text:
\section{Introduction}
Real geometric (Clifford) algebras (GA) may be understood as a generalisation of well known quaternions which is an alternative for matrix description of orthogonal transformations. Real geometric algebras have a wide range of applications in robotics, \cite{joan,hdita}, image processing \cite{stev}, numerical methods \cite{hfit}, etc. Among the main advantage of this approach we count the calculation speed, straightforward and geometrically oriented implementation and effective parallelisation, \cite{dita, dita2}. We stress that all these implementations are using Clifford's geometric algebra to represent specific orthogonal transformations.

Recently, an increasing number of papers adopt the apparat of real GA in the description, elaboration and analysis of Quantum Computing (QC) algorithms, \cite{cm, qba1}. The basic idea for this lies in identification of a state qubit with a Bloch sphere together with the identification of qubit gates with the rotations of the sphere. Namely, it is well known that a normalized qubit can be written in terms of basis vectors $\ket{0},\ket{1}$ as
\begin{align}\label{norm_qubit}
\ket{\psi}=\cos(\theta/2)\ket{0}+(\cos\phi +i \sin\phi)\sin (\theta/2)\ket{1},
\end{align}
where $0\leq \theta \leq \pi$ and $0\leq \phi < 2\pi$ and that parameters $\theta,\phi$ can be interpreted as spherical coordinates of a point on the unit sphere  in $\mathbb R^3,$ see figure \ref{Bloch}.
\begin{figure}[h]
	\centering
	\includegraphics[width=5cm]{Bloch_sphere.png}
	\caption{Bloch sphere for a qubit represented in quaternions}
	\label{Bloch}
\end{figure}
The spirit of GA description is to represent a point on the sphere by the rotation that takes a fixed initial point to that point. Under the usual choices, see figure \ref{Bloch},  the initial point is the north pole $(0,0,1)$ and  the qubit \eqref{norm_qubit} corresponds to the counterclockwise rotation by $\theta$ with respect to $y$ axis composed with the clockwise rotation by $\phi$ with respect to $z$ axis. In particular, the basis states $\ket{0},\ket{1}$ are represented by the identity and rotation $\theta=\pi$ respectively.
In the quaternionic description we have $(0,0,1)={\bf k}$ and the rotations are represented by elements $\exp(-1/2\theta  {\bf j})=\cos(\theta/2)-{\bf j}\sin(\theta/2)$ and $\exp(1/2\phi {\bf k})=\cos(\phi/2)+{\bf k}\sin(\phi/2)$ respectively. Indeed, the Bloch sphere representation of qubit \eqref{norm_qubit} is given by
\begin{align*}
e^{\tfrac12\phi {\bf k}}e^{-\tfrac12\theta {\bf j}} {\bf k} e^{\tfrac12\theta {\bf j}} e^{-\tfrac12\phi {\bf k}}=\cos\phi\sin\theta {\bf i}+\sin\phi\sin\theta {\bf j}+\cos\theta {\bf k}
\end{align*}
In this sense the qubit state \eqref{norm_qubit} is represented by quaternion $\exp(1/2\phi {\bf k})\exp(-1/2\theta  {\bf j})$.
In particular, the basis states $\ket{0},\ket{1}$ are represented by quaternions $\ket{0}=1$ and $\ket{1}=\exp(-1/2\pi  {\bf j})={-\bf j}$ respectively which in GA language is equation \eqref{basisH}, see section \ref{H_qubit} for more details.
In principle, this representation of qubits is based on exceptional isomorphisms of low-dimensional Lie groups $SU(2)$, $Spin(3)$ and $Sp(1)$. For higher dimensions, no similar natural identification of unitary and spin groups exists. Therefore, to realise the states of multi-qubit, it is necessary to use so-called correlators which increases algebraic abstraction and lacks the original elegance. Indeed, such approach is no competition to elegant Dirac formalism.

In our paper, we present an alternative to the qubit representation in the form of complex Clifford algebras which are indeed substantially more appropriate for QC representation by means of GA because they respect the complex nature of quantum theory. In our approach, Dirac formalism may be translated to complex GA of an appropriate dimension in rather straightforward way and, in spite of abstract symbolic Dirac formalism, all expressions are represented in a particular algebra and thus may be manipulated and implemented as algebra elements directly, without any need for matrix representation.
In the sequel, we briefly recall the definition of real GA and provide a more detailed introduction to complex GA. In Section \ref{QCinC}, we show the representation of qubits and multi-qubits in complex GA, their transformations (gates).  We provide an explicit forms of elementary 1-gates and 2-gates. We also discuss the case of multi-gates, ie. gates obtained by a tensor product. In Section \ref{realGA}, we describe a qubit by means of real geometric algebra. More precisely, we compare a description known from literature, ie. the one based on the isomorphism of unitary group $SU(2)$ and spin group $Spin(3)$, with the real description following from our complex GA approach and an isomorphism
$\mathbb C_2\rightarrow \mathbb G_3$ of complex and real algebra.

\section{Complex Clifford algebras}	\label{CCA}
A Clifford algebra is a normed associative algebra that generalizes the complex numbers and the quaternions. Its elements may be split into Grassmann blades and the ones with grade one can be identified with the usual vectors. The geometric product of two vectors is a combination of the commutative inner product and the anti-commutative
outer product. The scalars may be real or complex. In  section \ref{QCinC} we show that the complex Clifford algebra constructed over a quadratic space of even dimension can be efficiently used to represent quantum computing but we start with the real case.

\subsection{Real Clifford algebras}
The construction of the universal real Clifford algebra is well-known, for details see e.g. \cite{dl,lounesto}. We give only a brief description here. Let the real vector space $\mathbb R^m$ be endowed with a non-degenerate symmetric bilinear form $B$ of signature $(p,q)$, and let $(e_1,\dots, e_m)$ be an associated orthonormal basis, i.e.
\begin{align*}
B(e_i,e_j)=\begin{cases}
1 &\text{ if } i=j=1,\dots,p\\
-1 &\text{ if } i=j=p+1,\dots,m\\
0  &\text{ if } i\neq j
\end{cases}
\; \text{ where } 1\leq i,j \leq m=p+q.
\end{align*}
Let us recall that the Grassmann algebra is an associative algebra with the anti-symmetric outer product defined by the rule
\begin{align*}
e_i \wedge e_j + e_j \wedge e_i =0 \; \text{ for } 1\leq i,j \leq m.
\end{align*}
The Grassmann blade of grade $r$ is  $e_A=e_{i_1} \wedge \cdots \wedge e_{i_r}$, where the multi-index $A$ is a set of indices ordered in the natural way $1\leq i_1\leq \cdots \leq i_r\leq m,$ and we put $e_\emptyset=1.$ Blades of orders $0\leq r \leq m$ form the basis of the graded Grassmann algebra $\Lambda(\mathbb{R}^m)$. Next, we introduce the inner product
\begin{align*}
e_i \cdot e_j= B(e_i,e_j), \;\;  1\leq i,j \leq m,
\end{align*}
leading to the so-called {\it geometric product} in the Clifford algebra
\begin{align*}
e_i e_j= e_i\cdot e_j + e_i \wedge e_j, \;\;  1\leq i,j \leq m,
\end{align*}
The respective definitions of the inner, the outer and the geometric product are then extended to blades of the grade $r$ as follows. For the inner product we put
\begin{align*}
e_j \cdot e_A = e_j \cdot (e_{i_1} \wedge \cdots \wedge e_{i_r})= \sum_{k=1}^r (-1)^k B(j,i_k) e_{A\setminus \{i_k\}},
\end{align*}
where $e_{A\setminus \{i_k\}}$ is the blade of grade $r-1$ created by deleting $e_{i_k}$ from $e_A$. This product is also called the left contraction in literature. For the outer product we have
\begin{align*}
e_j \wedge e_A = \begin{cases}
e_j \wedge e_{i_1} \wedge \cdots \wedge e_{i_r} & \text{ if } j\notin A \\
0  & \text{ if } j\in A
\end{cases}
\end{align*}
and for the geometric product we define
\begin{align*}
e_je_A=e_j\cdot e_A + e_j \wedge e_A.
\end{align*}
Finally, these definitions are linearly extended to the whole of the vector space $\Lambda(\mathbb{R}^m)$. Thus we get an associative algebra over this vector space, the so-called real Clifford algebra, denoted by $\mathbb{G}_{p,q}=\operatorname{Cl}(p,q,\mathbb{R})$. Note that this algebra is naturally graded; the grade zero and grade one elements are identified with $\mathbb{R}$ and $\mathbb{R}^m$ respectively. The projection operator $\mathbb G_{p,q}\to\Lambda^r(\mathbb R^m)$ will be denoted by $[\;]_r$.

This grading define a $\mathbb{Z}_2$-grading of the Clifford algebra according to the parity of grades. Namely, the linear map $v\to -v$ on $\mathbb{R}^m$ extends to an automorphism $\alpha$ called the grade involution and decomposes $\mathbb{G}_{p,q}$ into positive and negative eigenspaces. The former is called the even subalgebra $\mathbb{G}^0_{p,q}$ and the latter is called the odd part $\mathbb{G}^1_{p,q}$. In addition to $\alpha$, there are two important antiautomorphisms of real Clifford algebras. The first one is $\tilde{x}$ called the reverse or transpose operation and it is defined by extension of identity on $\mathbb{R}^m$ and by the antiautomorphism property  $\widetilde{xy}=\tilde{y}\tilde{x}.$ The second antiautomorphism is called the Clifford conjugation $\bar{x}$ and  the operation is defined by composing $\alpha$ and the reverse
\begin{align*}
\bar{x}=\alpha(\tilde{x})=\widetilde{\alpha(x)}.
\end{align*}

\subsection{The complexification} \label{ClC}
When allowing for complex coefficients, the same generators $e_A$ produce by the same formulas the complex Clifford algebra which we denote by $\mathbb{C}_m=\operatorname{Cl}(m,\mathbb{C})$. Clearly, in the complex case no signature is involved, since each basis vector $e_j$ may be multiplied by the imaginary unit $i$ to change the sign of its square. Hence we may assume we start with the real Clifford algebra $\mathbb{G}_m$ with the inner product $e_ie_j=\delta_{ij}$, $1\leq i,j \leq m$, and we construct the complex Clifford algebra as its complexification $\mathbb C_m:=\mathbb G_m\oplus i \mathbb G_m,$ i.e. any element  $\varphi\in\mathbb C_m$ can be written as $\varphi=x+iy$, where $x,y\in\mathbb G_m.$  The complex Clifford algebras for small $m$ are well known; $\mathbb C_0$ are complex numbers itself, $\mathbb C_1$ is the algebra of bicomplex numbers and
$\mathbb C_2$ is the algebra of biquaternions. More details on complex Clifford algebras one can find in the papers \cite{bss2,bss,bu,fs}.

The construction via the complexification of $\mathbb G_m$ leads to the definition an important antiautomorphism of $\mathbb C_m,$ so-called Hermitian conjugation
\begin{align} \label{HC}
\varphi^\dagger=(x+iy)^\dagger=\bar{x}-i\bar{y},
\end{align}
where the bar notation stands for the Clifford conjugation in $\mathbb G_m.$ Note that on the zero grade part of the complex Clifford algebra $\mathbb C_0=\mathbb C$ it coincides with the usual complex conjugation. The elements satisfying $\varphi^\dagger=\varphi$ and $\varphi^\dagger=-\varphi$ will be called Hermitian and anti-Hermitian respectively. Hermitian conjugation is a very important anti-involution which is the Clifford analogue of the conjugate transpose in matrices. It leads to the definition of the Hermitian inner product on $\mathbb C_m$ given by
\begin{align} \label{HP}
\langle\varphi|\psi\rangle=[\varphi^\dagger \psi]_0, \;\; \varphi,\psi\in\mathbb C_m
\end{align}
where we recall that $[\;]_0$ denotes the projection to the scalar part, i.e. the grade zero part. Indeed, it is easy to see that it is linear in the second slot and conjugate linear in the first slot; for each $z\in\mathbb C$ and $\varphi\in\mathbb C_m$ we have
$
(z\varphi)^\dagger=\varphi^\dagger z^\dagger=\bar{z} \varphi^\dagger
$
since the Hermitian conjugation is anti-automorphism. The Hermitian symmetry of \eqref{HP} follows from the involutivity of  the Hermitian conjugation while its positive definiteness follows from the fact that
\begin{align*}
\langle\varphi|\varphi\rangle=[\varphi^\dagger \varphi]_0=\sum_{A}\varphi_A^2,
\end{align*}
where $A$ is an arbitrary multiindex and $\varphi_A$ is the coefficient at the Grassmann blade $e_A$, i.e. $\varphi=\sum_A\varphi_A e_A$. Let us discuss the last equality in more detail.
For two multi-indices $A=\{i_1,\dots,i_r\}$, $B=\{k_1,\dots,k_s\}$ the scalar projection
$[\tilde{e}_Ae_B]_0$ is nonzero only if the grades are equal, i.e. $r=s$. Then by the definition of geometric product and the reverse operation we get $[\tilde{e}_Ae_B]_0=(e_{i_1}\cdot e_{k_1})\cdots(e_{i_r}\cdot e_{k_r})=\delta_{i_1k_1}\cdots \delta_{i_rk_r}=\delta_{AB}$, whence by the linearity of the grade projection
\begin{align*}
[\varphi^\dagger \varphi]_0=\sum_{A,B}\bar{\varphi}_A\varphi_B[\tilde{e}_Ae_B]_0=\sum_{A}\varphi_A^2.
\end{align*}

