {\rm and} \quad H_q Q =  {\rm Ker} {\cal A}_q .
\end{equation}
The map $T_q \pi : H_q \to T_{ \pi (q)} M $ is an isomorphism from
the horizontal space to the tangent space of the base space. Thus
a curve $q(t) \in Q$ defines a curve in the base by specifying a
tangent vector at each point $\pi(q(t)) = p(t)$. The properties of
the connection and the uniqueness theory of ODE's provide the
reverse procedure of reconstructing a unique curve in the
horizontal space given a curve in the base called the
\emph{horizontal lift} \cite{nak}.  Denote the horizontal lift of
the vector $X \in T_p M$ where $p = \pi(q)$ by $X_q^h$ and $ h \,
X_q$ as the horizontal component of a vector in $T_q Q.$

Let $\sigma$ be a local section over a neighborhood $U$ of $M$
that contains a curve $\gamma.$ Then the horizontal lift
$\widetilde{\gamma}$ can be expressed as $\widetilde{\gamma}(t) =
\sigma ( \widetilde{\gamma}(t)) g(t),$ where $g(t) = g(\gamma(t))
\in G.$ The local connection one-form $A$ is defined as the
pullback $A \equiv \sigma^* ( \cal{A}) . $ The local connection
may be written in terms of group elements by the equation
\begin{equation}
\frac{d g(t)}{dt} = - A \,  g(t).
\end{equation}
Choosing the initial condition, $g(0) = \mathrm{id}_G , $
the solution may be obtained by the path ordered integral
\begin{equation}
g(\gamma(t)) = {\bf P}\,  \exp \int_\gamma -A .
\end{equation}

Given a connection on a principal bundle, we may define the
covariant derivative of sections of associated bundles. Let $
E_\mathfrak{g} = Q \times_{\rm Ad} \mathfrak{g}$ be the adjoint
bundle associated to $Q$ as defined above.  Let $\gamma$ be a base
curve with tangent vector $X$ at $\gamma(0).$  If $ \{
\cal{T}_\alpha \} $ is a basis of $\mathfrak{g},$ then a section
$s$ of $E_\mathfrak{g}$ may be obtained by forming $s = [ ( \sigma
(p), {\cal T} (p) ) ]$, where ${\cal T}(p) = f^{\alpha} (p) {\cal
T}_{\alpha}$ and $\sigma$ is a local section over a neighborhood
$U$ of $p.$   The covariant derivative $D_X s$ of $s$ along $X \in
TM$ is defined by \cite{nak}
\begin{equation}
D_X s \equiv [ ( \widetilde{\gamma} (0) \, , \, \, \frac{d}{dt}
\mathrm{Ad}_{g^{-1} (t)} {\cal T} (t)  ] \Big|_{t=0} ) ]
\end{equation}
\noindent The previous definitions imply,
\begin{align}
D_X s &= [ ( \sigma (0) g (0) \, , \, \,  \frac{d}{dt} \mathrm{Ad}_{g^{-1} (t)}
{\cal T} (t)  ] \Big|_{t=0} ) ]  \\
&=  [ ( \sigma (0) \, , \, \,  \mathrm{Ad}_{g}\frac{d}{dt} \mathrm{Ad}_{g^{-1} (t)}
{\cal T} (t)  ] \Big|_{t=0} ) ] \\
&=  [( \sigma (0) \, , \, \, \frac{d {\cal T}}{dt} \, + \, [ A(X) \, , \, {\cal T} (t)]
\Big|_{t=0})] \,.
\end{align}

Covariant derivatives of sections of the adjoint bundle provide a way of computing Lie
derivatives of Ad-equivariant functions on the total space by the correspondence
\begin{equation}\label{covlie}
D_{X} s_{f} = X^h f
\end{equation}
\noindent where $s_f$ is the section of the adjoint bundle corresponding to the Ad-equivariant
function $f$ \cite{kn}.

\subsection{Infinitesimal holonomy }
The horizontal lift of any closed curve in the base space with
initial point $p$ maps the fiber over $p$ to itself and thus
corresponds to the group translation $ q g , q \in \pi^{-1}(p)$
and  $ g \in G.$  In this way, we can associate to closed curves
in the base space  elements of the structure group $G.$  The set
of all such group elements forms a subgroup of $G$ called the {\it
holonomy group}.  With the restriction to sufficiently small loops
homotopic to zero, we may define the {\it infinitesimal} holonomy
group.  The infinitesimal holonomy group is in general a subgroup
of the full holonomy group, however for analytic connections over
connected base manifolds, these notions coincide \cite{kn}

The holonomy group is determined from the curvature. The {\it curvature} ${\cal F}$
associated to  ${\cal A}$  is by definition the Lie algebra-valued two-form on Q such that
\begin{equation}
{\cal F} (X_1, X_2) = {\rm {\bf d}} \,{\cal A} ( h ~ X_1 \, , \, h
~ X_2 )
\end{equation}
where {\bf d} denotes exterior differentiation.  Using the identity
\begin{equation}
{\rm {\bf d}} {\cal A} (X_1 , X_2) = X_1 {\cal A}(X_2) - X_2 {\cal
A}(X_1) - {\cal A} ( [ X_1 ~ , X_2] )
\end{equation}
and the fact that the connection form evaluates to zero on
horizontal vectors, we get the relation ${\cal F}(X_1,X_2) = -
{\cal A} (  [ \, h ~ X_1 \, , \, h ~ X_2  \, ] ). $ Roughly, the
curvature measures the non-commutativity in the total space of
horizontally lifted vector fields defined in the base space . Note
that ${\cal F}$ is Ad-equivariant since ${\cal A}$ is
Ad-equivariant and pullback and exterior differentiation commute.


