\section{Introduction}

\section{Clifford algebras.}

\subsection{Preliminaries.}

Generally, Clifford algebra is defined \cite{ClDir} for linear vector space
$V$ with arbitrary quadratic form $Q(\mbi x)$ as some algebra $A$ with map
\EQ{\alpha: V \to A}, \EQ{\alpha(\mbi x)^2 = -Q(\mbi x){\qe0}} (here ${\qe0}$ is
{\em unit} of the algebra), but let us consider first Euclidean case with
\EQ{Q(\mbi x)=x_1^2+\dots+x_n^2}. The Clifford algebra, \EQ{\Cl(n)},
is generated by $n$ elements \EQ{{\mbi{e}}_1,\dots,{\mbi{e}}_n} with property:
\begin{equation}
 {\mbi{e}}_i {\mbi{e}}_j + {\mbi{e}}_j {\mbi{e}}_i = -2\delta_{ij}; \implies
 {\mbi{e}}_i {\mbi{e}}_j = - {\mbi{e}}_j {\mbi{e}}_i \ (i \neq j),\ \  {\mbi{e}}_k^2 = -{\qe0}
 \label{ClEu}
\end{equation}
The eq.(\ref{ClEu}) defines $2^n$-dimensional real algebra. It is clear,
because product of any number of ${\mbi{e}}_i$ can be
simplified to product\footnote{Let the product is ${\qe0}$ for $k=0$}
with up to $n$ different ${\mbi{e}}_i$:
\begin{equation}
 {\mbi{e}}_{i_1}{\mbi{e}}_{i_2} \cdots {\mbi{e}}_{i_k};\quad 0 \leq k \leq n;
 \quad i_1 < i_2 < \dots < i_k
 \label{compos}
\end{equation}
There are $2^n$ such terms because every combination corresponds to
$n$-digits binary number with units in positions \EQ{i_1,i_2,\dots,i_k} :
\[
\underbrace{\underdots{\underbrace{\underbrace{%
00\dots01}_{i_1}00\dots01\vrule}_{i_2}0\dots0}1\vrule}_{i_k}00\dots00
\]

Elements of vector space $V$ map to $n$-dimensional subspace of the algebra:
\EQ{\mbi{v} = \sum_{i=1}^n v_i {\mbi{e}}_i}. Elements with unit norm \EQ{\|\mbi v\|=1}
\footnote{More generally \EQ{|Q(\mbi v)|=1}} have inverse, (for \EQ{\Cl(n)},
\EQ{\mbi v^{-1} = -\mbi v} because \EQ{-\mbi v{\cdot}\mbi v = {\qe0}}) and product of
even number of such elements is some group. It is called {\em spinor group}
\EQ{Spin(n)} and it has 2-1 isomorphism with group of
$n$-dimensional rotations, $SO(n)$.

\smallskip

In the definition of spinor group is used only multiplication of {\em even}
number of elements of algebra \EQ{\Cl(n)}. It is possible to build $2^{n-1}$
different compositions eq.(\ref{compos}) with even number of generators of
\EQ{\Cl(n)} and all such compositions produce {\em even subalgebra} \EQ{\Cl^e(n)}
of Clifford algebra and really spinor group is subspace of the smaller algebra.
But it can be shown that \EQ{\Cl^e(n)} is isomorphic with \EQ{\Cl(n-1)} and the
algebra also may be used for representation of \EQ{Spin(n)} group. The property
is mentioned here to explain why it is important sometime to consider Clifford
algebras with dimension reduced by unit, for example \EQ{\Cl(2)} is enough to
build spinor group for 3D rotations.

\medskip

It is possible also to introduce yet another algebra, \EQ{\Cl_+(n)}, if to
change sign of square of all ${\mbi{e}}_k$ to $+{\qe0}$ :
\begin{equation}
 {\mbi{e}}_i {\mbi{e}}_j + {\mbi{e}}_j {\mbi{e}}_i = 2\delta_{ij}; \implies
 {\mbi{e}}_i {\mbi{e}}_j = - {\mbi{e}}_j {\mbi{e}}_i \ (i \neq j),\ \  \underline{{\mbi{e}}_k^2 = {\qe0}}
\end{equation}
The two real algebras are not equivalent.

