Circuit model is a sequence of some building blocks that carry out our computations, called gates

Single qubit gates

  • classical example: NOT

  • quantum example: as quantum theory is unitary, quantum gates are represented by unitary matrices: $u_+ u = \vec I$

direct notation: \(\begin{pmatrix} u_{00} & u_{01} \\ u_{10} & u_{11} \end{pmatrix} = u_{00} \vert 0 \rangle \langle 0 \vert + u_{01}\vert 0 \rangle \langle 1 \vert + u_{10}\vert 1 \rangle \langle 0 \vert + u_{11}\vert 1 \rangle \langle 1 \vert\)

  • \[\sigma_x = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} = \vert 0 \rangle \langle 1 \vert + \vert 1 \rangle \langle 0 \vert\] \[\sigma_x \vert 0 \rangle = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \cdot \begin{pmatrix} 1 \\ 0 \end{pmatrix} = \begin{pmatrix} 0 \\ 1 \end{pmatrix} = \vert 1 \rangle\] \[\sigma_x \vert 1 \rangle = ( \vert 0 \rangle \langle 1 \vert + \vert 1 \rangle \langle 0 \vert ) \cdot \vert 1 \rangle = \vert 0 \rangle \langle 1 \vert 1 \rangle + \vert 1 \rangle \langle 0 \vert 1 \rangle = \vert 0 \rangle\]

    => bit flip $\cong$ NOT-gate. e.g. => rotation around the X axis by $\pi$.

  • \[\sigma_z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} = \vert 0 \rangle \langle 0 \vert - \vert 1 \rangle \langle 1 \vert\] \[\sigma_z \vert + \rangle = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} \cdot \frac{1}{\sqrt{2}} \begin{pmatrix} 1 \\ 1 \end{pmatrix} = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 \\ -1 \end{pmatrix} = \vert - \rangle\] \[\sigma_z \vert - \rangle = ( \vert 0 \rangle \langle 0 \vert - \vert 1 \rangle \langle 1 \vert ) \cdot \frac{1}{\sqrt{2}} ( \vert 0 \rangle - \vert 1 \rangle ) = \frac{1}{\sqrt{2}} ( \vert 0 \rangle + \vert 1 \rangle ) = \vert + \rangle\]

    => phase flip => rotation around $z\cdot$axis by $\pi$

  • \(\sigma_y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix} = i \cdot \sigma_z \cdot \sigma_x\) => bit & phase flip

  • $\sigma_x, \sigma_y, \sigma_z$ are the so-called Poly matrices \({\sigma_i}^2 = \vec I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}\) (dose nothing)

    together with identify $\vec I$ they form a basis of 2x2 matrices.

    any 1-qubit rotation can be written as linear combination of them.

  • Hadamard gate:

    \[H = \frac{1} {\sqrt{2}} \begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix} =- \frac{1} {\sqrt{2}} ( \vert 0 \rangle \langle 0 \vert + \vert 0 \rangle \langle 1 \vert + \vert 1 \rangle \langle 0 \vert - \vert 1 \rangle \langle 1 \vert )\] \[H \vert 0 \rangle = \frac{1} {\sqrt{2}} \begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix} \cdot \begin{pmatrix} 1 \\ 0 \end{pmatrix} = \frac{1} {\sqrt{2}} \begin{pmatrix} 1 \\ 1 \end{pmatrix} = \vert + \rangle\] \[H \vert 1 \rangle = ( \vert 0 \rangle \langle 0 \vert + \vert 0 \rangle \langle 1 \vert + \vert 1 \rangle \langle 0 \vert - \vert 1 \rangle \langle 1 \vert ) \cdot \vert 1 \rangle = \frac{1}{\sqrt{2}} ( \vert 0 \rangle - \vert 1 \rangle ) = \vert - \rangle\]

    H can user for creates superposition -> $H\vert + \rangle = \vert 0 \rangle , H \vert - \rangle = \vert 1 \rangle$ => used to change between X & Z basis.

