QElephant/docs/Gate.md

Gates

Here is the list of all the main gates in quantum algorithm available in the library. In the following formulas, i represents the complex number.

H(q: QuBit) -> None

Hadamard's gate.

• q: the qubit manipulated by the gate. This function is in-place.

• matrix:

\begin{pmatrix}
\frac {1} {\sqrt 2} & \frac {1} {\sqrt 2}\\
\frac {1} {\sqrt 2} & -\frac {1} {\sqrt 2}
\end{pmatrix}

X(q: QuBit) -> None

Pauli-X gate or NOT gate. Inplementes a rotation aroud the x-axis of \pi radians.

• q: the qubit manipulated by the gate. This function is in-place.

• matrix:

\begin{pmatrix}
0 & 1\\
1 & 0
\end{pmatrix}

Y(q: QuBit) -> None

Pauli-Y gate. Inplementes a rotation aroud the y-axis of \pi radians.

• q: the qubit manipulated by the gate. This function is in-place.

• matrix:

\begin{pmatrix}
0 & -i\\
i & 0
\end{pmatrix}

Z(q: QuBit) -> None

Pauli-Z gate. Inplementes a rotation aroud z-axis of \pi radians.

• q: the qubit manipulated by the gate. This function is in-place.

• matrix:

\begin{pmatrix}
1 & 0\\
0 & -1
\end{pmatrix}

S(q: QuBit) -> None

NOT gate. Invert the state |0> and |1> of the QuiBit q.

• q: the qubit manipulated by the gate. This function is in-place.

• matrix:

\begin{pmatrix}
1 & 0\\
0 & i
\end{pmatrix}

T(q: QuBit) -> None

NOT gate. Invert the state |0> and |1> of the QuiBit q.

• q: the qubit manipulated by the gate. This function is in-place.

• matrix:

\begin{pmatrix}
1 & 0\\
0 & e^{i\frac\pi4}
\end{pmatrix}

Rx(q: QuBit, phi: float) -> None

NOT gate. Inplementes a rotation aroud x-axis of \phi radians.

• q: the qubit manipulated by the gate. This function is in-place.

• matrix:

\begin{pmatrix}
\cos(\frac\phi2) & -\sin(\frac\phi2)\\
\sin(\frac\phi2) & \cos(\frac\phi2)
\end{pmatrix}

Ry(q: QuBit, phi: float) -> None

NOT gate. Inplementes a rotation aroud y-axis of \pi radians.

• q: the qubit manipulated by the gate. This function is in-place.

• matrix:

\begin{pmatrix}
e^{-i\frac\phi2} & 0\\
0 & e^{i\frac\phi2}
\end{pmatrix}

R1(q: QuBit, phi: float) -> None

NOT gate. Invert the state |0> and |1> of the QuiBit q.

• q: the qubit manipulated by the gate. This function is in-place.

• matrix:

\begin{pmatrix}
1 & 0\\
0 & e^{i\frac\phi2}
\end{pmatrix}

CNOT(q: MuBit, n1: int, n2: int) -> None

X-controlled gate. Invert the state |0> and |1> of the QuiBit in n2 if the state of the QuBit in n1 is 1.

• q: the qubit manipulated by the gate. This function is in-place.

• n1: the first QuBit to manipualte.

• n2: the second QuBit to manipulate

• matrix:

\begin{pmatrix}
1 & 0 & 0 & 0\\
0 & 1 & 0 & 0\\
0 & 0 & 0 & 1\\
0 & 0 & 1 & 0
\end{pmatrix}

SWAP(q: MuBit, n1: int, n2: int) -> None

NOT gate. Invert the state |0> and |1> of the QuiBit q.

• q: the qubit manipulated by the gate. This function is in-place.

• n1: the first QuBit to manipualte.

• n2: the second QuBit to manipulate

• matrix:

\begin{pmatrix}
1 & 0 & 0 & 0\\
0 & 0 & 1 & 0\\
0 & 1 & 0 & 0\\
0 & 0 & 0 & 1
\end{pmatrix}

Cu(q: MuBit, u: list[list[float]], n1: int, n2: int) -> None

controlled-u gate. Applies the gate u to the QuBit in n2 if the Qubit in n1 is 1.

• q: the qubit manipulated by the gate. This function is in-place.

• the list respresentation of a matrix of the size 2 by 2.

• n1: the first QuBit to manipualte.

• n2: the second QuBit to manipulate

• matrix:

\begin{pmatrix}
1 & 0 & 0 & 0\\
0 & 1 & 0 & 0\\
0 & 0 & u_{00} & u_{01}\\
0 & 0 & u_{10} & u_{11}
\end{pmatrix}