Clifford algebra \EQ{\Cl(m,l)} corresponds to more general, pseudo-Euclidean
case (\EQ{{{\mbi{e}}_k}^2 = \pm {\qe0}}):
\begin{equation}
 {\mbi{e}}_i {\mbi{e}}_j = - {\mbi{e}}_j {\mbi{e}}_i \ (i \neq j), \quad
 {\mbi{e}}_k^2 = -{\qe0} \ (k \leq m), \quad
 {\mbi{e}}_k^2 = {\qe0} \ (k > m)
\end{equation}
Two previous examples are special cases of the definition:
\EQ{\Cl(n) = \Cl(n,0)}, \EQ{\Cl_+(n) = \Cl(0,n)}.

It is possible also to consider `degenerated' case ( \EQ{Q(x) \equiv 0 } ) :
\begin{equation}
 {\mbi{e}}_i {\mbi{e}}_j + {\mbi{e}}_j {\mbi{e}}_i = 0 \quad \forall i,j \ ({\mbi{e}}_k^2=0)
\end{equation}
It is known also as {\em Grassmann algebra} $\Lambda^n$ (with notation
\EQ{a \mathbin\land b} is used instead of \EQ{a\,b}, \EQ{{\mbi{e}}_i\mathbin\land{\mbi{e}}_j = -{\mbi{e}}_j\mathbin\land{\mbi{e}}_i}) and usually
it is considered separately. One of applications of Grassmann algebras is
related with description of subspaces of $n$-dimensional vector space.

\medskip

It was real Clifford algebras and the complexification of any algebra
\EQ{\Cl(l,n-l)} produces the same universal Clifford algebra \EQ{\Cl(n,\C)}
because it is always possible to `correct' sign of any ${\mbi{e}}_k^2$ by
substitution \EQ{\tilde {\mbi{e}}_k = i {\mbi{e}}_k}, \EQ{\tilde{\mbi{e}}_k^2 = -{\mbi{e}}_k^2} .

Let us use notation ${\mbi{e}}_k$ for elements of basis with \EQ{{\mbi{e}}_k^2 = -{\qe0}} and
${\stackrel{_+}{\e}}{}_k$ for \EQ{{\stackrel{_+}{\e}}{}_k^2 = {\qe0}}, \EQ{{\stackrel{_+}{\e}}{}_k = i{\mbi{e}}_k}.

\noindent ---

 The following constructions of Grassmann algebras are used further:
Let us consider even-dimensional
universal Clifford algebra \EQ{\Cl(2n,\C)}. It is possible to build a
Grassmann algebra $\Lambda^n$ with using $n$-generators:
\begin{equation}
 \mbi d_l = {\mbi{e}}_{2l} + i {\mbi{e}}_{2l+1}; \implies
 \mbi d_k \mbi d_j = -\mbi d_j \mbi d_k \quad \forall k,j \quad (\mbi d_k^2=0)
\end{equation}