The Lie algebra corresponding to the infinitesimal holonomy group is spanned by the
$\mathfrak{g}$-valued Ad-equivariant function $f$ given by
\begin{equation} \label{inflie}
f \, = \, X^h_{i_k}\,X^h_{i_{k-1}}\dots X^h_{i_3}{\cal F} ( X^h_{i_2}, X^h_{i_1})
\end{equation}
\noindent where $X_i \in TM$ and $ 2 \leq k < \infty$ \cite{ozeki}. By the correspondence
(\ref{covlie}), the pullback  through a local section $\sigma$ of this function
 may written in terms of covariant derivatives as
\begin{equation} \label{infcov}
D_{X_{i_k}} \, D_{X_{i_{k-1}}} \dots  \, D_{X_{i_3}} F (X_{i_2},X_{i_1})
\end{equation}
\noindent where $F$ is the local form of the curvature defined with respect to a local
section $F \equiv \sigma^* ({\cal F}).$
\section{HQC with squeezed coherent states}

In this section, we revisit the mathematical framework of Optical
Holonomic Computation. See \cite{pach1, pach2, fujii1, fujii2} for
the many details omitted here. We refer to this setup as  {\it HQC
with squeezed coherent states} since the properties of interest in
quantum computation, e.g. universality, entanglement, etc., rely
only on the squeezing and displacing operators and not directly on
the degenerate Hamiltonian used to encode the qubits. Therefore,
the mathematics describing this computing scheme will be shared by
any quantum system whose states can be displaced and squeezed. See
for example the remarks concerning trapped ions in \cite{pach2}.

Let ${\cal H}_i = \{ | \alpha_i \rangle \, , \, \alpha \in {\bf N} \} $ be a Fock space with
number operator $n_i \, . $  The Hamiltonian $H^i = n_i (n_i -1) $ encodes the $i$-th
qubit in the degenerate subspace $\{ | 0_i \rangle \, , \, |  1_i \rangle \} \, . $  Single mode
squeezing and displacing operators control the single qubit
\begin{equation}
S^i(\mu_i) \, = \, \mathrm{exp} \, (\mu_i a_i^{{\dag}{2}} - \bar{\mu_i} a_i^{2}) \quad \quad
D^i(\lambda_i) \, = \, \mathrm{exp} \, (\lambda_i a_i^{\dag} - \bar{\lambda_i}a_i)
\end{equation}
\noindent where $\mu_i \, , \, \lambda_i \in {\bf C}$ and $a_i$ and $a_i^{\dag}$ are the
annihilation and creation operators respectively.  To obtain controllable interactions among
product states we employ the Hamiltonian $H^{12} = n_1(n_1-1) + n_2(n_2-1)$
with degenerate subspace $ \{ | 0 0 \rangle , | 0 1 \rangle , | 1 0 \rangle , | 11 \rangle \}$
and two mode squeezing and displacing operators
\begin{equation}
M(\zeta) = \mathrm{exp} \, (\zeta a_1^{\dag}a_2^{\dag} - \bar{\zeta} a_1a_2) \quad \quad
N(\xi) = \mathrm{exp} \, (\xi a_1^{\dag}a_2 - \bar{\xi}a_1a_2^{\dag})
\end{equation}
where $\xi \, , \, \zeta \, \in {\bf C} . $  Set ${\cal U}_2 = D(\lambda)S(\mu)$ and
${\cal U}_4 = N(\xi)M(\zeta).$

The geometry of the $U(n)$ holonomy problem \cite{fujii1,fujii2}
is briefly described as follows. Define the manifolds
\begin{align}
\mathrm{St}_n ({\cal{H}}) &= \{ V = (v_1, \, \dots \, , v_n) \in {\cal{H}} \times \, \dots \,
\times {\cal{H}} \,\, | \,\, V^{\dag}V = \mathrm{Id}_{ n \times n} \}   \\
\mathrm{Gr}_n ({\cal{H}}) &= \{ X \in B({\cal H}) \,\, | \,\, X^2=X \, , \, X^{\dag} = X \, , \,
\mathrm{tr}X = n \} \, .
\end{align}
\noindent where $B({\cal H})$ denotes the set of bounded linear operators on ${\cal H}$.
These manifolds are known as the Stiefel and Grassmann manifolds respectively.
They form a principal bundle with the (right) $U(n)$ action on $\mathrm{St}_n({\cal H})$
and the projection $ \pi: \mathrm{St}_n({\cal H}) \to \mathrm{Gr}_n({\cal H})$ given by
$\pi(V) = V V^{\dag}$.  Denote this $U(n)$-bundle by $P_n$.  Let $M \subset {\bf C^2} $ be
a parameter space and define the classifying map $\Pi_n : M \to \mathrm{Gr}_n({\cal H})$ given by
\begin{equation}
\Pi_n(\eta)  \equiv  {\cal U}_n(\eta) \Big( \sum_{j = i}^n v_j v_j^\dag \Big) {\cal U}_n^\dag(\eta)
\end{equation}
Now form the pullback bundle $Q_n = \Pi_n^* P_n ~.$  The quantum
code is then realized as the ${\bf C}^n$-vector bundle associated to $Q_n$
\cite{fujii1,fujii2,nak}.  To ease the notation, we consider only
the manifolds $P_n$ and $Q_n ~.$  So then, the full two-qubit
system consisting of single qubit rotations on each qubit and
interactions among them is modelled by the pullback bundles
corresponding to the product bundle
\begin{equation}
\label{unibund}
 \begin{array}{ccccc}
     \mathrm{St}_{2}({\cal H}_1 ) &
      \times  &
      \mathrm{St}_{4}({\cal H}_1 \otimes {\cal H}_2) &
      \times  &
      \mathrm{St}_{2}({\cal H}_2 )  \\
        \Big\downarrow  & &
      \Big\downarrow     & &
      \Big\downarrow    \\
      \mathrm{Gr}_{2}({\cal H}_1 ) &
       \times &
      (\mathrm{Gr_{4}})^{\rm int}({\cal H}_1 \otimes {\cal H}_2) &
       \times &
      Gr_{2} ({\cal H}_2 )
 \end{array}
\end{equation}
The local connection one-form is given by the Wilczek-Zee
connection \cite{zee}  with matrix elements
\begin{equation}
A_{\eta_i}^{\bar{\varrho} \varrho} \equiv \langle \bar{\varrho} |
{\cal U}^\dag (\eta) \frac{\partial}{\partial \eta_i} {\cal U}
(\eta) | \varrho \rangle.
\end{equation}
The connection components are listed in Appendix A with respect to the labelling of parameters
$\lambda_1 = x_1 + i y_1 \, , \, \mu_1 = r_1 e^{i \theta_1} \, , \,
\lambda_2 = x_2 + i y_2 \, , \, \mu_2 = r_4 e^{i \theta_4} \, , \,
 \zeta = r_2 e^{ i \theta_2} \, \mathrm{and} \, \xi= r_3 e^{i\theta_3} \, . $