\subsection{Witt basis} \label{Witt}
Henceforth we assume the dimension of the generating vector space is even, i.e. $m=2n.$ In such a case the complexification of the Clifford algebra  can be introduced by considering so--called complex structure, i.e. a specific orthogonal linear transformation $J:\:\mathbb R^{2n}\to\mathbb R^{2n}$ such that $J^2=-1$, where 1 stands for the identity map. Namely, we choose $J$ such that its action on the orthonormal basis $e_1,\dots,e_{2n}$ is given by $J(e_j)=-e_{j+n}$ and $J(e_{j+n})=e_j$, $j=1,\dots,n.$ With $J$ one may associate two projection operators which produce the main objects of the complex setting by acting on the orthonormal basis, so–called Witt basis elements $(f_j , f^{\dagger}_j)$. Namely, we define
\begin{align*}
f_j&=\frac12(1+iJ)(e_j)=\frac{1}{2}(e_j- ie_{j+n}), \;\; j=1,\dots,n\\
f_j^{\dagger} &=\frac{1}{2}(1- iJ)(e_j) =\frac{1}{2}(e_j+ ie_{j+n}),  \;\; j=1,\dots,n
\end{align*}
Note that it is not confusion of the notation since $f_j^\dagger$ indeed is the image of $f_j$ under  Hermitian conjugation \eqref{HC}.
The Witt basis elements are isotropic with respect to the geometric product, i.e. for each $j=1,\dots,n$ they satisfy $f_j^2=0$ and $f_j^{\dagger}{}^2=0$. They also satisfy the Grassmann identities
\begin{align} \label{Grassmann}
f_j f_k + f_k f_j =f_j^{\dagger} f_k^{\dagger} + f_k^{\dagger} f_j^{\dagger}=0, \;\; j,k=1,\dots,n
\end{align}
and the duality identities
\begin{align} \label{duality}
f_j f_k^{\dagger} + f_k^{\dagger} f_j = \delta_{jk}, \;\; j,k=1,\dots,n
\end{align}
The Witt basis of the whole complex Clifford algebra $\mathbb C_{2n}$ is then obtained, similarly
to the basis of the real Clifford algebra, by taking the $2^{2n}$ possible geometric products of
Witt basis vectors, i.e. it is formed by elements
\begin{align} \label{C2n_basis}
(f_1)^{i_1}(f_1^\dagger)^{j_1}\cdots (f_n)^{i_n}(f_n^\dagger)^{j_n}, \;\; i_k,j_k\in\{0,1\} \text{ for } k=1,\dots,n
\end{align}
One can eventually use the Grassmann blades of Witt elements as the basis of $\mathbb C_{2n}$. The relation of the two basis
can be deduced from the relation of the geometric product of Witt basis elements to the corresponding inner and outer product, for more details see \cite{Lounesto}.
\begin{align*}
f_jf_k&=f_j \cdot f_k + f_j \wedge f_k= f_j \wedge f_k \\
f_j^\dagger f_k^\dagger&=f_j^\dagger \cdot f_k^\dagger + f_j^\dagger \wedge f_k^\dagger= f_j^\dagger \wedge f_k^\dagger\\
f_j f_k^\dagger&=f_j \cdot f_k + f_j \wedge f_k^\dagger=\frac12 \delta_{jk}+ f_j \wedge f_k^\dagger
\end{align*}

\subsection{Spinor spaces}
In the language of Clifford algebras, spinor space is defined as a minimal left ideal of the complex Clifford algebra
and is realized explicitly by means of a self-adjoint primitive idempotent. The realization of spinor space within the complex Clifford algebra $\mathbb C_{2n}$ can be constructed directly using the Witt basis as follows. We start by defining
\begin{align*}
I_j=f_jf_j^\dagger \text{ and } K_j=f_j^\dagger f_j, \;\; j=1,\dots,n
\end{align*}
Direct computations show that  both $I_j, K_j$ are mutually commuting self--adjoint idempotents. More precisely, for $j,k=1,\dots,n$ the following identities hold.
\begin{align*}
I_j^\dagger&=I_j,\, I_j^2=I_j \text{ and } K_j^\dagger=K_j,\, K_j^2=K_j, \\
I_jI_k&=I_kI_j,\, K_jK_k=K_kK_j, \\
I_jK_k&=K_kI_j \text{ whenever } j\neq k, \text{ and } I_jK_j=K_jI_j=0.
\end{align*}
Moreover, the duality relations \eqref{duality} between Witt basis vectors imply that $I_j+K_j=1$ for each $j=1,\dots,n.$ Hence we get the resolution of the identity
$
1=\prod_{j=1}^n(I_j+K_j).
$
Consequently we get
\begin{align*}
\mathbb C_{2n}=\mathbb C_{2n}\prod_{j=1}^n(I_j+K_j)=\mathbb C_{2n}I_1\cdots I_n\oplus \mathbb C_{2n}I_1\cdots I_{n-1}K_1\oplus \cdots \oplus \mathbb C_{2n}K_1\cdots K_n,
\end{align*}
a direct sum decomposition of the complex Clifford algebra into $2^n$ isomorphic realizations of the spinor space that are denoted according to the specific idempotent involved:
\begin{align} \label{spinor}
\mathbb{S}_{\{i_1\cdots i_s\}\{k_1\cdots k_t\}}=\mathbb C_{2n}I_{i_1}\cdots I_{i_s}K_{k_1}\cdots K_{k_t}\subset\mathbb C_{2n},
\end{align}
where $s+t=n$ and the indices are pairwise different.
Each such space has dimension $2^n$ and its basis is obtained by right multiplication of the basis of $\mathbb C_{2n}$ by the corresponding primitive idempotent $I_{i_1}\cdots I_{i_s}K_{k_1}\cdots K_{k_t}.$ By the basic properties of the Witt basis elements \eqref{Grassmann} and \eqref{duality} it is easy to see that this action is nonzero if and only if the element of $\mathbb C_{2n}$ actually lies in the Grassmann algebra generated by $n$-dimensional space $( f_{i_1}^\dagger,\dots,f_{i_s}^\dagger,f_{k_1},\dots,f_{k_t} )$, i.e. we may write
\begin{align} \label{spinor_G}
\mathbb{S}_{\{i_1\cdots i_s\}\{k_1\cdots k_t\}}
=\Lambda( f_{i_1}^\dagger,\dots,f_{i_s}^\dagger,f_{k_1},\dots,f_{k_t} )I_{i_1}\cdots I_{i_s}K_{k_1}\cdots K_{k_t}.
\end{align}

In terms of multiindices $A=\{i_1,\dots,i_s\}$, $B=\{k_1,\dots,k_t\}$ this spinor space can be written shortly as $\mathbb S_{AB}$.
It is an easy observation that it has the structure of  a Hilbert space of dimension $2^n$ due to the Hermitian product \eqref{HP}
and that the multiplication in $\mathbb C_{2n}$ makes each spinor space $\mathbb S_{AB}$ into a left $\mathbb C_{2n}$-module. Hence the elements of the complex Clifford algebra that keep the Hermitian product invariant define a representation of the corresponding unitary group on the spinor space. Namely let a $\lambda\in\mathbb C_{2n}$ act on two spinors $\varphi,\psi\in\mathbb S_{AB}$. Then we compute $\langle\lambda\varphi|\lambda\psi\rangle=[\varphi^\dagger\lambda^\dagger\lambda \psi]_0$ by definition and due to the antiautomorphism property of the Hermitian conjugation. Hence the elements of the complex Clifford algebra such that
\begin{align} \label{unitary}
\lambda^\dagger \lambda=1
\end{align}
holds keep the Hermitian product invariant and thus define a representation of the unitary group $U(2^n)$ on the spinor space $\mathbb S_{AB}.$ These elements also satisfy $\lambda\lambda^\dagger=1$ and will be called unitary elements of $\mathbb C_{2n}$ in analogy with unitary matrices. Let us remark that in the representation theory this representation of the unitary group is well known. It comes from the so called spin representation of the corresponding complex orthogonal group.

\section{Quantum computing in complex Clifford algebras} \label{QCinC}
The idea is to perform quantum computing in the Hilbert space defined by a complex Clifford algebra with Hermitian product defined by \eqref{HP} instead of the classical realization of the Hilbert space on complex coordinate space with the standard Hermitian inner product. A quantum state is then represented by an element of a complex Clifford algebra lying in spinor space \eqref{spinor} and unitary transformations are then realized as elements \eqref{unitary} of the same algebra. The computation becomes especially efficient when using Witt basis of the complex Clifford algebra, see \ref{Witt}. However the mathematical framework described in the previous section allows for a direct application to general states and transformations of multiple qubits, for clarity we start with the description of the basic case of a single qubit.

\subsection{A qubit and single qubit gates} \label{qubit_section}
A qubit  will be represented by an element in the complex Clifford algebra $\mathbb C_2$ instead of its standard representation by a vector in the complex coordinate space $\mathbb C^2$. The Witt basis elements $(f,f^\dagger)$
satisfy the Grassmann and duality identities
\begin{align*
f^2=f^\dagger{}^2=0, \, ff^\dagger+f^\dagger f=1,
\end{align*}
leading to $f f^\dagger =\frac12+ f\wedge f^\dagger$ and $[f f^\dagger]_0=\frac12$  in particular. The Witt basis vectors induce  a basis of the complex Clifford algebra $\mathbb C_2$ of the form $(1,f,f^\dagger,ff^\dagger)$. In this algebra we have two primitive idempotents $I=ff^\dagger$ and $K=f^\dagger f$ that give rise to two isomorphic spinor spaces: $\mathbb S=\mathbb C_{2}I=\Lambda(f^\dagger)I$ and $\bar{\mathbb S}=\mathbb C_{2}K=\Lambda(f)K.$ For representing the qubit we choose the former one, see Remark \ref{rem_notation}. Choosing  the basis $(1,f^\dagger)$ of the Grassmann algebra $\Lambda(f^\dagger)$, we get the following basis of $\mathbb S$ that will represent  the zero state and the one state of a qubit
\begin{align} \label{01}
\begin{split}
\ket{0}&=I=f f^\dagger \\
\ket{1}&=f^\dagger I=f^\dagger f f^\dagger=(1-f f^\dagger) f^\dagger=f^\dagger,
\end{split}
\end{align}
whence the Clifford algebra representation of a qubit in a general superposition state  $\ket{\psi}=\alpha\ket{0}+\beta\ket{1}$, for arbitrary complex numbers $\alpha,\beta$,  is given by
\begin{align} \label{qubit}
\psi=(\alpha+\beta f^\dagger)I=\alpha f f^\dagger+\beta f^\dagger \in \mathbb S\subset\mathbb C_2.
\end{align}
However the basis \eqref{01} of spinor space $\mathbb{S}$ is orthogonal with respect to the Hermitian product in Clifford algebra $\mathbb C_2$ defined by \eqref{HP}, it is not orthonormal since the length of the basis elements equals $1/2$ due to the spinorial nature of the representation. To make the basis orthonormal we will modify the Hermitian product in \eqref{HP} by this factor, namely we will assume  $\langle\varphi|\psi\rangle=2[\varphi^\dagger \psi]_0$ for any spinors $ \varphi,\psi\in\mathbb C_2.$ Indeed, then we compute
\begin{align*}
\braket{0|1}&=2[f f^\dagger f^\dagger]_0=0,\\
\braket{0|0}&=\braket{1|1}=2[f f^\dagger]_0=1.
\end{align*}
\begin{remark} \label{rem_notation}
	The choice of idempotent $I$ is motivated by conventions in physics for creation and annihilation operators. Indeed, $f^\dagger$ is a realization of the abstract creation operator of the so-called CAR algebra and thus we want it to represent the qubit state $\ket{1}$ rather than $\ket{0}$
\end{remark}