It is possible also to define two different algebras: $\Lambda^n_{_+}$
with generators~$a_l$ and $\Lambda^n_{_-}$ with $a_l^\dag$:
\begin{equation}
 \a_l = \frac{{\stackrel{_+}{\e}}{}_{2l} + i {\stackrel{_+}{\e}}{}_{2l+1}}{2} \,, \quad
 \a^\dag_l = \frac{{\stackrel{_+}{\e}}{}_{2l} - i {\stackrel{_+}{\e}}{}_{2l+1}}{2}
\label{aadag}
\end{equation}
with properties:
\begin{equation}
 \fbox{$\quad%
 \{ \a_i, \a_j \} = 0,\quad \{ \a^\dag_i, \a^\dag_j \} = 0, \quad
 \{ \a^\dag_i, \a_j \} = \delta_{ij}
 \quad$}
\label{aacomm}
\end{equation}
where \EQ{\{a,b\} \equiv (a\,b + b\,a)}.
A nontrivial case is
\EQ{\{\a^\dag_l,\a_l\} = {\qe0}} also can be simply checked:
\begin{align*}
  4\,\a_l\,\a^\dag_l & = ({\stackrel{_+}{\e}}{}_{2l} + i {\stackrel{_+}{\e}}{}_{2l+1})({\stackrel{_+}{\e}}{}_{2l} - i {\stackrel{_+}{\e}}{}_{2l+1}) = \\
& = {\stackrel{_+}{\e}}{}_{2l}^2 + i {\stackrel{_+}{\e}}{}_{2l+1}{\stackrel{_+}{\e}}{}_{2l} - i {\stackrel{_+}{\e}}{}_{2l}{\stackrel{_+}{\e}}{}_{2l+1} + {\stackrel{_+}{\e}}{}_{2l+1}^2
 = 2 - 2i\,{\stackrel{_+}{\e}}{}_{2l}{\stackrel{_+}{\e}}{}_{2l+1} \\
  4\,\a^\dag_l\,\a_l & = ({\stackrel{_+}{\e}}{}_{2l} - i {\stackrel{_+}{\e}}{}_{2l+1})({\stackrel{_+}{\e}}{}_{2l} + i {\stackrel{_+}{\e}}{}_{2l+1}) = \\
&= {\stackrel{_+}{\e}}{}_{2l}^2 - i {\stackrel{_+}{\e}}{}_{2l+1}{\stackrel{_+}{\e}}{}_{2l} + i {\stackrel{_+}{\e}}{}_{2l}{\stackrel{_+}{\e}}{}_{2l+1} + {\stackrel{_+}{\e}}{}_{2l+1}^2
 = 2 + 2i\,{\stackrel{_+}{\e}}{}_{2l}{\stackrel{_+}{\e}}{}_{2l+1}
\end{align*}

\subsection{Two-dimensional case.}


Let us consider real Clifford algebras first. The 4D real algebra \EQ{\Cl(2)} is
described by elements \EQ{{\qe1} = {\mbi{e}}_1}, \EQ{{\qe2} = {\mbi{e}}_2},
\EQ{{\qe3} = {\mbi{e}}_1 {\mbi{e}}_2} and unit element ${\qe0}$. The elements satisfy relations:
\EQ{{\qe1}^2 \stackrel{\text{def}}= -{\qe0}}, \EQ{{\qe2}^2 \stackrel{\text{def}}= -{\qe0}}, \EQ{-{\qe2}{\qe1} \stackrel{\text{def}}= {\qe1}{\qe2} \stackrel{\text{def}}= {\qe3}},
\EQ{{\qe3}^2 = {\qe1}{\qe2}{\qe3} = -{\qe0}}, \EQ{{\qe1}{\qe3} = {\qe1}\I{\qe2} = -{\qe2} = {\qe2}{\qe1}\I = -{\qe3}{\qe1}},
\EQ{{\qe2}{\qe3} = -{\qe2}\J{\qe1} = {\qe1} = -{\qe1}{\qe2}\J = -{\qe3}{\qe2}} and the equations define well known
4D {\em quaternions algebra} $\H$
introduced by Hamilton at 1843 as a generalization of complex numbers.

\smallskip

Another 4D real Clifford algebra \EQ{\Cl_+(2)} is generated by elements ${\me1}$,
${\me2}$, \EQ{{\me3} = {\me1}{\me2}} with defining relations are:  \EQ{{\me1}^2 \stackrel{\text{def}}= {\qe0}},
\EQ{{\me2}^2 \stackrel{\text{def}}= {\qe0}}, \EQ{{\me1}{\me2} \stackrel{\text{def}}= -{\me2}{\me1} \stackrel{\text{def}}= {\me3}} and with implications:
\EQ{{\me3}^2 = {\me1}{\me2}{\me3} = -{\qe0}}, ${\me2}{\me3} = {\me2}\V{\me1} = {\me1} = {\me1}{\me2}\V = -{\me3}{\me2}$,
${\me1}{\me3} = -{\me1}\U{\me2} = -{\me2} = -{\me2}{\me1}\U= -{\me3}{\me1}$.