\subsection{Reduction}
The local curvature components are listed in Appendix B.  The holonomy group of the two-qubit
interaction model is $SU(2) \times SU(2) \times U(1)$ \cite{luc}.  These transformations
leave invariant the subspaces  $\{ | 0 0 \rangle \, , \, | 1 1 \rangle \} \, \, \mathrm{and} \, \,
\{ | 0 1 \rangle \, , \, | 1 0 \rangle \}$ in ${\cal H}_1 \otimes {\cal H}_2 \, .$  By the
reduction theorem for connections \cite{kn}, the connection is reducible to a $su(2) \times
su(2) \times u(1)$-valued connection and we may reduce the total space to
$\mathrm{St}_{2,4}({\cal H}_1 \otimes {\cal H}_2) ~. $  To determine the reduced base manifold, we
form the quotient
\begin{equation}
\frac{U(4)}{SU(2) \times SU(2) \times U(1)}  \cong  \frac{SU(4)}{SU(2) \times SU(2)}
\cong \rm{SGr}_{2,4} \equiv (\rm{Gr}_{2,4})^{\rm int} ~.
\end{equation}
In a similar manner, we can reduce the bundles ${\rm St}_2 ({\cal
H}_i) \mapsto {\rm Gr}_2 ({\cal H}_i)$ corresponding to the single
qubit rotations. The $U(2)$ holonomies act in the product space
${\cal H}_1 \otimes {\cal H}_2 $ as $U(2) \otimes {\bf 1}$ and $
{\bf 1} \otimes U(2) .$  The bundles reduce to $ {\rm
St}_{2,4}({\cal H}_1 \otimes {\cal H}_2) \mapsto {\rm
Gr}_{2,4}({\cal H}_1 \otimes {\cal H}_2). $  For the full
two-qubit system, we have the reduced product bundle
\begin{equation}
\label{unibundreduced}
 \begin{array}{ccccc}
     \mathrm{St}_{2,4}({\cal H}_1 \otimes {\cal H}_2 ) &
      \times  &
      \mathrm{St}_{2,4}({\cal H}_1 \otimes {\cal H}_2) &
      \times  &
      \mathrm{St}_{2,4}({\cal H}_1 \otimes {\cal H}_2 )  \\
        \Big\downarrow  & &
      \Big\downarrow     & &
      \Big\downarrow    \\
      \mathrm{Gr}_{2,4}({\cal H}_1  \otimes{\cal H}_2) &
       \times &
      (\mathrm{Gr_{2,4}})^{\rm int}({\cal H}_1 \otimes {\cal H}_2) &
       \times &
      Gr_{2,4} ({\cal H}_1 \otimes{\cal H}_2 )
 \end{array}
\end{equation}
This structure is similar to the product bundle for the generic two-qubit system
proposed by Pachos \cite{pach3}.
\subsection{Control algebra}
The standard argument to prove universality of a quantum system is
to demonstrate the ability to perform all single qubit rotations
on each qubit and a non-trivial two-qubit transformation.  The
implicit argument is then that the {\it control algebra} formed by
these matrices and their Lie brackets generates the Lie algebra
$su(n)$ or $u(n) ~.$  In this section, we carry out this argument
for HQC with squeezed coherent states.  The matrices we refer to
can be found in Appendices B and C.

\vspace{12pt}
\noindent The holonomy algebra for each single qubit system is $u(2) ~.$
\begin{align}
u(2) \otimes {\bf 1} \, \, &: \, \, {\rm span}_{\bf R} \{ F_{x_1 y_1} \otimes {\bf 1} ~,
F_{r_1 \theta_1} \otimes {\bf 1} ~,  F_{y_1 r_1} \otimes
{\bf 1} ~, F_{x_1 r_1} \otimes {\bf 1} \} \\
{\bf 1} \otimes u(2) \, \, &: \, \,{\rm span}_{\bf R} \{ {\bf 1} \otimes F_{x_2 y_2} ~,
{\bf 1} \otimes F_{r_4 \theta_4} ~, {\bf 1} \otimes F_{y_2 r_4 } ~,
{\bf 1} \otimes F_{x_2 r_4 } \}
\end{align}
\noindent The holonomy algebra for two qubit interactions system is
\newline
$ su(2) \times su(2) \times u(1) ~.$
\begin{multline}
{\rm span}_{\bf R}
\{ F_{r_2 r_3} ~, F_{r_2 \theta_2} ~, F_{r_2 \theta_3} ~, F_{r_3 \theta_3} ~,
D_{\frac{\partial}{\partial r_2}} F_{r_{2}\theta_{2}} ~,
D_{\frac{\partial}{\partial \theta_2}} F_{r_{2}\theta_{2}} ~,
D_{\frac{\partial}{\partial \theta_2}} D_{\frac{\partial}{\partial \theta_2}} F_{r_{2}\theta_{2}} \}
\end{multline}