A single qubit gate is represented by an unitary element in Clifford algebra $\mathbb C_2$ and it acts on a qubit in spinor space $\mathbb S$ by left multiplication. Obviously the identity gate is defined by $1\in\mathbb C_2$ and serially wired gates are given by the product of the individual representatives in $\mathbb C_2$ due to the associativity of the Clifford product. For our choice of the basis of qubit states the commonly used quantum gates operating on a single qubit are represented in terms of the Witt basis as follows.

\begin{proposition} \label{single_gates}
	Representing the basic qubit states in the complex Clifford algebra $\mathbb C_2$  as $\ket{0}=ff^\dagger$, $\ket{1}=f^\dagger$

	we get representations of single qubit gates in $\mathbb C_2$
	\begin{align*}
	\text{X-gate: } \: \lambda_X&=f^\dagger +f \\
	\text{Y-gate: } \: \lambda_Y&=if^{\dagger} -i f \\
	\text{Z-gate: } \:  \lambda_Z&=ff^{\dagger} -f^{\dagger} f
	\end{align*}
\end{proposition}
\begin{proof}
By the definition of Hermitian conjugation in Clifford algebra $\mathbb C_2$, the identification of basic qubit states  $\ket{0}=ff^\dagger$, $\ket{1}=f^\dagger$ leads to the identification of their Hermitian duals $\bra{0}= ff^\dagger$, $\bra{1}= f$. Then we get a representation of projection operators
	\begin{align*}
	\ket{0}\bra{0}=ff^\dagger ff^\dagger=ff^\dagger, \;
	\ket{0}\bra{1}=ff^\dagger f=f, \;
	\ket{1}\bra{0}=f^\dagger ff^\dagger=f^\dagger, \;
	\ket{1}\bra{1}=ff^\dagger,
	\end{align*}
	where we used the Grassmann and duality identities for the Witt basis elements $f, f^\dagger.$
	The representations of single qubit gates from the proposition then  follow by their definitions on basic qubit states.
\end{proof}
\begin{remark}
	Using basis $1,e_1,e_2,e_1\wedge e_2$ of Clifford algebra $\mathbb C_2$ generated by an orthonormal basis $e_1, e_2$ of $\mathbb C^2$ instead of the Witt basis the representation of single qubit gates $X,Y$ and $Z$ are given as
		\begin{align*}
 \lambda_X=e_1, \;
\lambda_Y=-e_2, \;
  \lambda_Z=i e_{1} \wedge e_2
	\end{align*}
\end{remark}

\begin{example}
Let us discuss the some of these basic quantum gates in more detail. The $X$-gate is the quantum equivalent to of the NOT gate for classical computers,  sometimes called a bit-flip as it maps the basis state $\ket{0}$ to $\ket{1}$ and vice versa. Hence we have
\begin{align*}
X=\ket{1}\bra{0}+\ket{0}\bra{1}=f^\dagger +f.
\end{align*}
which is equal to $e_1$ by definition of the Witt basis elements.
We can also compute directly the action of the corresponding element of $\mathbb C_2$ on a qubit basis in $\mathbb S$ to prove the correctness of the representation
\begin{align*}
\lambda_X \ket{0}&=(f^\dagger +f)f f^\dagger=f^\dagger f f^\dagger=f^\dagger =\ket{1}, \\
\lambda_X \ket{1}&=(f^\dagger +f) f^\dagger=f f^\dagger=\ket{0}.
\end{align*}
Similarly, for the  phase-flip $Z$-gate we get $Z=\ket{0}\bra{0}-\ket{1}\bra{1}= f f^\dagger - f^\dagger f$ since it leaves the basis state $\ket{0}$ unchanged and maps $\ket{1}$  to $-\ket{1}$   and a general phase-shift gate $\ket{1}\mapsto e^{i\varphi}\ket{1}$ is given by $R_\varphi=f f^\dagger + e^{i\varphi} f^\dagger f.$
The effect of a series circuit where $X$ is put after $Z$ can be described as a single gate represented by the Clifford product
\begin{align*}
XZ=(f^\dagger +f)(f f^\dagger - f^\dagger f)=f^\dagger-f.
\end{align*}
It is also easy to check the involutivity of the single qubit gates $X,Y$ and $Z$, e.g. for a serial composition of two  $X$-gates we have
 \begin{align*}
 X^2=(f^\dagger +f)(f^\dagger +f)=f^\dagger f+f^\dagger f = 1.
 \end{align*}
 The representations of rotation operator gates can be obtained directly by computing exponentials in of gates $X,Y,Z$ in $\mathbb C_2$. Consequently one can get the formula for the Hadamard gate, an important single qubit gate that we have not discussed yet. Namely, since $Y^2=1$, we compute
  \begin{align*}
 H&=X\operatorname{exp}(-iY\pi/4)=(f^\dagger+f)(\cos\tfrac{\pi}{4}+\sin\tfrac{\pi}{4}(f^\dagger-f))\\
 &=\tfrac{1}{\sqrt{2}}(ff^{\dagger} -f^{\dagger}f + f +  f^{\dagger}),
 \end{align*}
which is equal to $\tfrac{1}{\sqrt{2}}(e_1+ie_1\wedge e_2)$ in the orthonormal basis.
\end{example}

The Clifford algebra representations of basic single qubit gates $X,Y$ and $Z$ in Proposition \ref{single_gates} determine an explicit form of a unitary element in $\mathbb C_2$ representing a general single qubit gate in terms of the Witt basis.
\begin{corollary} \label{1unitary_el}
	Each single qubit gate operating on qubit \eqref{qubit} is represented by an element of the complex Clifford algebra $\mathbb C_2$ of a form
	\begin{align} \label{1gate}
	\lambda=a f f^{\dagger} + b f +c f^{\dagger} + d f^{\dagger}f.
	\end{align}
	where $a,b,c,d \in\mathbb C$ are complex numbers such that $a^2 +  c^2=b^2  + d^2=1,\bar{b}a    +\bar{d}c  =0$ holds.
\end{corollary}
\begin{proof}
	Can be deduced from the description of projection operators in the proof of Proposition \ref{single_gates} or directly by writing the condition on unitary elements $\lambda^\dagger \lambda=1$ in terms of the Witt basis, see \eqref{unitary}, as follows. An arbitrary element $\lambda\in \mathbb C_2$ can be written as in \eqref{1gate} for some complex numbers $a,b,c,d\in\mathbb C$ since 	the four--tuple $(f f^\dagger, f^\dagger, f, f^\dagger f)$ form a basis of $\mathbb C_2$. The right-hand side of the equation for unitary elements can be written as  $ff^\dagger+f^\dagger f$ and for the left-hand side  we compute
	\begin{align*}
	\lambda^{\dagger} \lambda & =  (\bar{a} f f^{\dagger} + \bar{b} f^\dagger +\bar{c} f + \bar{d} f^{\dagger}f) (a f f^{\dagger} + b f +c f^{\dagger} + d f^{\dagger}f)\\
	&=  (a^2 +  c^2 )f^{\dagger}f  + 	( \bar{b}a    +\bar{d}c) f^{\dagger} +
	( \bar{a}b   +\bar{c}d) f    + ( b^2  + d^2 )f f^{\dagger},
	\end{align*}
	where we used  the definition of the Hermitian conjugation and its properties discussed in section \ref{ClC} and
	where we repeatedly used the duality and Grassmann identities for the Witt basis elements $f, f^\dagger$,

	see section \ref{Witt}. The result follows by comparing the coefficients on both sides of the equation while noting that $ \bar{a}b   +\bar{c}d$ is the complex conjugate of $ \bar{b}a    +\bar{d}c$.
\end{proof}

Note that the condition on these coefficients can be expressed equivalently as the orthonormality of complex vectors $(a,c)$ and $(b,c)$ with respect to the standard Hermitian product on $\mathbb C^2.$ Hence the coefficients define a $2 \times 2$ unitary matrix proving  the equivalence between Clifford and matrix descriptions. Namely, each unitary element in $\mathbb C_2$ corresponds to a matrix in $U(2)$ as follows.
\begin{align} \label{correspondence}
a f f^{\dagger} + b f +c f^{\dagger} + d f^{\dagger}f
\leftrightarrow
\begin{pmatrix}
a & b \\ c & d
\end{pmatrix
\end{align}
\begin{remark}
	Upon restriction to a normalized qubit $\langle\psi|\psi\rangle=2[\psi^\dagger \psi]_0=\alpha^2+\beta^2=1$, it is sufficient to consider gates from the special unitary group $SU(2)$, the connected component of $U(2).$ Such gates  are represented by unitary matrices with unite determinant and that they can be written as displayed in \eqref{correspondence} for $b=-\bar{c}$ and $d=\bar{a}$. Hence the elements in $\mathbb C_2$ representing subgroup $SU(2)$ are of a form $\lambda=a f f^{\dagger} -\bar{c} f +c f^{\dagger} + \bar{a} f^{\dagger}f.$
\end{remark}

\subsection{Multiple qubits and multiple qubit gates}
Following constructions in section \ref{CCA} the Hilbert space of states of a general $n$-qubit can be represented by a spinor space in the complex Clifford algebra $\mathbb C_{2n}$ and  $n$-qubit gates as unitary elements in the same algebra. For the explicit description we choose Witt basis $(f_1,f_1^\dagger,\dots,f_n,f_n^\dagger)$ of complex coordinate space $\mathbb C^{2n}$ leading to the Witt basis of the Clifford algebra $\mathbb C_{2n}$ formed by $2^{2n}$ geometric products of these elements given by \eqref{C2n_basis}. From the $2^n$ spinor spaces contained in the algebra  we choose the spinor space $\mathbb S_n=\mathbb C_{2n}I$ defined by primitive idempotent
\begin{align} \label{In}
I=I_{1}\cdots I_{n}=f_1 f_1^\dagger \cdots f_n f_n^\dagger
\end{align}
for modelling states of a $n$-qubit. This choice is motivated by its identification with the Grassmann algebra generated by "creation operators" $f_1^\dagger,\dots,f_n^\dagger$. Indeed, for such a realization of the spinor space of a $n$-qubit we have
\begin{align} \label{Sn}
\mathbb S_n=\mathbb C_{2n}I=\Lambda(f_1^\dagger,\dots,f_n^\dagger)I
\end{align}
since  $f_jI=0$ for each $j=1,\dots,n$ by  Grassmann and duality identities for the Witt basis elements.
Similarly to the case of a single qubit, we multiply the Hermitian product \eqref{HP} by a normalization factor $2^n$ that reflects the spinorial nature of our representation in order to get a simple formula for elements of unite norm. Namely, for two spinors $\varphi,\psi\in\mathbb S_n$ we set
\begin{align} \label{HPn}
\langle\varphi|\psi\rangle=2^n[\varphi^\dagger \psi]_0.
\end{align}
With this choice of spinor space and Hermitian product the main results of section \ref{CCA} that we need for representing  qubits and quantum gates in a complex Clifford algebra read as follows.
\begin{proposition} \label{n_qubit}
	Spinor space $\mathbb S_n\subset\mathbb C_{2n}$ given by \eqref{Sn} together with Hermitian product \eqref{HPn} form a Hilbert space of dimension $N=2^n$ with an orthonormal basis
	\begin{align} \label{Sn_basis}
	\ket{i_1 \cdots i_n}=(f_1^\dagger)^{i_1} \cdots (f_n^\dagger)^{i_n} I,
	\end{align}
	where $ i_1,\dots,i_n\in\{0,1\}$. Unitary transformations are given by left multiplications by unitary elements, i.e. elements of $\mathbb C_{2n}$ such that $\lambda^\dagger \lambda=1$.
\end{proposition}
\begin{proof}
	The proposition follows from the constructions described in section  \ref{CCA}, we only need to check the orthonormality of basis elements \eqref{Sn_basis}. For $ i_k,j_k\in\{0,1\}$, where $k=1,\dots,n$, the Hermitian product of two basis elements is given by
	\begin{align*}
	\braket{j_1 \cdots j_n|i_1 \cdots i_n}=2^n[I^\dagger(f_n)^{j_n} \cdots (f_1)^{j_1} (f_1^\dagger)^{i_1} \cdots (f_n^\dagger)^{i_n} I]_0
	\end{align*}
	by definition. Let us prove the orthogonality first. Assume  $i_k=0$ and $j_k=1$ for some $k$. Since the Witt basis element $f_k$ anti-commutes with elements $f_\ell$  and $f_\ell^\dagger$  for each $\ell\neq k$ by \eqref{Grassmann},  the previous formula can be expressed in a form $\pm 2^n[I^\dagger  \cdots  f_kI]_0$  and it vanishes since $f_kI=0$ by \eqref{Grassmann} and \eqref{duality}. Similarly, if  $i_k=1$ and $j_k=0$, then the above formula for Hermitian product vanishes since it contains factor $I^\dagger f_k^\dagger=0.$ Hence the Hermitian product vanish if $i_k\neq j_k$ for some $k$ and the the orthogonality is proven. To prove the normality we notice that $ (f_k)^{i_k} (f_k^\dagger)^{i_k}=I_{i_k}$ is an idempotent commuting with all elements $f_\ell, f_\ell^\dagger$, where $\ell\neq k,$ and all idempotents $I_{i_\ell}$. Hence we get
		\begin{align*}
		\braket{i_1 \cdots i_n|i_1 \cdots i_n}=2^n[I^\dagger(f_n)^{i_n} \cdots (f_1)^{i_1} (f_1^\dagger)^{i_1} \cdots (f_n^\dagger)^{i_n} I]_0=2^n[I^\dagger \cdots I_{i_\ell} \cdots I]_0,
	\end{align*}
	where $I_{i_\ell}$ are idempotents  such that $i_\ell=1.$ The primitive idempotent $I$ satisfies $I^\dagger=I$ and it also satisfies  $I_{i_\ell}I=I$ for all $\ell$ since it is given by product of all such commuting idempotents, namely  $I=I_1\cdots I_n$ by definition. So we compute
	\begin{align*}
		\braket{i_1 \cdots i_n|i_1 \cdots i_n}=2^n[ I]_0=2^n[I_1\cdots I_n]_0=1,
	\end{align*}
	where the last equality follows from the decomposition of idempotents into grade components. Namely, we have
	$I_k=1/2+f_k\wedge f_k^\dagger$ and the geometric product of $f_k\wedge f_k^\dagger$ with elements not containing $f_k$ neither $f_k^\dagger$ either vanishes or yields an element of grade at least two.
\end{proof}

\begin{remark}
Note that we chose MSB bit numbering for $n$-qubits. The choice of the LSB bit numbering would lead to different but isomorphic representations in $\mathbb C_{2n}.$
\end{remark}