It is possible to consider ${\me2}$ and ${\me3}$ as generators of some new Clifford
algebra and because \EQ{{\me3}^2 = -{\qe0}} it is \EQ{\Cl(1,1)} and so the algebra is
isomorphic with \EQ{\Cl_+(2) \equiv \Cl(0,2)}.

The \EQ{\Cl_+(2)} is isomorphic with 4D algebra \EQ{\mR{2}} of all \EQ{2\times2} real
matrices. It is enough to choose: \EQ{{\me1} = \smx (0110)}, \EQ{{\me2} = \smx (100{-1})},
\EQ{{\me3} = {\me1}{\me2} =-{\me2}{\me1} = \smx (0{-1}10)} and \EQ{{\qe0} = \smx (1001)} is unit matrix.

\medskip

The complexification of the algebras, \EQ{\Cl(2,\C)} is isomorphic with
algebra \EQ{\mC{2}} of all complex \EQ{2\times2} matrices and representation
by Pauli matrices is: \EQ{{\stackrel{_+}{\e}}{}_1 = \sgm_{\!x}}, \EQ{{\stackrel{_+}{\e}}{}_2 = \sgm_{\!y}}, \EQ{{\stackrel{_+}{\e}}{}_1{\stackrel{_+}{\e}}{}_2 = i\sgm_{\!z}},
\begin{equation}
 \sgm_{\!x} = \mx(0110), \quad \sgm_{\!y} = \mx(0{-i}i0), \quad \sgm_{\!z} = \mx(100{-1}).
\end{equation}
with standard relations \EQ{\sgm_{\!x}^2=\sgm_{\!y}^2=\sgm_{\!z}^2={\qe0}}, \EQ{\sgm_{\!x}\sgm_{\!y}=i\sgm_{\!z}=-\sgm_{\!y}\sgm_{\!x}},
$\sgm_{\!z}\sgm_{\!x} = i\sgm_{\!y} = -\sgm_{\!x}\sgm_{\!z}$, \EQ{\sgm_{\!y}\sgm_{\!z} = i\sgm_{\!x} = -\sgm_{\!z}\sgm_{\!y}}.

\smallskip

The real Clifford algebras \EQ{\Cl(2) \equiv \Cl(2,0) \iso \H} and
$\Cl_+(2) \equiv \Cl(0,2) \iso \Cl(1,1) \iso \mR{2}$ are subalgebras
of \EQ{\Cl(2,\C) \iso \mC{2}}:
\begin{align}
 \Cl_+(2) \MapsIn \Cl(2,\C):&\quad {\me1} \to \sgm_{\!x},\ {\me2} \to \sgm_{\!z},\ {\me3} \to \sgm_{\!y}/i
 \label{RPa}\\
 \Cl(2) \MapsIn \Cl(2,\C):&\quad {\qe1} \to i\sgm_{\!x},\ {\qe2} \to i\sgm_{\!y},\ {\qe3} \to i\sgm_{\!z}
 \label{quPa}
\end{align}
Really, the eq.(\ref{quPa}) corresponds to usual representation of quaternions
by complex \EQ{2\times2} matrices: \EQ{{\qe1} = \smx (0ii0)}, \EQ{{\qe2} = \smx (01{-1}0)},
\EQ{{\qe3} = \smx (i00{-i})}, but it is useful sometime to do not use some
particular matrix notation and consider quaternions as an {\em abstract}
linear algebra defined by relation with ${\qe1}$, ${\qe2}$ and ${\qe3}$ introduced
earlier:
\begin{equation}
  {\qe1}^2 = {\qe2}^2 = -{\qe0},\ {\qe1}\cdot{\qe2}=-{\qe2}\cdot{\qe1} = {\qe3};
  \ q_0{\qe0}+q_1{\qe1}+q_2{\qe2}+q_3{\qe3} \in \H;\ (q_k \in \R)
\end{equation}
For example, such definition makes more clear difference between 4D {\em real}
algebra of quaternions and 4D complex Pauli algebra.