\noindent From these matrices, we choose the subset linearly
independent over the reals
\begin{align}
\mathcal{C}_1 \, &= \, \{
F_{r_1 \theta_1} \otimes {\bf 1} ~,  F_{y_1 r_1} \otimes {\bf 1} ~,
F_{x_1 r_1} \otimes {\bf 1} \} \\
\mathcal{C}_2 \, &= \, \{{\bf 1} \otimes F_{x_2 y_2} ~,
{\bf 1} \otimes F_{y_2 r_4 } ~, {\bf 1} \otimes F_{x_2 r_4 } \} \\
\mathcal{C}_{12} \, &= \, \{  F_{r_2 r_3} ~, F_{r_2 \theta_2} ~, F_{r_2 \theta_3} ~,
D_{\frac{\partial}{\partial \theta_2}} F_{r_{2}\theta_{2}} ~,
D_{\frac{\partial}{\partial \theta_2}} D_{\frac{\partial}{\partial \theta_2}} F_{r_{2}\theta_{2}} \}
\end{align}
\noindent Additional independent matrices can be obtained by
taking Lie brackets from the set $ \{ \mathcal{C}_1 ~ \cup ~
\mathcal{C}_2 ~ \cup ~ \mathcal{C}_{12} \} ~.$  In principle,
these transformations can be constructed in the usual
infinitesimal way. For example, to construct $\big[\,  F_{x_1 r_1}
\otimes {\bf 1} \, , \, F_{r_2  r_3} \, \big]$ traverse a small
loop in the plane $(x_1 ~, r_1)$, then traverse a loop in the
plane $(r_2 ~, r_3)$, then traverse the loop backwards in $(x_1 ~,
r_1)$, and finally traverse the loop backwards in  $(r_2 ~, r_3 )
~. $ One choice is given by
\begin{multline}
\mathcal{C}_{12}^2 = \{
\big[\,  F_{x_1 r_1} \otimes {\bf 1} \, , \, F_{r_2 r_3} \, \big] ~,
\big[\,  {\bf 1} \otimes F_{x_2 r_4} \, , \, F_{r_2 r_3} \, \big] ~,
\big[\,  F_{y_1 r_1} \otimes {\bf 1} \, , \, F_{r_2 \theta_3} \, \big] ~, \\
\big[\,  {\bf 1} \otimes F_{y_2 r_4} \, , \, F_{r_2 \theta_3} \, \big] ~,
\Big[\,  {\bf 1} \otimes F_{y_2 r_4} \, , \,
\big[\, F_{x_1 r_1} \otimes {\bf 1 } \, ,\, F_{r_2 r_3} \big] \, \Big] \}
\end{multline}

\noindent Note that these matrices are independent with each other
and the set 
\newline
$ \{ \mathcal{C}_1 ~ \cup ~ \mathcal{C}_2 ~ \cup ~
\mathcal{C}_{12} \}$ over the reals.

The 16 matrices $ \mathcal{C} =  \{ \mathcal{C}_1 ~ \cup ~
\mathcal{C}_2 ~ \cup ~ \mathcal{C}_{12} ~ \cup ~
\mathcal{C}_{12}^2 \}$ span the Lie algebra $u(4) ~.$  Thus, at
least in principle, we have obtained complete controllability over
the two-qubit system.
\newpage
\appendix
\begin{center}
\bf{\Large{Appendix}}
\end{center}
\section{Connection components}
\subsection{Single qubit}
\begin{gather}
A_{x_{1}} \, = \,
\left[
\begin{matrix}
- i y_1 & -(\cosh2 r_1 - e^{i \theta_1 } \sinh2r_1 ) \\
\cosh2 r_1 - e^{-i \theta_1 } \sinh2r_1 &  -i y_1
\end{matrix}
\right]         \notag \\
A_{y_{1}} \, = \,
\left[
\begin{matrix}
i x_1 & i(\cosh2 r_1 + e^{i \theta_1 } \sinh2r_1 ) \\
i(\cosh2 r_1 + e^{-i \theta_1 } \sinh2r_1) &  i x_1
\end{matrix}
\right]         \notag \\
A_{\theta_{1}} \, = \,
\left[
\begin{matrix}
1 & 0 \\
0 & 3
\end{matrix}
\right]\frac{i}{4} ( \cosh4r_1 - 1)
\quad
A_{r_{1}} \, = \,
\left[
\begin{matrix}
0&0\\0&0
\end{matrix}
\right] \notag
\end{gather}

\begin{gather}
A_{x_{2}} \, = \,
\left[
\begin{matrix}
- i y_2 & -(\cosh2 r_4  - e^{i \theta_4 } \sinh2r_4 ) \\
\cosh2 r_4 - e^{-i \theta_4 } \sinh2r_4 &  -i y_2
\end{matrix}
\right]       \notag   \\
A_{y_{2}} \, = \,
\left[
\begin{matrix}
i x_2 & i(\cosh2 r_2 + e^{i \theta_4 } \sinh2r_4 ) \\
i(\cosh2 r_4 + e^{-i \theta_4 } \sinh2r_4) &  i x_2
\end{matrix}
\right]         \notag \\
A_{\theta_{4}} \, = \,
\left[
\begin{matrix}
1 & 0 \\
0 & 3
\end{matrix}
\right]\frac{i}{4} ( \cosh4r_4 - 1)
\quad
A_{r_{4}} \, = \,
\left[
\begin{matrix}
0&0\\0&0
\end{matrix}
\right]  \notag
\end{gather}