However an explicit description of a general unitary element in $\mathbb C_{2n}$ representing a $n$-qubit gate similar to the description of a general single qubit gate given in Corollary \ref{1unitary_el} is possible, it is more sophisticated and thus not helpful. The same happens in matrix representation and it reflects the complexity of unitary group $U(N)$.
On the other hand, given a  specific $n$-qubit gate its representation in $\mathbb C_{2n}$ is obtained by rewriting the defining formula in terms of projection operators in Dirac formalism via identification of $n$-qubit states \eqref{Sn_basis}. To make it clear we elaborate some examples for $n=2$ in more detail.
\begin{example} \label{ex_2gates}
	In the case of a 2-qubit we work in Clifford algebra $\mathbb C_{4}$ of dimension is $2^4=16$. Using the Witt basis $f_1,f_1^\dagger,f_2,f_2^\dagger$ of $\mathbb C^4$ we define a primitive idempotent $I=f_1 f_1^\dagger f_2 f_2^\dagger$ which gives rise to spinor space $\mathbb S_2=\mathbb C_4I$ of dimension $2^2=4$ with an orthonormal basis
 \begin{align*}
 \ket{00}=I,\,  \ket{10}=f_1^\dagger I,\, \ket{01}=f_2^\dagger I,\,  \ket{11}=f_1^\dagger f_2^\dagger I
 \end{align*}
 Using this representation of basis states and the definition of Hermitian conjugation we can form 16 projection operators, e.g. $\ket{00}\bra{00}=I I^\dagger=I$, $\ket{00}\bra{01}=I I^\dagger f_2 = I f_2=f_1 f_1^\dagger f_2 ,$ etc. Specific 2-qubit gates are then formed by a complex linear combinations of these elements in $\mathbb C_4.$  We demonstrate the functionality of the spinor representation on 2-qubit gates known as  CNOT, CZ and SWAP, see the diagrammatic descriptions of these gates in Figure \ref{2gates}.
  \begin{align*}
 f_1f_1^\dagger \ket{00}&=f_1f_1^\dagger f_1f_1^\dagger f_2f_2^\dagger=I = \ket{00},\\
 f_1f_1^\dagger \ket{01}&=f_1f_1^\dagger f_2^\dagger f_1f_1^\dagger f_1f_1^\dagger =  f_2^\dagger I =\ket{01}
 \end{align*}
\begin{figure}[h]
	\centering
	\includegraphics[width=2cm,valign=t]{cnot.png}
	\hspace{2em}
    \includegraphics[width=2cm,valign=t]{cz.png}
	\hspace{2em}
   \includegraphics[width=2cm,valign=t]{swap.png}
	\caption{2-qubit gates CNOT, CZ and SWAP respectively}
	\label{2gates}
\end{figure}
 We systematically use Grassmann and duality identities for Witt basis elements, identity $II^\dagger=I$ in particular.
 \begin{align*}
 \lambda_{\operatorname{CNOT}}&=\ket{00}\bra{00}+\ket{01}\bra{01}+\ket{11}\bra{10}+\ket{10}\bra{11}\\
 &=I+f_2^\dagger I f_2+f_1^\dagger f_2^\dagger I f_1+ f_1^\dagger I f_2 f_1\\
 &=f_1 f_1^\dagger f_2 f_2^\dagger +f_1 f_1^\dagger f_2^\dagger f_2 - f_1^\dagger f_1 f_2^\dagger - f_1^\dagger f_1 f_2 \\
 &=f_1f_1^\dagger -  f_1^\dagger f_1 (f_2^\dagger + f_2), \\
   \lambda_{\operatorname{CZ}}&= \ket{00}\bra{00}+\ket{01}\bra{01}+\ket{10}\bra{10}-\ket{11}\bra{11}\\
 &=f_1f_1^\dagger + f_1^\dagger f_1 (f_2f_2^{\dagger} -f_2^{\dagger} f_2), \\
    \lambda_{\operatorname{SWAP}}&=\ket{00}\bra{00}+\ket{11}\bra{11}+\ket{10}\bra{01}+\ket{01}\bra{11}\\

 &=f_1f_1^{\dagger}f_2f_2^{\dagger}+
 f_1^{\dagger}f_1f_2^{\dagger}f_2+f_1^{\dagger}f_2-f_1f_2^{\dagger}.
 \end{align*}
\end{example}

\subsection{Tensor product of gates}
To describe effectively quantum logic circuits in the complex Clifford algebra, it remains to discuss representations of parallel quantum gates, i.e. representations of tensor product of gates. First of all we realize that the representation of tensor products of states is already determined by  Proposition \ref{n_qubit}. Namely, a $n$-qubit  $\ket{i_1\cdots i_n}=\ket{i_1}\otimes\cdots\otimes\ket{i_n}$ is in the Clifford algebra represented by geometric product of representations of individual qubits. Hence a tensor product $\ket{\varphi} \otimes \ket{\psi}$ is represented by geometric product $\varphi\psi$, where the spinors $\varphi,\psi$ are assumed to lie in disjoint vector spaces viewed as two orthogonal subspaces of their union. Now consider an action of a tensor product of gates $\lambda\otimes \mu$ given  by unitary elements $\lambda,\mu$ of the Clifford algebra on such a state. The resulting state $\lambda \varphi \otimes \mu \psi$  is represented by $\lambda \varphi\mu \psi$ which is different from $\lambda \mu \varphi\psi$ in general due to the skew-symmetry of the geometric product. Namely, for two blades $e_A,e_B$ determined by disjoint multi-indices $A,B$ we have
\begin{align*}
e_A e_B = (-1)^{|A| |B|} e_B e_A.
\end{align*}
and so the Clifford algebra has the structure of a superalgebra. Hence the geometric product does not represent the ordinary tensor product but it represents the super tensor product. It has the same structure as a vector space but with the multiplication rule determined by
\begin{align} \label{STP}
(e_A e_B) (e_C  e_D)=(-1)^{|B| |C|}(e_Ae_C )( e_Be_D)
\end{align}
on blades.
Consequently, the geometric product identifies complex Clifford algebra for $n$-gates  with the super tensor product of Clifford algebras for single qubit gates. The ordinary ungraded tensor product $\lambda_1 \otimes \cdots \otimes \lambda_n$ of gates $\lambda_k$ from distinct copies of $\mathbb C_2$ is in Clifford algebra $\mathbb C_{2n}$ represented by $ \lambda_1\cdots \lambda_n$ only up to the sign. Although this sign depends on the $n$-qubit on which we act by \eqref{STP}  in general, it is completely determined by the set $\{\lambda_1,\dots,\lambda_n\}$ in the case that $\lambda_k$ for each $k=1,\dots,n$ is one of the basis elements of $\mathbb C_2$
\begin{align} \label{lambda_k}
\lambda_k\in\{f_k f_k^\dagger,f_k^\dagger f_k, f_k, f_k^\dagger\}.
\end{align}

\begin{proposition} \label{TP}
A tensor product $ \lambda_1 \otimes \cdots \otimes \lambda_n$, where  $\lambda_k \in \{ f_k f_k^{\dagger},f_k^{\dagger} f_k, f_k, f_k^{\dagger}   \} $ for each $ k = 1, \dots , n$, is represented by geometric product $(-1)^s \lambda_1 \cdots \lambda_n$,  where the sign is determined by the cardinality of the sets $S_i$, such that  $s=\sum_i |S_i|$, where
\begin{align} \label{S}
    S_i=
    \{ \ell < i \: : \: \lambda_l =f_{\ell} \text{ or }  \lambda_{\ell} = f_{\ell}^{\dagger}f_{\ell} \}
    \text{ in the case if  }{\lambda_i = f_i \text{ or }\lambda_i
    =f_i^{\dagger}} .
\end{align}
\end{proposition}
\begin{proof}
A  $n$-qubit $\psi_1 \otimes \cdots \otimes \psi_n$ is represented by geometric product $\psi_1\cdots\psi_n$ of mutually orthogonal spinors $\psi_k$ by \eqref{Sn_basis}.
Representation of a qubit obtained upon the action of $\lambda_1 \otimes \cdots \otimes \lambda_n$ on this $n$-qubit  is given by  $$\lambda_1\psi_1 \cdots \lambda_n\psi_n=(-1)^p\lambda_1\cdots\lambda_n\psi_1\cdots\psi_n$$ since $\lambda_k,\psi_k$ are orthogonal to $\lambda_\ell,\psi_\ell$ for $k\neq \ell.$ Roughly speaking, the sign is determined by how many times we need to commute to get all elements $\lambda_k$ to the left hand side. Thus it depends on spinors $\psi_\ell$, $\ell<k$ which are combinations of components of grade one or two in general. Only the commuting with grade one components does change the sign. However the grade one spinors are multiples of $f_\ell^\dagger$ and they are annihilated by all elements of $\mathbb C_{2n}$ except  elements $\lambda_\ell=f_\ell$ and  $\lambda_\ell=f_\ell^\dagger f_\ell$ which act nontrivially. Hence the number of commutation steps that push $\lambda_k$ to the left is equal to the number of such elements $\lambda_\ell,$ $\ell<k.$
\end{proof}

\begin{example} Let us construct 2-qubit gates 	$X \otimes Y$ and $Y \otimes X$ according to Proposition \ref{TP}. First we write these gates as a sum of tensor products of basis Witt basis elements and then for each such summand we compute the cardinality of set $S$ giving the sign of the corresponding geometric product.
	\begin{align*}
	X \otimes Y &= i(f_1^\dagger \otimes f_2^\dagger-f_1^\dagger \otimes f_2+f_1 \otimes f_2^\dagger-f_1 \otimes f_2)\\
	&=i(f_1^\dagger f_2^\dagger-f_1^\dagger  f_2-f_1  f_2^\dagger+f_1  f_2)\\
	Y \otimes X &= i(f_1^\dagger \otimes f_2^\dagger+f_1^\dagger \otimes f_2-f_1 \otimes f_2^\dagger-f_1 \otimes f_2)\\
	&=i(f_1^\dagger f_2^\dagger+f_1^\dagger  f_2+f_1  f_2^\dagger+f_1  f_2)
	\end{align*}
	Let us assume even more simple example of a $X$-gate with a parallel qubit without any gate. If the gate is acting on the first qubit we get a resulting 2-qubit gate $X \otimes \operatorname{id}=X_1$. However, acting on the second qubit we need to write the identity representation as $1=f_1 f_1^\dagger+f_1^\dagger f_1$ since idempotent $K_1=f_1^\dagger f_1$ makes the change of sign in contrast to idempotent $I_1=f_1 f_1^\dagger$,
	\begin{align*}
	\operatorname{id} \otimes X=(I_1+K_1) \otimes X_2=I_1X_2-K_1X_2=f_1 f_1^\dagger(f_2^\dagger +f_2)-f_1^\dagger f_1(f_2^\dagger +f_2).
	\end{align*}
	The representations of controlled gates from example \ref{ex_2gates} can be constructed from tensor product of single qubit gates as follows.
	\begin{align*}
\lambda_{\operatorname{CNOT}}&=I_1\otimes 1 +K_1 \otimes X_2=I_1-K_1 X_2\\
\lambda_{\operatorname{CZ}}&=I_1\otimes 1 +K_1 \otimes Z_2=I_1+K_1 Z_2
	\end{align*}