It should be mentioned, that the Pauli algebra can be considered also as 8D
{\em real} Clifford algebra \EQ{\Cl_+(3)} with relations: \EQ{\sgm_{\!x}^2=\sgm_{\!y}^2=\sgm_{\!z}^2={\qe0}}.
An element \EQ{\mathbf{\dot{\Id}} \equiv \sgm_{\!x}\sgm_{\!y}\sgm_{\!z}} of the algebra commutes with any
other element and corresponds to \EQ{i{\qe0}} of complex algebra \EQ{\Cl(2,\C)}.

\subsection{Constriction of \EQ{\Cl(2n,\C)}.}

Let us recall that for $n$-dimensional linear space $V$ with basis
\EQ{e_1,\dots,e_n} and $m$-dimensional linear space $W$ with basis
\EQ{e'_1,\dots,e'_m} the {\em tensor product} \EQ{V \ox W} of the spaces
can be introduced as \EQ{m{\cdot}n}-dimensional space with basis: \EQ{e_i\ox e'_j}.

Similarly, for $n$-dimensional linear space $V$ with basis \EQ{e_1,\dots,e_n}
the {\it tensor power} \EQ{V^{{\ox}k}} of $k$ copies of such space
can be introduced as $n^k$-dimensional space with basis:
\EQ{e_{i_1}\ox e_{i_2}\ox\dots\ox e_{i_k}}, $1 \leq i_l \leq n$.

If the linear spaces have structure of some algebras $A$ and $B$ it is
possible to build new \EQ{nm}-dimensional algebra \EQ{A \ox B} with composition is
defined as: \EQ{(a \ox b) \cdot (c \ox d) \equiv (ac) \ox (bd)}
for elements of basis and extended to all elements by distributivity. The
$n^k$ dimensional tensor power \EQ{A^{{\ox}k}} is defined analogously.

\medskip

Let us show direct construction of \EQ{\Cl(2n,\C)} as algebra \EQ{\mC{2^n}} of all
\EQ{2^n\times2^n} matrices. It can be done by two steps with using Pauli
matrices. First step is isomorphism \EQ{\Cl(2n,\C) \iso \mC{2}^{{\ox}n}} and
second is \EQ{\mC{2}^{{\ox}n} \iso \mC{2^n}}.

It is possible to show first isomorphism by direct construction of \EQ{2n}
generators of \EQ{\Cl(2n,\C)} :
\begin{equation}
\begin{split}
 {\stackrel{_+}{\e}}{}_{2k} = &
 \underbrace{{\qe0}\ox\dots\ox{\qe0}}_{n-k-1}\ox\sgm_{\!x}\ox\underbrace{\sgm_{\!z}\ox\dots\ox\sgm_{\!z}}_k
 \\ {\stackrel{_+}{\e}}{}_{2k+1} = &
 \underbrace{{\qe0}\ox\dots\ox{\qe0}}_{n-k-1}\ox\sgm_{\!y}\ox\underbrace{\sgm_{\!z}\ox\dots\ox\sgm_{\!z}}_k
\end{split}
\label{genClnnC}
\end{equation}
To check that all the \EQ{2n} generators are anticommutative it is possible to
introduce the construction inductively. For \EQ{n=1} the \EQ{\Cl(2,\C)} has two
generators $\sgm_{\!x}$ and $\sgm_{\!y}$. If the suggestion is shown for some \EQ{l \geq 1}
and we have \EQ{2l} generators ${\stackrel{_+}{\e}}{}_1,\dots,{\stackrel{_+}{\e}}{}_{2l} \in \mC{2}^{{\ox}l} \iso
\Cl(2l,\C)$ then for \EQ{l+1} it is also possible to choose \EQ{2l+2} generators
in \EQ{\mC{2}^{{\ox}l+1}}: {\em i.e.} \EQ{2l} anticommutative elements
\EQ{{\stackrel{_+}{\e}}{}_1\ox\sgm_{\!z},\dots,{\stackrel{_+}{\e}}{}_{2l}\ox\sgm_{\!z}} together with two elements
\EQ{{\qe0}^{{\ox}l}\ox\sgm_{\!x}} and \EQ{{\qe0}^{{\ox}l}\ox\sgm_{\!y}}, where
\EQ{{\qe0}^{{\ox}l} = {\underbrace{{\qe0}\ox\dots\ox{\qe0}}_l}}
is unit of \EQ{\Cl(2l,\C)}. So \EQ{\Cl(2l+2,\C) \iso \mC{2}^{{\ox}l+1}}.