\subsection{Two-qubit}
\begin{gather}
A_{r_{2}} =
\left[
\begin{matrix} 0 & 0 & 0 & -e^{-i\theta_2} \\
0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ e^{i\theta_2}& 0 & 0 & 0
\end{matrix}
\right]  \quad
A_{r_{3}} =
\left[
\begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & -e^{-i\theta_3} & 0 \\
0 & e^{i\theta_3} & 0 & 0 \\ 0 & 0 & 0 & 0 \\
\end{matrix}
\right] (2\cosh^2r_2 - 1) \notag
\end{gather}

\begin{gather}
A_{\theta{_2}}  \, = \,
\left[
\begin{matrix}
0 & 0 & 0 & e^{-i\theta_2} \\
0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 \\ e^{i\theta_2}& 0 & 0 & 0
\end{matrix}
\right] \frac{i}{2} \sinh2r_2 \, + \,
\left[
\begin{matrix}
1 & 0 & 0 & 0 \\
0 & 2 & 0 & 0 \\ 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & 3
\end{matrix}
\right]  \frac{i}{2} ( \cosh2r_2 - 1 ) \,   \notag \\
A_{\theta{_3}} \, = \,
\left[
\begin{matrix}
0 & 0 & 0 & 0 \\ 0 & 0 & e^{-i\theta_{3}} & 0 \\
0 & e^{i\theta{3}} & 0 & 0 \\ 0 & 0 & 0 & 0
\end{matrix}
\right]
\frac{i}{2} \cosh 2r_2 \sin 2r_3 \, + \,
\left[
\begin{matrix}
0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & 0
\end{matrix}
\right]
i \sin^2 r_3  \notag
\end{gather}
\section{Holonomy algebra}
\subsection{$u(2) \otimes {\bf 1}$}
\begin{align}
F_{x_1 y_1} \, &= \,
\left[
\begin{matrix}
0&0 \\ 0&1
\end{matrix}
\right] 4i  \quad \quad
F_{r_1 \theta_1} \, = \,
\left[
\begin{matrix}
1&0 \\ 0&3
\end{matrix}
\right] i \sinh 4 r_1 \notag \\
F_{y_1 r_1}\, &= \,
\left[
\begin{matrix}
0&-1\\
0&0 \\
\end{matrix}
\right] 2 i \big(  e^{i \theta_1} \cosh2 r_1  + \sinh2 r_1 \big)  \notag \\
&+ \quad
\left[
\begin{matrix}
0&0\\
-1&0 \\
\end{matrix}
\right]  2 i \big(  e^{-i \theta_1} \cosh2 r_1  + \sinh2 r_1 \big)  \notag \\
F_{x_1 r_1}\, &= \,
\left[
\begin{matrix}
0&-1\\
0&0 \\
\end{matrix}
\right]  2 \big( e^{i \theta_1} \cosh2 r_1  - \sinh2 r_1 \big) \notag \\
&+ \quad \left[
\begin{matrix}
0&0\\
1&0 \\
\end{matrix}
\right]  2  \big( e^{-i \theta_1} \cosh2 r_1   - \sinh2 r_1 \big) \notag
\end{align}
\begin{gather}
F_{x_1 y_1} \otimes {\bf 1} \, = \,
\left[
\begin{matrix}
0&0&0&0 \\ 0&0&0&0 \\ 0&0&1&0 \\ 0&0&0&1
\end{matrix}
\right] 4i  \quad \quad
F_{r_1 \theta_1} \otimes {\bf 1} \, = \,
\left[
\begin{matrix}
1&0&0&0 \\ 0&1&0&0 \\ 0&0&3&0 \\ 0&0&0&3
\end{matrix}
\right] i \sinh 4 r_1 \notag
\end{gather}
\begin{gather}
\begin{align}
F_{y_1 r_1} \otimes {\bf 1} \, &= \,
\left[
\begin{matrix}
0&0&-1&0 \\
0&0&0&-1 \\
0&0&0&0 \\
0&0&0&0
\end{matrix}
\right] 2 i \big(  e^{i \theta_1} \cosh2 r_1  + \sinh2 r_1 \big)  \notag  \\
&+ \quad
\left[
\begin{matrix}
0&0&0&0 \\
0&0&0&0 \\
-1&0&0&0 \\
0&-1&0&0
\end{matrix}
\right] 2 i \big( e^{-i \theta_1}  \cosh2 r_1  + \sinh2 r_1 \big)  \notag  \\
F_{x_1 r_1} \otimes {\bf 1} \, &= \,
\left[
\begin{matrix}
0&0&-1&0 \\
0&0&0&-1 \\
0&0&0&0 \\
0&0&0&0
\end{matrix}
\right] 2 \big( e^{i \theta_1} \cosh 2r_1 - \sinh2 r_1 \big) \notag  \\
&+ \quad \, \, \left[
\begin{matrix}
0&0&0&0 \\
0&0&0&0 \\
1&0&0&0 \\
0&1&0&0
\end{matrix}
\right] 2 \big( e^{-i \theta_1} \cosh 2r_1 - \sinh2 r_1 \big) \notag
\end{align}
\end{gather}