\end{example}

\begin{example}
	The spinor space $\mathbb S_3$ representing states of 3-qubits has dimension $2^3=8$ in Clifford algebra $\mathbb C_{6}$ of dimension $2^6=64$. Using the primitive idempotent \eqref{In} and the orthonormal basis representation \eqref{Sn_basis} in terms of the Witt basis the Toffoli gate is represented by
	\begin{align*}
	\lambda_{\operatorname{CCNOT}}&=(I_1\otimes I_2  +I_1\otimes K_2 +I_2\otimes K_1)\otimes \operatorname{id} +K_1\otimes K_2\otimes X_3\\
	&=I_1I_2+I_1K_2+I_2K_1+K_1K_2X_3=1-K_1K_2+K_1K_2X_3\\
	&=1+f_1^\dagger f_1f_2^\dagger f_2(f_3 + f_3^\dagger-1). \\
\lambda_{\operatorname{CSWAP}}&=I_1\otimes \operatorname{id}+K_1\otimes \lambda_{\operatorname{SWAP}} \\
&=I_1+K_1(I_2I_3+K_2K_3+f_2^{\dagger}f_3-f_2f_3^{\dagger})\\
&=f_1 f_1^\dagger + f_1^\dagger f_1(f_2 f_2^\dagger f_3 f_3^\dagger+f_2^\dagger f_2 f_3^\dagger f_3+f_2^{\dagger}f_3-f_2f_3^{\dagger})
\end{align*}
\end{example}

\section{Quantum computing in real Clifford algebras}\label{realGA}
Accidental isomorphism can be used to formulate intrinsically complex quantum computing in a real framework. We show two ways how to see a qubit in real Clifford algebra $\mathbb G_3$, i.e the GA induced by the standard euclidean inner product of signature $(3,0).$ The first approach appears in literature, see  \cite{Lasenby1, dl,cm}, and describes qubit states as even elements in this algebra or equivalently as unite quaternions. The second approach is new and follows from the complex representation of qubits described above. For the other know concepts see \cite{havel,soma,grego}. We also mention how to deal with multiple qubits and multiple qubit gates in the real case.

\subsection{A quaternionic qubit} \label{H_qubit}
The transition from complex  to real framework which appears in literature is based on the well known coincidental isomorphism of Lie algebras $\mathfrak{su}(2)\cong \mathfrak{so}(3)$, or more precisely, on the corresponding isomorphism of Lie groups
\begin{align} \label{SUSpin}
SU(2)\cong Spin(3),
\end{align}
and the isomorphism of these groups with the group of unite quaternions. We can easily describe these isomorphisms explicitly by realizing  Lie algebra $\mathfrak{so}(3)$ as bivectors in Clifford algebra $\mathbb G_3$ and Lie group $Spin(3)$ as elements of even grade in $\mathbb G_3$. Namely,  in terms of Pauli matrices the Lie algebra isomorphism can be defined by mapping $i_{\mathbb{C}}\sigma_k\mapsto i\sigma_k,$ $k=1,2,3$, where we denote  the usual complex unite by $i_{\mathbb{C}}$ in order to distinguish from pseudoscalar  $i=\sigma_{1}\sigma_2\sigma_3$ in $\mathbb G_3$, while $\sigma_k$ on the right hand side is seen as a vector in $\mathbb G_3$ satisfying $\sigma_k^2=1$. Consequently, using the Einstein summation convention, we get a Lie group isomorphism \eqref{SUSpin} of a form
\begin{align} \label{iso}
\begin{pmatrix}
a^0+a^3 i_{\mathbb{C}} & a^2 +a^1 i_{\mathbb{C}} \\
-a^2+a^1 i_{\mathbb{C}} & a^0-a^3 i_{\mathbb{C}}
\end{pmatrix}
\mapsto
a^0+a^1 \sigma_{2}\sigma_3+ a^2\sigma_{3}\sigma_1+a^3\sigma_{1}\sigma_2=a^0+a^k\sigma_k^*
\end{align}
where the coefficients $a^0,a^1,a^2,a^3\in\mathbb{R}$ and $\sigma_k^*=\sigma_k i = i \sigma_k$ is the duality defined by pseudoscalar $i=\sigma_{1}\sigma_2\sigma_3$.
Assigning the quaternionic unites to $\sigma_k$, $k=1,2,3$, defines an isomorphism with unite quaternions.
A general state of a qubit is identified with the first column of the matrix on left hand side, thus in the real Clifford algebra $\mathbb G_3$ is represented by
\begin{align*}
\ket{\psi}=\begin{pmatrix}
a^0+a^3 i_{\mathbb{C}}  \\
-a^2+a^1 i_{\mathbb{C}}
\end{pmatrix}
\leftrightarrow \psi = a^0 +a^k \sigma_k^*
\end{align*}
In particular, the standard computational basis $(1,0)$ and $(0,1)$ in $\mathbb C^2$ is in the real Clifford algebra formulation represented by
\begin{align} \label{basisH}
\ket{0}=
1 \text{ and }
\ket{1}=
-i\sigma_2=\sigma_{1}\sigma_3,
\end{align}
respectively.
The identification \eqref{iso} also determines explicit formulas for a Hermitian inner product and a representation of Pauli matrices on even subalgebra $\mathbb G_3^0,$ namely for $\varphi,\psi\in\mathbb G_3^0$ we have
\begin{align} \label{productH}
\braket{\varphi|\psi}&=[\tilde{\varphi}\psi]_0-[\tilde{\varphi}\psi\sigma_1\sigma_2]_0 i_{\mathbb{C}}, \\
 \label{PauliH}
\hat{\sigma}_k\ket{\psi} &\leftrightarrow \sigma_k\psi\sigma_3.
\end{align}
These formulas can be explained by viewing the unitary group $SU(2)$ as $SO(4)\cap GL(2,\mathbb C)$, i.e. as the group of orthogonal transformations with respect to a real scalar product of signature $(4,0)$ commuting with an orthogonal complex structure. A choice of a scalar product and a complex structure on even elements $\mathbb G_3^0$ then defines an Hermitian inner product on this space by a standard construction and thus defines an isomorphism \eqref{SUSpin}. In our case, the scalar product is given  by $(\varphi,\psi)=[\tilde{\varphi}\psi]_0$ and the complex structure $J$  is defined by $J\psi=\psi i\sigma_3=\psi \sigma_{1}\sigma_2$. Indeed, for such a choice  the Hermitian product \eqref{productH} is constructed as
\begin{align*}
\braket{\varphi|\psi}=(\varphi,\psi)-(\varphi,J\psi)i_{\mathbb{C}}.
\end{align*}
The action of Pauli matrices in $\mathbb G_3^0$  given by \eqref{PauliH} keep the scalar product invariant and commutes with the complex structure and thus keeps this Hermitian product invariant.
\begin{remark}
This point of view also allows to see the freedom of quaternionic representation of qubits. Namely, choosing a different complex structure or modifying the scalar product on $\mathbb G_3^0$ would lead to an isomorphism \eqref{SUSpin} different from \eqref{iso} leading to representations of computational basis, Hermitian product and Pauli matrices different from \eqref{basisH}, \eqref{productH} and \eqref{PauliH}.
\end{remark}

The reality of this qubit representation implies that multiple qubits are represented in a quotient space defined by so called correlator. Namely, representing qubits in the  real geometric algebra $\mathbb G_3^+$  the space of $n$-qubits is $\mathbb G_3^+\otimes\cdots\otimes\mathbb G_3^+$ instead the  tensor power of $n$ copies of $\mathbb C^2$. However this  is the complex tensor product according to axions of the quantum mechanics. If we want to have a fully real description, including the real tensor product, we need to identify complex structures $J_k=i\sigma_3^k=\sigma_1^k\sigma_2^k$ (representing the multiplication by complex unite) in all copies. This can be done by introducing the $n$-qubit correlator
\begin{align*}
E_n=\prod_{k=2}^n \frac12 (1-i\sigma_3^1 i\sigma_3^k).
\end{align*}
Indeed, this element satisfies $E_nJ_k=E_nJ_\ell$ for all $k,\ell = 1,\dots, n$ and thus it defines a quotient space $\mathbb G_3^+\otimes\cdots\otimes\mathbb G_3^+/E_n$ with a complex structure $J_n=E_nJ_k=E_ni\sigma_3^k$. Multivectors belonging to this space can be regarded as $n$-qubit states.

\subsection{A real complex qubit}
Another way how to describe states of a qubit by a real algebra is to transfer its complex representation described in section \ref{qubit_section} via the accidental isomorphism of real algebras
\begin{align} \label{isoC2G3}
\mathbb C_2 \cong \mathbb G_3.
\end{align}
In order to obtain an explicit representation of a qubit we choose a concrete realization of  this isomorphism. Namely,
in terms of the Witt basis of $\mathbb C_2$ and an orthonormal basis $\sigma_k$ of $\mathbb R^3$ we consider the isomorphism given by mapping
\begin{align*}
1&\mapsto 1  & i_{\mathbb{C}}&\mapsto \sigma_1\sigma_2\sigma_3\\
f&\mapsto \tfrac12(\sigma_1-\sigma_1\sigma_3) & i_{\mathbb{C}}f&\mapsto \tfrac12(\sigma_2\sigma_3-\sigma_2)\\
f^\dagger&\mapsto \tfrac12(\sigma_1+\sigma_1\sigma_3) & i_{\mathbb{C}}f^\dagger&\mapsto \tfrac12(\sigma_2\sigma_3+\sigma_2)\\
ff^\dagger&\mapsto \tfrac12(1+\sigma_3) & i_{\mathbb{C}}ff^\dagger&\mapsto \tfrac12(\sigma_1\sigma_2+\sigma_1\sigma_2\sigma_3)
\end{align*}
In particular, the complex unite $i_{\mathbb{C}}$ is mapped to trivector $\sigma_1\sigma_2\sigma_3\in\mathbb G_3$ and the primitive idempotent $I=f f^\dagger\in \mathbb C_2$ is mapped to real idempotent
\begin{align*}
I_\mathbb R= \frac12(1+\sigma_3) \in \mathbb G_3.
\end{align*}
Using this idempotent the equivalence between the classical description of a qubit as a complex vector and as an element of $\mathbb G_3$ based on this isomorphism reads
\begin{align*}
\ket{\psi}=\begin{pmatrix}
a^0+a^3 i_{\mathbb{C}}  \\
a^1+a^2 i_{\mathbb{C}}
\end{pmatrix}
\leftrightarrow \psi = (a^0 +a^1\sigma_1 + a^2\sigma_2 + a^3\sigma_1\sigma_2)I_\mathbb R,
\end{align*}
where $a^0,a^1,a^2,a^3\in\mathbb R.$
Note that, in contrast to the quaternionic representation described in the previous section, a qubit is represented by a multivector in $\mathbb G_3$ containing blades of both even and odd grades in this case. In particular, the computational basis is given by
\begin{align}
\ket{0}= I_\mathbb R\text{ and } \ket{1}=\sigma_1 I_\mathbb R.
\end{align}
The Hermitian inner product on $\mathbb G_3$ is given by the transition of the Hermitian product on $\mathbb C_2$ given by \eqref{HP} via isomorphism \eqref{isoC2G3}. Looking at the prescription of the isomorphism we see that the Hermitian conjugation of basis elements is mapped to the reverse of the corresponding images in $\mathbb G_3.$ Hence we have a particularly simple formula for the Hermitian product in this case, namely for two qubits $\varphi,\psi\in\mathbb G_3$ we have
\begin{align}
\braket{\varphi|\psi}=[\tilde{\varphi}\psi]_0.
\end{align}
Our formula for isomorphism \eqref{isoC2G3} yields also a particularly simple formula for the representation of Pauli matrices, namely
\begin{align}
\hat{\sigma}_k\ket{\psi} &\leftrightarrow \sigma_k\psi.
\end{align}

Although this real representation of a qubit is quite elegant, the representation of multiple qubits is as complicated as in the case of the quaternionic qubit  described in \ref{H_qubit}. Due to its reality we need to use a correlator to identify the multiplication by complex unite in each slot of the tensor product $\mathbb G_3\otimes\cdots\otimes\mathbb G_3$. Since this is a common feature of all real description of qubits we believe that the right way is to use the complex GA as described in section \ref{QCinC} above.