\medskip

The isomorphism of \EQ{\mC{2}^{{\ox}n} \iso \mC{2^n}} can be also shown by
induction. It is true for \EQ{n=1} and if it is proven for some \EQ{l \geq 1}
then any \EQ{A \in \mC{2}^l \iso \mC{2^l}} is represented by some
\EQ{2^l\times2^l} matrix \EQ{\Mat A}. It is possible to check that
\EQ{\mC{2^{l+1}} \iso \mC{2} \ox \mC{2^l}}, if to define operation:
\begin{equation}
\begin{split}
 &\Mat a = \mx({a_{11}}{a_{12}}{a_{21}}{a_{22}}) \in \mC{2},
  \qquad \Mat A \in \mC{2^l},  \\
 &\Mat a \mathbin{\stackrel{_\mathrm{mat}}{\ox}} \Mat A \equiv
  \mx\lgroup{(a_{11}\Mat A)}{(a_{12}\Mat A)}
            {(a_{21}\Mat A)}{(a_{22}\Mat A)}\rgroup  \in \mC{2^{l+1}}
\end{split}
\label{MatOx}
\end{equation}
It can be checked directly,
\EQ{(\Mat{a\mathbin{\stackrel{_\mathrm{mat}}{\ox}} A}) \cdot (\Mat{b\mathbin{\stackrel{_\mathrm{mat}}{\ox}} B}) = \Mat{ab\mathbin{\stackrel{_\mathrm{mat}}{\ox}} AB}} if $\Mat a,\Mat b$
are basis elements: ${\qe0},\sgm_{\!x},\sgm_{\!y},\sgm_{\!z}$ and it is extended for all elements by
distributivity.

\section{Quantum gates.}

Any quantum $n$-gate is described by some unitary \EQ{2^n\times2^n}
matrix. It acts on $n$-qubits in $2^n$-dimensional complex vector space.

Matrices of all \EQ{2^{2n}=4^n} possible compositions of \EQ{2n} generators in
eq.(\ref{genClnnC}) are unitary. Really, any such element has form:
\begin{equation}
 {\mbi{e}}_{l_1}{\mbi{e}}_{l_2}\dotsm{\mbi{e}}_{l_k} \xrightarrow{0 \leqslant k \leqslant 2n}
 {\mbi{e}}_{\mbi I} \equiv {\mbi\sigma}_{\!i_1}\ox{\mbi\sigma}_{\!i_2}\ox\dots\ox{\mbi\sigma}_{\!i_n}
\label{composC}
\end{equation}
where \EQ{{\mbi\sigma}_{\!i_k}} is one of four basis elements: \EQ{{\qe0},\sgm_{\!x},\sgm_{\!y},\sgm_{\!z}}. In matrix
representation used here \EQ{{\mbi\sigma}_i^* = {\mbi\sigma}_i}, \EQ{{\mbi\sigma}_i^2 = {\qe0}}. For any
\EQ{{\mbi\sigma}_i} and matrix \EQ{\Mat A^* = \Mat A} the eq.(\ref{MatOx}) implies
\EQ{({\mbi\sigma}_i\mathbin{\stackrel{_\mathrm{mat}}{\ox}}\Mat A)^* = ({\mbi\sigma}_i\mathbin{\stackrel{_\mathrm{mat}}{\ox}}\Mat A)} and so
\EQ{{\mbi{e}}_{\mbi I}^* = {\mbi{e}}_{\mbi I}}. Because also \EQ{{\mbi{e}}_{\mbi I}^2 = {\qe0}}, all
the matrices for \EQ{4^n} elements satisfy equations:
\EQ{{\mbi{e}}_{\mbi I}^* = {\mbi{e}}_{\mbi I}= {\mbi{e}}_{\mbi I}^{-1}}, they
are not only Hermitian (\EQ{\Mat A^* = \Mat A}), but unitary too
(\EQ{\Mat A^* = \Mat A^{-1}}).

