Simple quantum gates

• These are some examples of useful single-qubit quantum state transformations. Because of linearity, the transformations are fully specified by their effect on the basis vectors. The associated matrix is also shown.
  
  $$\begin{array}{ll} \begin{array}{lrcl} I\ :& \vert{0}\rangle &... ...begin{array}{cc}1 & 0\\ 0 & -1\end{array}\right)\\ \end{array}$$
  
  
  I is the identity transformation, $\sigma_x$ is negation, $\sigma_z$ is a phase shift operation, and $\sigma_y = \sigma_z\sigma_x$ is a combination of both. All these gates are unitary. For example
  
  $$\sigma_y\sigma_y^\dag = \left(\begin{array}{cc}0 & -1\\ 1 & 0... ... \left(\begin{array}{cc}0 & 1\\ -1 & 0\end{array}\right) = I.$$
  
  

• Another important single-bit transformation is the Hadamard transformation defined by
  
  $$\begin{array}{lrcl} H:& \vert{0}\rangle & \to & \frac{1}{\sqr... ...{1}{\sqrt 2}(\vert{0}\rangle - \vert{1}\rangle).\\ \end{array}$$
  
  
  Applied to n bits each in the $\vert{0}\rangle$ state, the transformation generates a superposition of all 2ⁿ possible states.
  
  $$\begin{eqnarray*}& &(H\otimes H \otimes \dots \otimes H)\vert{00\dots 0}\rangle\... ...t)\\ &=&\frac{1}{\sqrt {2^n}}\sum_{x=0}^{2^n-1}\vert{x}\rangle. \end{eqnarray*}$$
  
  The transformation acting on n bits is called the Walsh or Walsh-Hadamard transformation W.

• An important example of a two qubit gate is the controlled- NOT gate, C_not, which complements the second bit if the first bit is 1 and leaves the bit unchanged otherwise.
  
  $$\begin{array}{ll}\begin{array}{lrcl} C_{not}:& \vert{00}\rang... ... 0 & 0 & 0 & 1\\ 0 & 0 & 1 & 0\end{array}\right)\\ \end{array}$$
  
  
  The transformation C_not is unitary since $C_{not}^\dag =C_{not}$ and C_notC_not= I. The C_not gate cannot be decomposed into a tensor product of two single-bit transformations.

• The bra/ket notation is useful in defining other unitary operations. Given two arbitrary unitary transformations U₁ and U₂, the ``conditional'' transformation $\vert{0}\rangle\langle{0}\vert \otimes U_1 + \vert{1}\rangle\langle{1}\vert\otimes U_2$ is also unitary. For example, the controlled- NOT gate can defined by
  
  $$C_{not} = \vert{0}\rangle\langle{0}\vert \otimes I + \vert{1}\rangle\langle{1}\vert\otimes X.$$
  
  

• The three-bit controlled-controlled- NOT  gate or Toffoli gate is also an instance of this conditional definition:
  
  $$T = \vert{0}\rangle\langle{0}\vert\otimes I \otimes I + \vert{1}\rangle\langle{1}\vert \otimes C_{not}.$$
  
  
  
  
  $$\begin{array}{lrcl} T: & \vert{000}\rangle & \to & \vert{000... ... & \vert{111}\rangle & \to & \vert{110}\rangle\\ \end{array}$$
  
  
  T can be used to construct a complete set of the classical boolean connectives and thus general combinatory circuits since it can be used to construct the not and and operators in the following way:
  
  $$\begin{eqnarray*}T\vert{1, 1, x}\rangle & = & \vert{1, 1, \sim x}\rangle\\ T\vert{x, y, 0}\rangle & = & \vert{x, y, x \wedge y}\rangle\\ \end{eqnarray*}$$