\subsection{${\bf 1} \otimes u(2) $}
\begin{align}
F_{x_2 y_2} \, &= \,
\left[
\begin{matrix}
0&0 \\ 0&1
\end{matrix}
\right] 4i  \quad \quad
F_{r_4 \theta_4} \, = \,
\left[
\begin{matrix}
1&0 \\ 0&3
\end{matrix}
\right] i \sinh 4 r_4 \notag \\
F_{y_2 r_4}\, &= \,
\left[
\begin{matrix}
0&-1\\
0&0 \\
\end{matrix}
\right] 2 i \big(  e^{i \theta_4} \cosh2 r_4  + \sinh2 r_4 \big)  \notag \\
 &\quad  + \, \,
\left[
\begin{matrix}
0&0\\
-1&0 \\
\end{matrix}
\right] 2 i \big(  e^{-i \theta_4} \cosh2 r_4  + \sinh2 r_4 \big)  \notag \\
F_{x_2 r_4}\, &= \,
\left[
\begin{matrix}
0&-1\\
0&0 \\
\end{matrix}
\right]  2  \big( e^{i \theta_4} \cosh2 r_4  - \sinh2 r_4 \big) \notag \\
&+ \quad \left[
\begin{matrix}
0&0\\
1&0 \\
\end{matrix}
\right]  2  \big( e^{-i \theta_4} \cosh2 r_4   - \sinh2 r_4 \big) \notag
\end{align}

\begin{gather}
{\bf 1} \, \otimes \,F_{x_2 y_2} \, = \,
\left[
\begin{matrix}
0&1&0&0 \\ 0&0&0&0 \\ 0&0&0&0 \\ 0&0&0&1
\end{matrix}
\right] 4i   \quad \quad
{\bf 1} \otimes  F_{r_4 \theta_4}  \, = \,
\left[
\begin{matrix}
1&0&0&0 \\
0&3&0&0 \\
0&0&1&0 \\
0&0&0&3
\end{matrix}
\right] i\sinh4r_4  \notag
\end{gather}
\begin{gather}
\begin{align}
{\bf 1} \,\otimes \,F_{y_2 r_4}  \, &= \,
\left[
\begin{matrix}
0&-1&0&0 \\
0&0&0&0 \\
0&0&0&-1 \\
0&0&0&0
\end{matrix}
\right] 2 i \big(  e^{i \theta_4} \cosh2 r_4 + \sinh2 r_4 \big)   \notag \\
&+   \quad \,\, \left[
\begin{matrix}
0&0&0&0 \\
-1&0&0&0 \\
0&0&0&0 \\
0&0&-1&0
\end{matrix}
\right]  2 i \big(  e^{- i \theta_4} \cosh2 r_4  + \sinh2 r_4 \big)  \notag  \\
{\bf 1} \, \otimes \, F_{x_2 r_4} \, &= \,
\left[
\begin{matrix}
0&-1&0&0 \\
0&0&0&0 \\
0&0&0&-1 \\
0&0&0&0
\end{matrix}
\right]   2  \big(  e^{i \theta_4} \cosh2 r_4  - \sinh2 r_4 \big) \notag  \\
&+ \quad \,\, \left[
\begin{matrix}
0&0&0&0 \\
1&0&0&0 \\
0&0&0&0 \\
0&1&0&0
\end{matrix}
\right]  2  \big(  e^{-i \theta_4} \cosh2 r_4  - \sinh2 r_4 \big) \notag
\end{align}
\end{gather}

\subsection{$su(2) \times su(2) \times u(1)$}
\begin{gather}
F_{r_{2}r_{3}} \, =  \,
\left[
\begin{matrix} 0 & 0 & 0 & 0 \\
0 & 0 & -e^{-i\theta_3} & 0 \\ 0 & e^{i\theta_3} & 0 & 0 \\ 0& 0 & 0 & 0
\end{matrix}
\right]  2 \, \sinh 2r_2   \, \, \, \,
F_{r_{2}\theta_{2}} \, =  \,
\left[
\begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 1 & 0 &  0 \\
0 &  0 & 1 & 0 \\ 0 & 0 & 0 & 2 \\
\end{matrix}
\right]
2 i \,  \sinh 2r_2 \,  \notag \\
F_{r_{2}\theta_{3}}   \, =  \,
\left[
\begin{matrix}
0 & 0 & 0 & 0 \\
0 & 0 & e^{-i \theta_{3}} & 0 \\ 0 & e^{i \theta_3}& 0 & 0 \\ 0& 0 & 0 & 0 \\
\end{matrix}
\right]  \,
i \sin 2r_3 \sinh 2r_2  \, \, \, \, \notag
F_{r_{3}\theta_{3}} \, =  \,
\left[
\begin{matrix} 0 & 0 & 0 & 0 \\ 0 & -1 & 0 &  0 \\
0 &  0 & 1 & 0 \\ 0 & 0 & 0 & 0 \\
\end{matrix}
\right]
 i \,  \sin 2 r_3 \sinh^2 2r_2 \, ,  \notag \\
D_{\frac{\partial}{\partial \theta_2}} F_{r_{2}\theta_{2}} \, =  \,
\left[
\begin{matrix}
0 & 0 & 0 &  - e^{-i\theta_2} \\ 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\ e^{i \theta_2} & 0 & 0 & 0
\end{matrix}
\right]
 2 \sinh^2 2 r_2   \notag \\
D_{\frac{\partial}{\partial r_2}} F_{r_{2}\theta_{2}} \, =  \,
\left[
\begin{matrix}
0 & 0 & 0 &  -e^{-i\theta_2} \\ 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\ -e^{i \theta_2} & 0 & 0 & 0
\end{matrix}
\right]
4i\sinh 2r_2
\, + \,
\left[
\begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 1 & 0 &  0 \\
0 &  0 & 1 & 0 \\ 0 & 0 & 0 & 2 \\
\end{matrix}
\right]
4i \cosh 2r_2 \, ,  \notag
\end{gather}
\begin{gather}
D_{\frac{\partial}{\partial \theta_2}} D_{\frac{\partial}{\partial \theta_2}}
F_{r_{2}\theta_{2}} \, =  \,
\left[
\begin{matrix}
1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1
\end{matrix}
\right]
2 i \sinh^3 r_2 \, + \,
\left[
\begin{matrix}
0 & 0 & 0 & e^{-i\theta_2} \\ 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\ e^{i \theta_2} & 0 & 0 & 0
\end{matrix}
\right]  \,
2 i \sinh^2 2r_2 \cosh 2r_2   \notag
\end{gather}