  • similary, as \(S = \begin{pmatrix} 1 & 0 \\ 0 & i \end{pmatrix}\) adds $90^{\circ}$ to the phase $\varphi$: $S\cdot \vert + \rangle = \vert +i \rangle, S \vert - \rangle = \vert -i \rangle$

    => used to change between X & Y basis,

  • $S \cdot H$ is applied to change Z to Y basis.

Multi patent quantum states

  • we use these products to describe multi patent states: $\vert a \rangle \otimes \vert b \rangle = $ \(\begin{pmatrix} a_1 \\ a_2 \end{pmatrix} \otimes \begin{pmatrix} b_1 \\ b_2 \end{pmatrix} = \begin{pmatrix} a_1 b_1 \\ a_1 b_2 \\ a_2 b_1 \\ a_2 b_2 \end{pmatrix}\)

  • example: system A is in state ${\vert 1 \rangle}_A$ and system B is in the state ${\vert 0 \rangle}_B$

    => the total(bipartite) state is ${\vert 1 0 \rangle}_{AB} := {\vert 1 \rangle}_A \otimes {\vert 0 \rangle}_B = $ \(\begin{pmatrix} 0 \\ 1 \end{pmatrix} \otimes \begin{pmatrix} 1 \\ 0 \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \\ 1 \\ 0 \end{pmatrix}\)

    => remark: states of this form are called uncorrelated(不相关), but there are alse states that cannot be written as ${\vert \psi \rangle}_A \otimes {\vert \varphi \rangle}_B$. These states are correlated() and sometimes even entangled(very strong corrlation) e.g.

    \({ {\vert \phi \rangle}^{(00)}}_{AB} = \frac{1}{\sqrt{2}} ( {\vert 00 \rangle}_{AB} + {\vert 11 \rangle}_{AB} ) = \frac{1}{\sqrt{2}} ( \begin{pmatrix} 1 \\ 0 \end{pmatrix} \otimes \begin{pmatrix} 1 \\ 0 \end{pmatrix} + \begin{pmatrix} 0 \\ 1 \end{pmatrix} \otimes \begin{pmatrix} 0 \\ 1 \end{pmatrix} ) = \frac{1}{\sqrt{2}} ( \begin{pmatrix} 1 \\ 0 \\ 0 \\ 0 \end{pmatrix} + \begin{pmatrix} 0 \\ 0 \\ 0 \\ 1 \end{pmatrix} )= \frac{1}{\sqrt{2}} \begin{pmatrix} 1 \\ 0 \\ 0 \\ 1 \end{pmatrix}\) a so-called Bell state, used for teleportation, cryptography, etc.

Two-qubit gates

  • classical exampl: XOR -> irreversible (-> given the output we cannot recover the input)

    BUT: as quantum theory is unitary, we only consider unitary gate and there are always reversible. $u^+ u = \vec{I} \Leftrightarrow u^{-1} = u^+$

  • quantum example:

    CNOT = \(\begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{pmatrix} = \vert 00 \rangle \langle 00 \vert + \vert 01 \rangle \langle 01 \vert + \vert 10 \rangle \langle 11 \vert + \vert 11 \rangle \langle 10 \vert +\) => $CNOT \cdot \vert 00 \rangle = $ \(CNOT \cdot \begin{pmatrix} 1 \\ 0 \\ 0 \\ 0 \end{pmatrix} = \begin{pmatrix} 1 \\ 0 \\ 0 \\ 0 \end{pmatrix} = {\vert 00 \rangle}_{xy}, \quad CNOT{\vert 10 \rangle}_{xy} = {\vert 11 \rangle}_{xy}\)

    input output
    x y x $x \otimes y$
    0 0 0 0
    0 1 0 1
    1 0 1 1
    1 1 1 0

    => circuit: (reversible XOR)

    => we can show that evet function f can described by a reversible circuit

    => quantum circuits can perform all function that can be calculated classically.