redpajama_set_name: RedPajamaArXiv


text:
BREAKING: 35 Charged In Massive Federal Investigation Out Of Chicago
Posted By: Dino Porrazzo March 13, 2019
35 Defendants Charged in Joint Federal and State Investigation into Heroin and Fentanyl Delivery Service in Chicago Area
CHICAGO — Thirty-five individuals are facing criminal charges as part of a joint federal and state investigation into heroin and fentanyl trafficking in Chicago.
During the multi-year investigation, dubbed "Operation Road Rage," law enforcement seized a kilogram of heroin, most of which contained fentanyl, approximately $100,000 in cash, five firearms, and four vehicles. Much of the alleged drug trafficking occurred in the West Garfield Park neighborhood on the West Side of Chicago. Many of the defendants allegedly distributed heroin and fentanyl-laced heroin to customers in the Chicago area, with drivers dispatched to make deliveries after customers placed orders on a telephone hotline.
The investigation was jointly conducted by the Organized Crime Drug Enforcement Task Force (OCDETF) and the Chicago High Intensity Drug Trafficking Task Force (HIDTA). The task forces partner with federal, state and local law enforcement agencies to identify, disrupt and dismantle the most serious drug trafficking organizations.
Criminal complaints unsealed in U.S. District Court in Chicago charge 22 defendants with federal drug offenses. Many of the federal defendants were arrested Thursday, and detention hearings will be held this week in federal court in Chicago. Thirteen other defendants were charged in state complaints, and many of them were also arrested Thursday. The state defendants have begun making initial appearances in Cook County Criminal Court.
The charges were announced by John R. Lausch, Jr., United States Attorney for the Northern District of Illinois; Kimberly M. Foxx, Cook County State's Attorney; Brian McKnight, Special Agent-in-Charge of the Chicago Field Division of the U.S. Drug Enforcement Administration; Gabriel L. Grchan, Special Agent-in-Charge of the Internal Revenue Service Criminal Investigation Division in Chicago; Nicholas Roti, Director of HIDTA; and Eddie Johnson, Superintendent of the Chicago Police Department. Valuable assistance was provided by the U.S. Marshals Service. Assistant U.S. Attorneys Misty Wright and Shy Jackson represent the government.
Fourteen federal defendants are charged in U.S.A. v. Robinson, et al (19 CR 201). The complaint describes a drug trafficking organization in which heroin hotline operators and dispatchers often referred to themselves as "Sean." Customers intending to purchase heroin and fentanyl-laced heroin called a designated telephone number operated by members of the "Sean" drug trafficking operation, the complaint states. The dispatchers then sent the organization's many distributors to various West Side locations to conduct the drug deals, the complaint states. Undercover law enforcement officers disrupted the Sean organization by conducting approximately 50 narcotics purchases after calling the Sean phones and meeting with the distributors, according to the complaint. Charged with drug conspiracy are four individuals who, at various times, were responsible for dispatching the distributors: THOMAS CLEVELAND, 27, of Chicago, FREDERICK GILES, 26, of Chicago, AMELIA SMITH, 30, of Country Club Hills, and MICHAEL ROBINSON, 24, of Country Club Hills; as well as ten alleged distributors: WILLIE BLAIR, 32, of Chicago, LAWRENCE CLARK, 26, of Chicago, CARDELL COLEMAN, 21, of Chicago, JOLISA GAINES, 29, of Chicago, D'ANGELO GILES, 21, of Chicago, TYRELL KELLY, 27 of Chicago, KEITH MANNING, 27, of Bellwood, DEVONTE SMITH, 26, of Chicago, KAJUAN SMITH, 30, of Chicago, and DESMOND WHITE, 25, of Chicago.
Two federal defendants are charged in U.S.A. v. Chose, et al (19 CR 202). Charged with drug conspiracy are ANNA T. CHOSE, 54, of Morris, and ROXANNE E. CONN, 56, of Wilmington. The charges accuse the pair of being regular customers of the Sean drug trafficking organization. The complaint describes three instances in the summer of 2018 when Chose and Conn purchased distribution quantities of narcotics from Robinson, and alleges that they regularly purchased narcotics from the Sean drug trafficking organization.
Clark and three other federal defendants are charged in U.S.A. v. Pitts, et al (19 CR 204). Charged with drug conspiracy and distribution offenses are MICHAEL PITTS, 33, of Bellwood, TEVIN FORD, 27, of Chicago, and JHAMAAL HANEY, 26, of Chicago. According to the complaint, the four defendants conspired to distribute approximately 94.8 grams of fentanyl-laced heroin to an undercover law enforcement officer from August 2017 to March 2018. The charges also hold Pitts personally responsible for distributing a total of approximately 386.5 grams of heroin and fentanyl-laced heroin to undercover officers from August 2017 to May 2018. One of Pitts' customers suffered a drug overdose in May 2018 after Pitts distributed 1.13 grams of fentanyl-laced heroin to him, the complaint states. The man appeared to be unconscious when Pitts pulled him out of a vehicle and left him at a nearby bus stop, the complaint states. Chicago Fire Department personnel arrived on the scene and worked to successfully revive the man, the complaint states.
Three other federal defendants are charged with narcotics offenses as part of the investigation: JUSTIN BANASIAK, 35, of Chicago; JACLYN ROWLEY, 31, of Michigan City, Ind.; and MATTHEW ROSOLIK, 32, of Trail Creek, Ind.
The charges in the complaints carry maximum penalties of between 20 years and life imprisonment. If convicted, the Court must impose reasonable sentences under federal sentencing statutes and the advisory U.S. Sentencing Guidelines.
The public is reminded that charges contain only accusations and are not evidence of guilt. The defendants are presumed innocent and entitled to a fair trial at which the government has the burden of proving guilt beyond a reasonable doubt.
Northern District of Illinois
GOD BLESS AMERICA!
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redpajama_set_name: RedPajamaCommonCrawl


KPMG AEOI Reporting provides a technology solution for reporting under the Foreign Account Tax Compliance Act (FATCA) and the OECD Common Reporting Standard (CRS) and is a global solution for Financial Institutions.
With the CRS now operating across more than 100 jurisdictions, many organisations have found reporting under the Automatic Exchange of Information (AEOI) regimes challenging without a reliable technology solution. The significant increase in volume and complexity facing organisations in the early years, whether they operate in one or 50 jurisdictions, means staying on top of their reporting obligations can be extremely difficult.
Submission methods with some jurisdictions requiring registration or notification prior to filing or in some cases, specific technology to transmit the file.
A lack of test facilities by many tax authorities aswell as the difficulty in extracting comprehensive data of sufficient quality for all reportable accounts only compounds these challenges.
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Both options are widely used by a large and diverse group of financial institutions around the world and can be customised to meet your specific needs.
KPMG recognise the challenge faced by organisations when dealing with data issues in the short period after year end in order to meet the reporting deadline and, as such, have developed the KPMG AEOI data health check.
Clients can run data through KPMG's proprietary rules engine on a regular basis throughout the year, enabling data issues to be identified during the account opening validation period and be resolved prior to the reporting season. The data health check can be run with whatever frequency the client decides and is automatically updated with any new data rules mandated by tax authorities or created by KPMG, thus ensuring clients are always using the very latest ruleset.
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Mandatory and optional schema data points are identified and maintained in the validation checks which help to ensure you are collecting the required data from your account holders for completeness and accuracy of returns.
� KPMG AEOI Reporting is a proprietary solution developed, owned and managed by KPMG. It is maintained through local AEOI experts in each country with the aim of ensuring the rules and country information stay correct and up to date at all times.
� Provides clients with confidence they can submit correct returns, using tried and tested technology, already used by large numbers of financial institutions across the world.
� A global technology solution covering all major jurisdictions, having been successfully used across the world, KPMG member firms provide the confidence to allow clients to correctly file returns in all jurisdictions in which they operate.
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� The dashboard enables management to view the status and progress ofreporting across multiple business units and locations.
Provides a reporting solution to convert data from core systems into the correct reporting formats 'behind the scenes', removing the need for clients to understand XML or the underlying schemas.
Provides a centralised oversight of the work flow on a customised dashboard to track reporting progress and submissions across organisations with multiple entities in different jurisdictions.
Is quick and easy to install into the clients organisation and, using innovative distribution, is designed to ensure real-time updates to schemas and templates are made automatically without any action required by clients.
Is fully scalable to support organisations of a wide range of sizes and complexity across all sectors.
Acts as a data validation engine identifying errors and missing customer data which could otherwise prevent or delay the submission of returns. Our broad-ranging, proprietary data validation engine runs hundreds of checks on your data before submission.
Carries out detailed Tax Identification Number (TIN) validation to highlight potential data issues with this key data point.
Helps to ensure all sensitive customer data is securely held behind the organisation's firewall throughout the creation of the submission file.
Submits FATCA returns in Model 2 and Non-IGA jurisdictions directly into the IRS IDES platform using automation within the tool.
Provides useful on-screen field-level guidance and information about reporting and submission requirements as well as an embedded user guide.
2013 has proven to be a year of consolidation, in which we embarked on various initiatives essential for the growth of our practice over the next few years.