\section{Higher order brackets}
\begin{gather}
\big[\,  F_{x_1 r_1} \otimes {\bf 1} \, , \, F_{r_2 r_3} \, \big] =
\left[
\begin{matrix}
0 & -1 & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 \\
0 & 0 & 0 & 0
\end{matrix}
\right]
4 e^{i(\theta_1 + \theta_3 )} \sinh 2 r_2  \big( \cosh 2 r_1 - e^{-i \theta_1} \sinh 2r_1 \big)
\notag \\
\quad \quad \quad
\quad \quad \quad
+ \,\, \left[
\begin{matrix}
0 & 0 & 0 & 0  \\
1 & 0 & 0 & 0  \\
0 & 0 & 0 & 0 \\
0 & 0 & -1 & 0
\end{matrix}
\right]
4 e^{-i( \theta_1 + \theta_3 ) }   \sinh 2 r_2  \big( \cosh 2 r_1 - e^{i \theta_1} \sinh 2r_1 \big) \notag  \\
\big[ \, {\bf 1} \otimes F_{x_2 r_4}\, , \, F_{r_2 r_3} \, \big] =
\left[
\begin{matrix}
0 & 0 & 1 & 0  \\
0 & 0 & 0 & -1  \\
0 & 0 & 0 & 0  \\
0 & 0 & 0 & 0
\end{matrix}
\right]
 4 e^{-i (\theta_3 - \theta_4 ) } \sinh  2 r_2 \big( \cosh 2 r_4 - e^{- i \theta_4} \sinh 2r_4 \big) \notag \\
\quad \quad \quad
\quad \quad \quad
+ \, \, \left[
\begin{matrix}
0 & 0 & 0 &  0 \\
0 & 0 & 0 & 0 \\
-1 & 0 & 0 & 0  \\
0 & 1 & 0 & 0
\end{matrix}
\right]
4 e^{i (\theta_3 - \theta_4 ) } \sinh  2 r_2 \big( \cosh 2 r_4 - e^{i \theta_4} \sinh 2r_4 \big) \notag  \\
\big[\, F_{y_1 r_1} \otimes {\bf 1}    \, , \, F_{r_2 \theta_3} \, \big] =
\left[
\begin{matrix}
0 & 1 & 0 &  0 \\
0 & 0 & 0 &  0 \\
0 & 0 & 0 &  -1 \\
0 & 0 & 0 &  0
\end{matrix}
\right]
2e^{i ( \theta_1 + \theta_3 ) } \sin 2 r_3 \sinh 2 r_2 \big( \cosh 2 r_1 + e^{-i \theta_1} \sinh 2 r_1 \big) \notag \\
\quad \quad \quad
\quad \quad \quad
+ \left[
\begin{matrix}
0  & 0 & 0 & 0 \\
-1 & 0 & 0 & 0 \\
0  & 0 & 0 & 0 \\
0  & 0 & 1 & 0
\end{matrix}
\right]
2e^{- i ( \theta_1 + \theta_3 )} \sin 2r_3 \sinh 2r_2 \big( \cosh 2 r_1 + e^{i \theta_1} \sinh 2 r_1 \big) \notag
\\
\big[\,  {\bf 1} \otimes F_{y_2 r_4} \, , \, F_{r_2 \theta_3} \, \big] =  \notag
\left[
\begin{matrix}
0 & 0  & 1 & 0  \\
0 & 0  & 0 & -1  \\
0 & 0  & 0 & 0  \\
0 & 0  & 0 & 0
\end{matrix}
\right]
2 e^{- i ( \theta_3  - \theta_4 ) } \sin 2 r_3  \sinh 2 r_2  \big( \cosh 2 r_4 + e^{- i  \theta_4 }\sinh 2 r_4 \big) \notag \\
\quad \quad \quad
\quad \quad \quad
+ \left[
\begin{matrix}
 0 & 0 & 0  & 0 \\
 0 & 0 & 0  & 0 \\
-1 & 0 & 0  & 0 \\
 0 & 1 & 0  & 0
\end{matrix}
\right]
2 e^{i ( \theta_3  - \theta_4 ) } \sin 2 r_3  \sinh 2 r_2  \big( \cosh 2 r_4 + 
e^{i \theta_4 }\sinh 2 r_4 \big) \notag 
\end{gather}
\begin{gather}
\Big[{\bf 1 } \otimes F_{y_2 r_4}     \, , \, \big[     F_{x_1 r_1} \otimes {\bf 1 } \, ,
 \, F_{r_2 r_3} \big] \, \Big] =
\left[
\begin{matrix}
-1 &  0 & 0  & 0 \\
0  &  1 & 0  & 0 \\
0  &  0 & 1  & 0 \\
0  &  0 & 0  & -1
\end{matrix}
\right] \delta \notag \\
\delta =  16 i \sinh 2 r_2 \Big( - \sinh 2 r_1 \big( \cos(\theta_3 - \theta_4) \cosh 2 r_4 + \cos \theta_3 \sinh 2 r_4 \big)  \notag \\
 \quad \quad \quad  + \, \, \cosh 2 r_1 \big( \cos ( \theta_1 + \theta_3 - \theta_4) \cosh 2 r_4 + \cos(\theta_1 + \theta_3) \sinh 2 r_4 \big) \Big) \notag
\end{gather}

