SINGLE-QUBIT GATES

In a Quantum Computer, “gates” are unitary operations. The words gate and operator are used interchangebly in this context.

• Recall that a unitary operator $U$ is one where the adjoint is equal to the inverse, meaning $U^\dagger = U^{-1}$, giving us the defining relation for a unitary operator:

  $UU^\dagger = U^\dagger U = I$

• We also remember that, if $H$ is a hermitian opertator, then $U = e^{-iHt}$ is unitary.

• A quantum gate with $n$ inputs and outputs, can be represented by a matrix of degree $2^n$. Thus, for a single qubit, we require $2^1 = 2$ degree unitary matrices to reperesent our gates.

THE NOT GATE (X Pauli Matrix)

• Represented as $X$ or $U_{NOT}$:

  $X = U_{NOT} = \begin{pmatrix}0 & 1 \\ 1 & 0\end{pmatrix}$

• Further, recall the effect of this matrix on the standard computational basis:

  $\begin{matrix}U_{NOT}\ket{0} = \ket{1} & U_{NOT}\ket{1} = \ket{0}\end{matrix}$

• This action on an arbitray state $\ket{j}$ can be summarized as follows using $XOR$ operation: $U_{NOT}\ket{j} = \ket{j \oplus 1}$

THE HADAMARD MATRIX

• Given by

  $H = \frac{1}{\sqrt2} \begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix}$

  This matrix, helps transform a given state from the $\ket{0}-\ket{1}$ basis to the $\ket{\pm}$ basis.

  $\begin{matrix}H\ket{0} = \ket{+} & H\ket{1} = \ket{-} \end{matrix}$

APPLICABLE QUANTUM GATES:

• Pauli Matrices:

  $\begin{matrix}X = \begin{pmatrix}0 & 1 \\ 1 & 0 \end{pmatrix} & Y = \begin{pmatrix}0 & -i \\ i & 0 \end{pmatrix}\end{matrix}$

  • The $Z$ Pauli Operator, is also referred to as the PHASE FLIP GATE:

    $Z = \begin{pmatrix}1 & 0 \\ 0 & -1 \end{pmatrix}$

    Its operation on a given state $\ket{j}$, is given by:

    $\begin{matrix}Z\ket{j} = (-1)^j\ket{j} & j \in \{0, 1\}\end{matrix}$

• The Hadamard Gate ($H$), Phase Gate ($S$) & $\pi/8$ Gate ($T$):

  $\begin{matrix}H = \frac{1}{\sqrt2}\begin{pmatrix}1 & 1 \\ 1 & -1 \end{pmatrix} & S = \begin{pmatrix}1 & 0 \\ 0 & i \end{pmatrix}\end{matrix}$

  • $H = \Large \frac{X + Z}{\sqrt 2}$
  • $S = T^2$

  $T = \begin{pmatrix}1 & 0 \\ 0 & e^{\large\frac{i\pi}{4}} \end{pmatrix} = e^{\large\frac{i\pi}{8}}\begin{pmatrix}e^{\large\frac{-i\pi}{8}} & 0 \\ 0 & e^{\large\frac{i\pi}{8}} \end{pmatrix}$

• Square Root of NOT Gate ($\sqrt{NOT}$)

  $\sqrt{NOT} = \frac{1}{\sqrt2}\begin{pmatrix}1 & -1 \\ 1 & 1 \end{pmatrix}$

  note that:

  $\sqrt{NOT} \ne \sqrt{NOT}^\dagger$

CIRCUIT IDENTITIES:

• $Z = HXH$
• $X = HZH$
• $-Y = HYH$
• $-Y = XYX$

BLOCH SPHERE & BLOCH VECTOR

• Derivation of the Canonical Representation

• To Summarize, the Canonical Representation of a qubit is:

  $\ket{\psi} = cos(\large\frac{\theta}{2}\normalsize)\ket{0} + e^{i\phi}sin(\large\frac{\theta}{2}\normalsize)\ket{1}$

  Here,

  $\begin{matrix}\theta \in (0, \pi) & \phi \in (0, 2\pi)\end{matrix}$

• We also, note the following:

  • The $Z$ basis:

    $\theta = 0 \implies \ket{0}$

    $\theta = \pi \implies \ket{1}$

  • The $X$ basis:

    $\begin{matrix}\theta = \large \frac{\pi}{2} & \phi = 0 \end{matrix} \implies \ket{+}$

    $\begin{matrix}\theta = \large \frac{\pi}{2} & \phi = \large -\pi \end{matrix} \implies \ket{-}$