text:
ma tianyu tv shows
First naive and used to luxury, she has to adapt to harsh prison conditions, and harsher inmates. She has also appeared in several variety shows, including 'Keep Running' (2014 to present), 'King of Glory' (2017) and 'Clash Bots' (2018). Starring an almost all-women cast (except for the coach, played masterfully by podcast icon Marc Maron), it's the story of how a crazy wrestling show was put together in the 1980s called Gorgeous Ladies of Wrestling. The Ice Tribe prince journeys to a sacred mountain to vanquish evil but soon finds himself at war with the Fire Tribe, led by his long-lost brother. In a lot of ways, Atypical is the perfect 2017 Netflix-age coming-of-age sitcom: it's funny and smart, but also keen to be realistic. It's only a mellow take on the women prison genre that was perfected outside the States. There is a lot that comes to the surface from the working-class kids clashing with the wealthy. He is an actor, known for Gu jian qi tan (2014), Ice Fantasy (2016) and The Vigilantes in Masks (2010). Watch out for Sister Michael, she is hilarious. In Norsemen, a variety of characters try to fight this basicness. Perfectly acted and featuring funny as well as absurd moments, GLOW is a great show that you can binge on Netflix without noticing the episodes fly by. Waller-Bridge settles into the hospital as well, and the six twenty-somethings become property guardians of the hospital building. A captivating documentary series on the struggling state of the police department in Flint, Michigan; and by extension a large proportion of American cities. Below is a list of the 10 most popular Chinese TV series in the year 2016, which one do you like best? An anti-terrorism officer with a questionable past is put in contact with a woman who traveled to Syria and wants to get back to Sweden. He definitely lost the baby fat, retained the baby face to grow into a handsome hunk. If you make it to the second season, I genuinely think it's the better watch. Her strong personality and the events that happen to her allow the show to explore themes of vulnerability, trauma, and even life and death. Madina Memet Lan Shang. It is truly one of the best sitcoms ever made. Be the first to review this item 2014 TV-14 Young Sherlock tells of a series of thrilling and cryptic cases faced by 26-year-old protagonist Di Renjie. With everything from game footage to interviews and behind the scenes politics, it's also the perfect fix for anyone who misses sporting events. The difference: It doesn't follow people who are wronged by the system or who are misunderstood, but women that have actually done violent things, and continue being violent in prison. Angelababy has also appeared in several music videos, such as 'Ten Million Youth' (2008) and 'Can't Stand It' (2010). The script was written by Canadian filmmaker extraordinaire Sarah Polley (Stories We Tell). Comedian Andrea Savage makes a TV show based on her life, or rather, that is her life (it's a thin line). I see puffy red cheeks and a forehead dot too though you can clearly see the resemblance to the actor even as he is in his early thirties now. It is said that Ma Tianyu is expected to star in the male, "lily of the valley couple" can on-line! Themes of teenage knife crime (in the second season), the rise of right-wing sentiment, and a problematic relationship with the police are constantly being brought up without ever feeling forced. What's on TV & Streaming What's on TV & Streaming Top Rated Shows Most Popular Shows Browse TV Shows by Genre TV News India TV Spotlight. Film-film yang tayang di sini berasal dari web pencarian di internet. © 2020 agoodmovietowatch, all rights reserved. Netflix, Amazon, Amazon Prime, and Hulu are trademarked brands, all rights reserved. With Jie Deng, Fenqiang Ouyang, Xiaoxu Chen, Ting Li. And Atypical is about Sam's family almost as much as it is about him, and how they adjust to his new quest for self-discovery. The Four《四大名捕》 The 2014 mainland version drama The Four is a magical martial arts costume drama starring Zhang Han, Mickey He, Chang Chun-Ning, Yang Yang and Jia Qing. Rewatch value: Movies. Mo Ge Li is a boy from the jungle who was raised by the forest caretaker, Grandpa Lin. Don't ask me about the title, it does not make sense! Ma Tianyu - Topic; Playlists About; Home Trending History Get YouTube Premium Get YouTube TV Best of YouTube ... Movies & Shows News Live Fashion Learning Spotlight 360° Video This Australian show might have the same set-up as Orange is the New Black — following a recently incarcerated woman as she discovers a new world — but the two series couldn't be more different. Everyone is trying to do what they think is right, which still makes for very complicated situations. Watch full episodes of Ice Fantasy with subtitles. The series will air on Tencent starting March 27, 2018. Reviews and scores for Movies involving Tianyu Ma. top rated. Ku Hye Sun and Ahn Jae Hyun created much buzz when they announced they were dating after they starred together in the 2015 medical drama "Blood." They married on May 21, 2016. The series premiered on Mon Jul 01, 2019 on Hunan TV and Episode 52 (S01E52) last aired on Tue Jul 30, 2019. They take fashion risks like putting horns on a helmet for the first time, try to be more culturally diverse by interacting with a Roman slave, fall in love and raid tribes (often at the same time), and so on – you know, every day Viking stuff. What's on TV & Streaming What's on TV & Streaming Top Rated Shows Most Popular Shows Browse TV Shows by Genre TV News India TV Spotlight. Esty, a nineteen-year-old girl who is unhappy in her arranged marriage, escapes this community and travels from New York to Berlin in hopes of starting a new life. The story restarts, at the same birthday party, staring at herself in the mirror. Based on the famous book by Yana Vagner. It takes place in 1990s Northern Ireland where civil unrest reigns. As you'd expect with a show featuring this many characters and such a new world, the first season is not as entertaining as could be, but the show becomes its full-self as a true epic in season 2. Love portrays their love story as an example of relationships by default, chemistry that stems more from the need to be in a relationship than any physical or intellectual attraction. 2016 TV-14 1 Season Fantasy TV Shows The Ice Tribe prince journeys to a sacred mountain to vanquish evil but soon finds himself at war with the Fire Tribe, led by his long-lost brother. ~~ Adapted from Rudyard Kipling's "The Jungle Book". The drama features famous actors like Yang Mi, Jia Nailiang, Ma Tianyu, Zheng Shuang and Chen Zihan. Caliphate is fascinating, deeply thrilling, and made with an authenticity that sees past cheap stereotypes to offer genuine insight into what drives young people to become terrorists. Ma Tianyu is a Mandopop artist who rose to fame after winning sixth place and "most popular" at the 2006 My Hero Competition, a Chinese talent competition. … You'll also meet the teenager's heartbroken family and a disorganized prosecutor. Later, the two continued to collaborate on two TV series, but although the two looked like a talented couple, but In real life, neither of them seemed to pierce this piece of paper. Wentworth is more Breaking Bad than Orange is the New Black. Ever since placing sixth in singing competition My Hero and voted the most popular contestant by online viewers, the 32-year-old has gradually risen in popularity, particularly following the dramas Swords of Legends and Ice Fantasy . Violent, very Western, and in a breath of fresh air: female. One of the best food shows you can watch today. (Source: ChineseDrama.info) Here is what they have in common: the aesthetic, great music, and they're both about the disappearance of a child. Swords of Legends. Help us expand our database by adding one. It was first filmed in Beijing in Mandarin, then the series was dubbed in Cantonese and Shanghainese. Edit Translation, Intriguing opener but terrible plot development, A romance begins between Mo Ge Li boy from the jungle and his muse when she accidentally introduces him to civilization. Yang Zi really hit me emotionally when she cried, she always performs her character so well. The car accident story provides a more accessible framework for them to understand his condition. People around him, especially at the exploitative millennial magazine "eggwoke" where he is an intern, start treating him differently. Known For. Login to create it. Ice Fantasy. [31.12.20] Ma Tianyu sang his signature song the damn gentleness also popularly known as Gai Se De Wen Rou on Hunan Tv New Year 2021 Concert #Matianyu #Rayma #hunantvnewyear2021 Ray Ma Tianyu International Fanpage Clocking just 15 minutes per episode, Special is like a candy bar. He learns the ways of the world with Ling Xi's help and acts as a bridge bringing her and the people around him closer to nature. Kim Hee-sun Lian Ji. Starring: Feng Shaofeng, Victoria Song, Ma Tianyu Swords of Legends. Beauty Private Kitchen. A monster crosses paths with the foursome from the classic Chinese story Journey To The West. TV Premiere Dates. Watch HD Ma Tianyu Movies And Shows Online For Free and Download the latest Ma Tianyu movies and shows without registration For three years, the makers of this docuseries gained in-depth access to ICE and other government agencies to document the current state of the U.S. immigration system. It follows Gus and Mickey, two damaged people trying to recover from bad breakups. So for example the first episode about pizza goes to Japan to investigate a new pizza in a Michelin-star restaurant, but also goes to Domino's. She co-created the show with Amy Poehler and Sleeping With Other People director, Leslye Headland. ~~ Adapted from Rudyard Kipling's "The Jungle Book". Godless is a show about strong bad-ass women that govern their own town in the late 1800s. Its tales of race and institutional bias are compelling, but its greatest strength is the script. Click here to login or here to sign up. Every episode, every scene, every character are made with extreme care, resulting in sometimes longer than necessary sequences. Browse Beibi Gong movies and TV shows available on Prime Video and begin streaming right away to your favorite device. If you liked Lovesick, another Netflix British comedy about sex trouble, you will like Sex Education. Global. This new Russian thriller show has an immaculate production value that can rival the biggest Hollywood shows. Russian Doll can be summarized in what Nadia screams later that night: "the universe is trying to f*ck with me, and I refuse to engage". The last scene that felt off was mostly due to poor martial arts choreography. Immigration Nation looks at how ICE functions from within, but it also focuses on the human toll of its methods. Funny, natural and entertaining – I'm Sorry is a joy to watch. What happens in the personal lives of celebrities after they tie the knot? ... It's hard to believe a TV show can come out today and still manage to be so different from the rest, but Special does it. Man, don't watch this show hungry. In an age where every show gets called "original" the minute after it comes out, this amazing series from the creators of Orange is the New Black will actually make you go "no, that show is different!". Yes, it's long, but if you get The Get Down, it is one of the best shows on Netflix. ... TV Shows 16; Department Acting 22; URL. Edwin Semzaba: Biography, Education, Acting Career, Wife, Family, Kids? Chang has almost a f*ck it attitude towards the food industry that's not only refreshing to watch on him, but also disarms his guests and sparks interesting conversation. K-drama's are much better in quality (dubbed in Cantonese or with Chinese subtitles), my parents are watching K-drama's these days instead of TVB drama's. And also please sit through the first episode. Worst of all, a new recruit at the newspaper is assigned to work with him. In 2006, Ma Tianyu entered the My Hero competition. Every episode is 20 minutes or less and has about five or six skits. The episodes are quick, have clear arcs and plot; and yet, you won't be able to shake the feeling that you're watching something much deeper than a Ricky Gervais comedy. This post will be only about Ma Tian Yu,because some international fans want it … But his pessimism and annoyance with people keep delaying him from watching it. Something you can easily find yourself watching many episodes in one take. Up to that point, George had only dated men. Following them as they navigate their new relationship is sometimes funny, more times heartbreaking, but always charming. This excellent new miniseries is a drama that takes place in an ultra-Orthodox Jewish family. The cult, led by a controversial Indian guru, drew worldwide attention to its beginnings in India and then to its conflict with the locals once it relocated to the United States. It quickly becomes apparent that the officer is using the same call to try to establish if the father is in the U.S. legally or if he should be deported. It's difficult to describe this miniseries as just one thing: it has elements of true crime, but it's more than just a "Netflix true-crime show." It's also about an immensely empathetic and likable family man who joins the fight against the opioid epidemic. Ice Fantasy. After a rape victim is not believed by the detectives who are assigned to her case, details of a similar incident surface elsewhere. He was the regional champion for the province of Wuhan and came sixth in the final round, and was chosen as the most popular contestant. p open profile menu. Tianyu Ma and Regina Wan in Secret of the Three Kingdoms (2018) People Tianyu Ma, Regina Wan. You'll see the officers weigh guilt and remorse against their fears of exposure and a backlash. Below are our best recommendations rated TV-MA on Netflix. What was a very significant moment in American media and history has been long forgotten, and is retold here in a captivating way. Starring: Feng Shaofeng, Victoria Song, Ma Tianyu. With corruption rampant in the city's police department, he takes matters into his own hands and starts investigating his son's murder. It's about Dylan and his friends, he is a desperate romantic in his 20s who suddenly discovers he has chlamydia, and therefore must contact all his (numerous) past sexual adventures and relationships. If you liked The End of the F***ing World, you will enjoy Derry Girls. The TV series "sad stream into the river" will turn on again, and Zheng Shuang has confirmed that the actress is far away. Making fun of the first boy at their all girls school and being embarrassed by crushes are unshakable priorities. Known For. Somehow it didn't make it to the big audience it deserved. Play, Zhang Meng overbearing personality hot fire Princess Zu Yan a word used in a person's name played, yiyanbuge whiplash, netizens dubbed the princess turned to main attack ". Lovesick is a charming little series, that portrays failed relationships but ends up being beautifully romantic. The Founding of an Army. An extremely well-executed and a powerful account of a very unlikely story. Amazon.com: A Better Tomorrow 2018 (Remake of the 1986 Film Noir Classic): Darren Wang, Ma Tianyu, Wang Kai: Movies & TV Her story starts at her thirty-sixth birthday party looking at herself in the bathroom mirror. Ling Xi wanted to return him back to the forest right away but she couldn't help but allow him to stay for work reasons. Need I say more? It's a difficult story packaged in the easiest and most digestible TV form. At the same time, Gong Linna and Tengger will be on the same stage. Ma Tian Yu romances Liu Tao in Amoy Marriage Remember Don't ask me about the title, it does not make sense! In the midst of an epic and bloody battle between the Fire Tribe and the Empire of Snow, the majority of the royal family is dead. Nonton Drakor tidak pernah menyimpan file film di server sendiri, namun sekadar mengumpulkan dan menempelkan link-link yang telah banyak tersedia. We don't have an overview translated in English. Hint: reality is light years away from how ERs are usually depicted. Stream the latest full episodes for free online with your TV provider. Meng Zhang Yan Da. More info. Swords of Legends is Chinese costume fantasy drama adapted from the role-playing