Such construction of unitary matrices may be not very general, because
it produces finite number of gates, but it is possible to consider some
analogous of `Hamiltonian approach', if to use Hermitian matrices
$\Mat H$ and then \EQ{\Mat U = \exp (i\,\Mat H)} is unitary. It is useful
because whole \EQ{4^n} dimensional space of Hermitian matrices is
produced by combinations up to $4^n$ elements ${\mbi{e}}_{\mbi I}$ described
by eq.(\ref{composC}) with real coefficients:
\begin{equation}
 \mathcal H = \left\{ \Mat H \ : \ \Mat H =
 \sum\nolimits_{\mbi I}{c_{\mbi I} {\mbi{e}}_{\mbi I}};
 \quad c_{\mbi I} \in \R \right\}
\end{equation}
\EQ{\Mat H \in \mathcal H}, \EQ{\dim \mathcal H = 4^n}.

\medskip

It is necessary also to define action of an element of $\Cl(2n,\C)\iso\mC{2^n}$
on $n$-qubit register as an action of \EQ{2^n\times2^n} matrix on complex
vector in $2^n$-dimensional complex space. Let us define it for basis
of $2^n$ vector space
\begin{equation}
 e_L = e_{l_1}\ox e_{l_2}\ox\dots\ox e_{l_n},\quad l_k = 0,1
\end{equation}
or with notation used in quantum information science
\begin{gather}
 \ket{L} = \ket{l_1\,l_2 \dots l_n} =
 \ket{l_1}\ox \ket{l_2}\ox\dots\ox \ket{l_n},\quad l_k = 0,1
 \label{nqubasis}\\
 e_0=\ket{0} = \Mx({1\\0}),\quad e_1=\ket{1} = \Mx({0\\1})
 \label{qubasis}
\end{gather}
and $4^n$-dimensional basis eq.(\ref{composC}) of
Clifford algebra:
\begin{equation}
 \mbi{g}_I = {\mbi\sigma}_{\!i_1}\ox {\mbi\sigma}_{\!i_2}\ox\dots\ox {\mbi\sigma}_{\!i_n},
 \quad {\mbi\sigma}_{\!i_k} \in {\qe0},\sgm_{\!x},\sgm_{\!y},\sgm_{\!z}
\end{equation}
The action is defined for the elements of basis component-wise:
\begin{equation}
 \mbi{g}_I \ket{L} = \bigl({\mbi\sigma}_{\!i_1} \ket{l_1}\bigr) \ox
  \bigl({\mbi\sigma}_{\!i_2} \ket{l_2}\bigr)\ox
  \dots\ox \bigl({\mbi\sigma}_{\!i_n} \ket{l_n}\bigr)
 \label{eIeL}
\end{equation}
where \EQ{{\mbi\sigma}_{\!i_k} \ket{l_k}} is action of \EQ{2\times2} complex
matrix ${\mbi\sigma}$ on 2D complex vector $\0$ or $\1$. The action of arbitrary
element (gate) to a general vector ({\em entangled} state) is produced from
eq.(\ref{eIeL}) by linearity:
\begin{equation}
 \mbi G \ket{\psi} =
 \sum_I{a_I\mbi{g}_I}\, \sum_L{c_L\ket{L}} =
 \sum_{I,L}{(a_I c_L)\,\mbi{g}_I\ket{L}}
\end{equation}
\medskip


 So, any linear transformation of $2^n$-dimensional space can be represented
by elements of Clifford algebra with $2n$ generators
\begin{equation}
 \Cl(2n,\C) \iso \mC{2^n} : \C^{2^n} \to \C^{2^n}
\end{equation}
To build particular matrix for some quantum $n$-gate it is possible to
start with simple $2\times2$ ${\mbi\sigma}$-matrices for $\Cl(2,\C)$ and qubit
in basis eq.(\ref{qubasis}). After it, recursively $1,\dots,n$ on each
step with using eq.(\ref{MatOx}) it is considered matrices and vectors
with doubled size:
\begin{equation}
  \bigl(\square\bigr)\cdot\ket{...} \xrightarrow{k \to k+1}
  \mx\lgroup{(\square)}{(\square)}
            {(\square)}{(\square)}\rgroup \!\cdot
  \Mx|{\!\bigl(\ket{0...}\bigr)\!\! \\ \!\bigl(\ket{1...}\bigr)\!\!}>
\end{equation}