\vspace{12pt}

\noindent {\it Acknowledgements} :  I would like to thank SML and
MKM without whose cooperation this work would not have been
possible.

\begin{thebibliography}{99}
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(Singapore: World Scientific)
\bibitem{kelly} S. Kelly and R. Murray 1995 J. Robotic Syst. {\bf 12} 417
\bibitem{koon} W. Koon and J. Marsden 1997 SIAM J. Cont. Opt. {\bf 35} 901
\bibitem{mont} R. Montgomery 2002 \emph{A Tour of Subriemannian Geometries, Their Geodesics, and
Applications}, (Providence: American Mathematical Society)
\bibitem{zan} P. Zanardi and M. Rasetti 1999 Phys. Lett. A {\bf 264} 94 quant-ph/9904011
\bibitem{luc} D. Lucarelli {\it Chow's theorem and universal holonomic quantum computation}
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\bibitem{pach2} J. Pachos and P. Zanardi 2001 Int. J. Mod. Phys. B {\bf 15} 1257 quant-ph/0007110
\bibitem{fujii1} K. Fujii 2000 J. Math. Phys. {\bf 41} 4406 quant-ph/9910069
\bibitem{fujii2} K. Fujii 2001 Rep. Math. Phys. {\bf 48} 75 quant-ph/0004102
\bibitem{fujii3} K. Fujii 2001 {\it Mathematical foundations of holonomic quantum computer II}
quant-ph/0101102
\bibitem{chuang} M. Nielsen and I. Chuang, 2000 \emph{Quantum Computation and Quantum Information}
(New York: Cambridge University Press)
\bibitem{kn} S. Kobayashi and K. Nomizu 1969 {\it Foundations of Differential Geometry} vol. 1
(New York: Wiley Interscience)
\bibitem{found} K. Marathe and G. Martucci, 1991
\emph{The Mathematical Foundations of Gauge Theories} (Amsterdam: North Holland)
\bibitem{nak} M. Nakahara 1990 {\it Geometry, Topology, and Physics} (Bristol: IOP Publishing Ltd.)
\bibitem{ozeki} H. Ozeki 1956  Nagoya Math. J. {\bf 10}  106.
\bibitem{zee} F. Wilczek and A. Zee 1984 Phys. Rev. Lett. {\bf 52} 2111
\bibitem{pach3} J. Pachos 2000 {\it Quantum computation by geometrical means} quant-ph/0003150
\end{thebibliography}
\end{document}

quant-ph0202055.extracted_bib
\begin{thebibliography}{99}
\bibitem{phase} A. Shapere and F. Wilczek eds. 1989 {\it Geometric Phases in Physics}
(Singapore: World Scientific)
\bibitem{kelly} S. Kelly and R. Murray 1995 J. Robotic Syst. {\bf 12} 417
\bibitem{koon} W. Koon and J. Marsden 1997 SIAM J. Cont. Opt. {\bf 35} 901
\bibitem{mont} R. Montgomery 2002 \emph{A Tour of Subriemannian Geometries, Their Geodesics, and
Applications}, (Providence: American Mathematical Society)
\bibitem{zan} P. Zanardi and M. Rasetti 1999 Phys. Lett. A {\bf 264} 94 quant-ph/9904011
\bibitem{luc} D. Lucarelli {\it Chow's theorem and universal holonomic quantum computation}
quant-ph/0111078
\bibitem{pach1} J. Pachos and S. Chountasis 2000 Phys. Rev. A {\bf 62} 052318 quant-ph/9912093
\bibitem{pach2} J. Pachos and P. Zanardi 2001 Int. J. Mod. Phys. B {\bf 15} 1257 quant-ph/0007110
\bibitem{fujii1} K. Fujii 2000 J. Math. Phys. {\bf 41} 4406 quant-ph/9910069
\bibitem{fujii2} K. Fujii 2001 Rep. Math. Phys. {\bf 48} 75 quant-ph/0004102
\bibitem{fujii3} K. Fujii 2001 {\it Mathematical foundations of holonomic quantum computer II}
quant-ph/0101102
\bibitem{chuang} M. Nielsen and I. Chuang, 2000 \emph{Quantum Computation and Quantum Information}
(New York: Cambridge University Press)
\bibitem{kn} S. Kobayashi and K. Nomizu 1969 {\it Foundations of Differential Geometry} vol. 1
(New York: Wiley Interscience)
\bibitem{found} K. Marathe and G. Martucci, 1991
\emph{The Mathematical Foundations of Gauge Theories} (Amsterdam: North Holland)
\bibitem{nak} M. Nakahara 1990 {\it Geometry, Topology, and Physics} (Bristol: IOP Publishing Ltd.)
\bibitem{ozeki} H. Ozeki 1956  Nagoya Math. J. {\bf 10}  106.
\bibitem{zee} F. Wilczek and A. Zee 1984 Phys. Rev. Lett. {\bf 52} 2111
\bibitem{pach3} J. Pachos 2000 {\it Quantum computation by geometrical means} quant-ph/0003150
\end{thebibliography}