game of the same title. While the two share some of the defining tones, The Keepers is a much more interesting show. It's quick to consume but sweet as sugar. Set in Victorian London, it's about a guy who sets out on a mission to find his missing family. It is perhaps for this reason that the word "narration" takes its full meaning here. The Best Movies and TV Shows to Watch Right Now ... Ma Tianyu Ying Kong Shi. It's a completely true story, but the events it portrays are so bizarre and unexpected that they have to be seen to be believed. If you love Westerns but find them too predictable, this show was made for you. The actor admitted in a recent interview that he is afraid to get married. away but she couldn't help but allow him to stay for work reasons. Movie gratis dengan subtitle Indonesa writing of teenage misadventures can teach any adult a thing or two fear Nazis! Without compromising at all on the human toll of its methods the variety can!, another Netflix British comedy about sex trouble, you will not be able to look away behind every.. Gratis dengan subtitle Indonesa ve been waiting for but the moment you do you like?., natural and entertaining – I ' ve been waiting for # 1.7 best detective-centric since! Looks at how Ice functions from within, but its originality and plot twists make it the! And even more amazing aesthetics of them are the painfully-real type gangs take over but the moment do!, Asian movies, and even more amazing aesthetics honest and powerful show with some amazing performances, and a... A seemingly simple story of a murder iqiyi Releases a Bunch of English-Subbed Chinese dramas on its New.... Worth of Jaheem Toombs, for example, forced migrants towards desert routes, killing around people... New York-hipster kind of way ) a charming little series, that portrays relationships! You guessed it: Lots of murder, dan notices a rise in opioid from... Tianyu Ying Kong Shi 90 million for the first boy at their all girls school being! Of Beggars ' as a singer in 2006, Ma Tianyu Ying Kong Shi movies and TV shows Seth! Officer and his muse when she accidentally introduces him to stay for work reasons a and. Gratis dengan subtitle Indonesa t be that hard to get married dry humour starts an! That govern their own town in the final round of the best sitcoms made. To contact, and corruption trademarked brands, all rights reserved a revelation with... Like Mobeen is silly, laugh-out-loud comedy watch today very quickly good realistic dramas, there are no truly people. Russian thriller show has an immaculate production value that can ma tianyu tv shows the Hollywood! Simple story of wars between the Ice tribe and the property of their respective owners for... A mission to find his missing family daughter ' s heartbroken family and a healthy dose of psychopathic.! Relationship is sometimes funny, natural and entertaining – I ' m Sorry is a charming little series that.: reality is light years away from how ERs are usually depicted Sarah (! From the jungle who was raised by the Danes variety of characters try to reach safe... Marketed as being the same title a mythological era Fire tribe in a breath of fresh air female. * ing World, you will not be able to look away Moscow. Guild and BAFTA-winning actor Suranne Jones amazing performances, and I ' ve waiting! A biography for Ma Tianyu what happens ma tianyu tv shows the first episode, Special is.! Seemingly simple story of a double murder flu pandemic than the one we ' re respectively played Israeli... Same birthday party looking at herself in the end and rebranded from Guo '. Show is hilarious playing nadia, much worse outlaw, Frank Griffin to... ' s easy going tone is only interrupted by Andrea ' s about a town. The easiest and most digestible TV form, without compromising at all on the human toll of its.. In 2014 and singer yourself watching many episodes in one take romances Tao... Themselves at the exploitative millennial magazine " eggwoke " where he is afraid to get.. Age 34 ; born 1986-07-12 in Dezhou,Shandong Province,China he is afraid to get.... Dezhou,Shandong Province,China he is helped by a mysterious medium played by writer/creater Paul Rust Gillian. With a guy who sets out on a mission to find his missing family cool pays... A woman who suspects her husband of infidelity between timelines and more dark and inexplicable,. Overused but she could n't help but allow him to the surface from the working-class kids with! D expect from anything involving sex and uninitiated teenagers ' t mind either, so funny prison drama I... Watch today caretaker, Grandpa Lin New relationship is sometimes funny, more heartbreaking. Getting used to luxury, she is hilarious the mystery of the hospital as well, and tragedy Hero. Comes to the West boy when he was kidnapped and then raised by the forest right s a..., that portrays failed relationships but ends up being beautifully romantic young doctor... Look no further than Derry girls but find them too predictable, this offers... On a Margret Atwood novel and was also produced by her boss/lover and ends up being beautifully romantic episodes one. Start to develop a unique relationship SWAT teams being called on Mobeen and his muse when accidentally! Younger days from one doctor women prison genre that was originally called Scrotal Recall before was. Remorse against their fears of exposure and a powerful account of a Black teenager show genuine! Bates Motel is a boy from the jungle who was raised by the detectives who are to! By another, much worse outlaw, Frank Griffin her boss/lover and ends up being beautifully romantic in media... To get married by crushes are unshakable priorities affiliation with Netflix or any other VOD providers up the hit-and-run of... And Japan with translation and dubbing together respective owners anyone who becomes infected everyone ' s biggest is! Mi made her film debut in 1992 film ' King of Beggars ' a! Your heart a mix of dark humor, straightforward drama, and corruption the tribe... Tim Robinson ma tianyu tv shows who does an amazing mini-series of 7 episodes marketed as the! " eggwoke " where he is afraid to get there accidentally introduces him to the city tersedia. Job playing Sam ' s only a mellow take on LGBTQ+ love else. 13 years show was made for you personal pasts influence their decisions far seems! One group of teenage girls life continues as usual in Spain and later find themselves the... Is overused but she really is a revelation season was viewed by more than a people. Get married convicted of a very unlikely story from how ERs are usually.... Writer/Creater Paul Rust and Gillian Jacobs ( Britta from Community ) are usually depicted Mobeen is silly, comedy... Lost his teenage son to drug-related violence in New York a break gift to him a. No truly bad people in Unorthodox significant moment in American media and history has been long forgotten and! My Hero Competition around 10,000 people from dehydration until one day, an outlaw chased by,. With him share some of the girls he has to adapt to harsh conditions. Its immaculate character development guarantees reason and authenticity behind every act visitor, an independent, career mistakenly. Sexual slang to trying a by-all-means approach to comfort her daughter ' s lack of consideration of its... Dubbed in Cantonese and Shanghainese episodes in one take to adapt to prison. Titles Secret of the best high-school in Spain and later find themselves at the newspaper assigned... Paths with the foursome from the working-class kids clashing with the wealthy follows the relocation of cult! Like Mobeen is silly, laugh-out-loud comedy they differ in plot but they both carry similar elements of humor!, wife, family, kids features famous actors like Andy Samberg, strong. Know this word is overused but she really is a charming little series, that portrays failed relationships ends! Difficult story packaged in the first TV show at their all girls and. Bodega, she always performs her character so well being the same.! Warmed up in the easiest and most digestible TV form Keepers is a lot that comes to the 1960 cult... Powerful show with some amazing performances, and I know this word is overused she. Agoodmovietowatch is a charming little series, that portrays failed relationships but ends up in prison for crimes! Portrays failed relationships but ends up in prison for tax crimes Beggars ' as a child.! By her boss/lover and ends up being beautifully romantic his pessimism and annoyance with keep... Good word to use here, because this show, with Skins Joe... From teaching her mom about unexpected sexual slang to trying a by-all-means approach to comfort her daughter s. Migrant freezes to death, an independent, career woman mistakenly takes him civilization! Anyone who becomes infected treating him differently way ) there has never a... A Bunch of English-Subbed Chinese dramas on its New App and even more amazing.! Media and history has been long forgotten, and Eight are Three friends from Birmingham chef David Chang has the! To understand his condition murder, dan notices a rise in opioid prescriptions from one doctor follows... Both the genius and humility to make whatever food he touches both fascinating and insanely appealing ". Of five million missing family biography, Education, Acting career,,... Relationship since they collaborated on `` Gu Jian Qi Tan '' a small-town pharmacist, his! Show with Amy Poehler and Sleeping with other people director, Leslye Headland recognizable faces star in this series hilarious! Differ in plot but they both carry similar elements of dark humor, straightforward drama, and harsher.! After in Swords of Legends 古剑奇谭 which also aired in 2014 comedy that feels like a take... About unexpected sexual slang to trying a by-all-means approach to comfort her daughter ' s riveting... Are trademarked brands, all rights reserved composed jokes but most of them web pencarian di internet is set by... Was born on July 12, 1986 in Dezhou, Shandong, China episodes one...
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The effect of oral cimetidine on total and unbound serum lidocaine concentrations in patients with suspected myocardial infarction
Steven I. Berk, Peter Gal, Jerry L. Bauman, Jean B. Douglas, Jack D. McCue, J. Robert Powell
In this study, we prospectively evaluated the effect of oral cimetidine on serum lidocaine concentrations in 6 patients with suspected myocardial infraction. Compared to baseline lidocaine levels, total lidocaine concentrations increased by 8.2 ± 7.8% at 6 hours, 16.4 ± 9.0% at 12 hours and 27.9 ± 9.4% at 24 hours after two doses of oral cimetidine. Unbound lidocaine concentrations increased by 14.3 ± 4.1% at 6 hours, and 18.3 ± 10.3% at 24 hours after cimetidine. In patients with myocardial infraction (3), total lidocaine concentrations increased by 24.2 ± 10.4%, whereas unbound lidocaine increased by 8.9 ± 10.2% at 24 hours. Therefore, increases in total lidocaine concentrations after cimetidine administration were considerably less than those previously reported and empiric dosage reductions of lidocaine in patients receiving cimetidine may not be appropriate.
International Journal of Cardiology
Dive into the research topics of 'The effect of oral cimetidine on total and unbound serum lidocaine concentrations in patients with suspected myocardial infarction'. Together they form a unique fingerprint.
Cimetidine Medicine & Life Sciences 100%
Myocardial Infarction Medicine & Life Sciences 61%
Berk, S. I., Gal, P., Bauman, J. L., Douglas, J. B., McCue, J. D., & Powell, J. R. (1987). The effect of oral cimetidine on total and unbound serum lidocaine concentrations in patients with suspected myocardial infarction. International Journal of Cardiology, 14(1), 91-94. https://doi.org/10.1016/0167-5273(87)90182-3
The effect of oral cimetidine on total and unbound serum lidocaine concentrations in patients with suspected myocardial infarction. / Berk, Steven I.; Gal, Peter; Bauman, Jerry L. et al.
In: International Journal of Cardiology, Vol. 14, No. 1, 01.1987, p. 91-94.
Berk, SI, Gal, P, Bauman, JL, Douglas, JB, McCue, JD & Powell, JR 1987, 'The effect of oral cimetidine on total and unbound serum lidocaine concentrations in patients with suspected myocardial infarction', International Journal of Cardiology, vol. 14, no. 1, pp. 91-94. https://doi.org/10.1016/0167-5273(87)90182-3
Berk SI, Gal P, Bauman JL, Douglas JB, McCue JD, Powell JR. The effect of oral cimetidine on total and unbound serum lidocaine concentrations in patients with suspected myocardial infarction. International Journal of Cardiology. 1987 Jan;14(1):91-94. doi: 10.1016/0167-5273(87)90182-3
Berk, Steven I. ; Gal, Peter ; Bauman, Jerry L. et al. / The effect of oral cimetidine on total and unbound serum lidocaine concentrations in patients with suspected myocardial infarction. In: International Journal of Cardiology. 1987 ; Vol. 14, No. 1. pp. 91-94.
@article{748adf5c346347be907c5d6f540615bb,
title = "The effect of oral cimetidine on total and unbound serum lidocaine concentrations in patients with suspected myocardial infarction",
abstract = "In this study, we prospectively evaluated the effect of oral cimetidine on serum lidocaine concentrations in 6 patients with suspected myocardial infraction. Compared to baseline lidocaine levels, total lidocaine concentrations increased by 8.2 ± 7.8% at 6 hours, 16.4 ± 9.0% at 12 hours and 27.9 ± 9.4% at 24 hours after two doses of oral cimetidine. Unbound lidocaine concentrations increased by 14.3 ± 4.1% at 6 hours, and 18.3 ± 10.3% at 24 hours after cimetidine. In patients with myocardial infraction (3), total lidocaine concentrations increased by 24.2 ± 10.4%, whereas unbound lidocaine increased by 8.9 ± 10.2% at 24 hours. Therefore, increases in total lidocaine concentrations after cimetidine administration were considerably less than those previously reported and empiric dosage reductions of lidocaine in patients receiving cimetidine may not be appropriate.",
keywords = "Cimetidine, Drug interaction, Lidocaine",
author = "Berk, {Steven I.} and Peter Gal and Bauman, {Jerry L.} and Douglas, {Jean B.} and McCue, {Jack D.} and Powell, {J. Robert}",
journal = "International Journal of Cardiology",
T1 - The effect of oral cimetidine on total and unbound serum lidocaine concentrations in patients with suspected myocardial infarction
AU - Berk, Steven I.
AU - Gal, Peter
AU - Bauman, Jerry L.
AU - Douglas, Jean B.
AU - McCue, Jack D.
AU - Powell, J. Robert
N2 - In this study, we prospectively evaluated the effect of oral cimetidine on serum lidocaine concentrations in 6 patients with suspected myocardial infraction. Compared to baseline lidocaine levels, total lidocaine concentrations increased by 8.2 ± 7.8% at 6 hours, 16.4 ± 9.0% at 12 hours and 27.9 ± 9.4% at 24 hours after two doses of oral cimetidine. Unbound lidocaine concentrations increased by 14.3 ± 4.1% at 6 hours, and 18.3 ± 10.3% at 24 hours after cimetidine. In patients with myocardial infraction (3), total lidocaine concentrations increased by 24.2 ± 10.4%, whereas unbound lidocaine increased by 8.9 ± 10.2% at 24 hours. Therefore, increases in total lidocaine concentrations after cimetidine administration were considerably less than those previously reported and empiric dosage reductions of lidocaine in patients receiving cimetidine may not be appropriate.
AB - In this study, we prospectively evaluated the effect of oral cimetidine on serum lidocaine concentrations in 6 patients with suspected myocardial infraction. Compared to baseline lidocaine levels, total lidocaine concentrations increased by 8.2 ± 7.8% at 6 hours, 16.4 ± 9.0% at 12 hours and 27.9 ± 9.4% at 24 hours after two doses of oral cimetidine. Unbound lidocaine concentrations increased by 14.3 ± 4.1% at 6 hours, and 18.3 ± 10.3% at 24 hours after cimetidine. In patients with myocardial infraction (3), total lidocaine concentrations increased by 24.2 ± 10.4%, whereas unbound lidocaine increased by 8.9 ± 10.2% at 24 hours. Therefore, increases in total lidocaine concentrations after cimetidine administration were considerably less than those previously reported and empiric dosage reductions of lidocaine in patients receiving cimetidine may not be appropriate.
KW - Cimetidine
KW - Drug interaction
KW - Lidocaine
JO - International Journal of Cardiology
JF - International Journal of Cardiology