\medskip

For example of application the theory of Clifford algebras to quantum
information science let us use now other basis with dual Grassmann
elements introduced by eq.(\ref{aadag},\ref{aacomm}) to describe approach
with occupation numbers used by Feynman \cite{feynman:simul,feynman:comp}.

Let us apply eq.(\ref{aadag}) to expressions for Clifford generators
eq.(\ref{genClnnC}):
\begin{equation}
\begin{split}
 \a_k = & {\qe0}^{{\ox}n-k-1}\ox\frac{\sgm_{\!x}+i\sgm_{\!y}}2\ox\sgm_{\!z}^{{\ox}k} \\
 \a^\dag_k = & {\qe0}^{{\ox}n-k-1}\ox\frac{\sgm_{\!x}-i\sgm_{\!y}}2\ox\sgm_{\!z}^{{\ox}k}
\end{split}
\label{GenFeym}
\end{equation}

The matrices \EQ{\a = \frac{\sgm_{\!x}+i\sgm_{\!y}}2 = \smx (0100)} and
\EQ{\a^\dag = \frac{\sgm_{\!x}-i\sgm_{\!y}}2 = \smx (0010)} correspond to {\em annihilation}
and {\em creation} operators used by Feynman for description of quantum
bit and commutation relations in eq.(\ref{aacomm}) correspond to correct
extension of the operators for $n$ fermions.

\medskip

The approach is analogue of {\em second quantization} and some
advantages of the paradigm are discussed already in \cite{RQC}.

On the other hand, due to isomorphism \EQ{\Cl(2n,\C) \iso \mC{2^n}} described
above, the same elements produce basis for description of quantum gates by
\EQ{2^n\times2^n} complex matrices used in S-matrix approach originated
by Deutsch \cite{deutsch:gates} {\em et al}.

\smallskip

The operator $\a$ of annihilation and $\a^\dag$ of creation can be also
written as (here {\bf 0} is zero vector):
\begin{align}
 &\a = \ktbr 01; &&\a^\dag = \ktbr 10 \\
 &\a \1 = \0,\quad \a \0 = \mathbf 0; &
 & \a^\dag \0 = \1,\quad \a^\dag \1 = \mathbf 0
\end{align}

\section{Some remarks}

The construction of tensor product of algebras depends on {\em field
of scalars} (say $\R$ or $\C$) used in definition of the algebras. And if
algebras could be considered either as complex or as real, then it
is possible to use two kinds of tensor products:
\begin{equation}
 A \ox_{_\C} B \niso A \ox_{_\R} B
\end{equation}
For example,
for algebra of Pauli matrices considered as 8D real algebra the $n$-th
tensor power is \EQ{8^n = 2^{3n}}D real algebra and it is not equivalent
to construction used in the paper with $2^{2n}$ complex or $2^{2n+1}$
real dimension. The fact clarify difference between spaces used in the
paper and in \cite{Cory} there real tensor product was used.

\medskip

With using of real tensor product it is also possible to express space of
quantum $n$-gates and $\Cl(2n,\C)$ :
\begin{equation}
 \mbi{G}_n \in \C\ox\mR{2}^{{\ox}n}
\end{equation}
or:
\begin{equation}
 \mbi{G}_n \in \C\ox\H^{{\ox}n}
\end{equation}
where $\C$ is considered as 2D real and $\H$ as 4D real algebras.


\section*{Acknowledgements}
Author is grateful to organizers of {\em TMR Network School on
``Quantum Computation and Quantum Information Theory''} (Italy, Torino,
12--23 July 1999) for invitation, financial support and hospitality.

