Single Qubit Gates

Posted 2021-07-24 安徽思远

Logic gates:

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We saw that qubits could be represented by $2\mathrm{D}$ vectors, and that their states are limited to the form:
$$|q⟩=\cos \frac{\theta }{2}|0⟩+e^{i\varphi }\sin \frac{\theta }{2}|1⟩$$
Where $\theta$ and $\varphi$ are real numbers. In this section we will cover gates, the operations that change a qubit between these states. Due to the number of gates and the similarities between them, this chapter is at risk of becoming a list. To counter this, we have included a few digressions to introduce important ideas at appropriate places throughout the chapter.

The Pauli Gates

You should be familiar with the Pauli matrices from the linear algebra section. If any of the maths here is new to you, you should use the linear algebra section to bring yourself up to speed. We will see here that the Pauli matrices can represent some very commonly used quantum gates.

1. The $X$ -Gate

The X-gate is represented by the Pauli-X matrix:
$$X=\left[\begin{array}{ll}0& 1\\ 1& 0\end{array}\right]=|0⟩⟨1|+|1⟩⟨0|$$
To see the effect a gate has on a qubit, we simply multiply the qubit’s statevector by the gate. We can see that the X-gate switches the amplitudes of the states $|0⟩$ and $|1⟩$ :
$$X|0⟩=\left[\begin{array}{ll}0& 1\\ 1& 0\end{array}\right]\left[\begin{array}{l}1\\ 0\end{array}\right]=\left[\begin{array}{l}0\\ 1\end{array}\right]=|1⟩$$

X Gate can also be regarded as Not Gate…

# Let's do an X-gate on a |0> qubit
qc = QuantumCircuit(1)
qc.x(0)
qc.draw()

2. The Y & Z-gates

Similarly to the X-gate, the Y & Z Pauli matrices also act as the Y & Z-gates in our quantum circuits:
$$\begin{array}{c}Y=\left[\begin{array}{cc}0& -i\\ i& 0\end{array}\right]Z=\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right]\\ Y=-i|0⟩⟨1|+i|1⟩⟨0|Z=|0⟩⟨0|-|1⟩⟨1|\end{array}$$
And, unsurprisingly, they also respectively perform rotations by $\pi$ around the $y$ and $z$ -axis of the Bloch sphere.

3. Digression: The X, Y & Z-Bases

Eigensates:
We have seen that multiplying a vector by a matrix results in a vector:
$$M|v⟩=|v^{\prime }⟩\leftarrow \text{ new vector }$$
If we chose the right vectors and matrices, we can find a case in which this matrix multiplication is the
same as doing a multiplication by a scalar:
$$M|v⟩=\lambda |v⟩$$
(Above, $M$ is a matrix, and $\lambda$ is a scalar). For a matrix $M$, any vector that has this property is called an eigenvector of $M$. For example, the eigenvectors of the Z-matrix are the states $|0⟩$ and $|1⟩$ :
$$\begin{array}{l}Z|0⟩=|0⟩\\ Z|1⟩=-|1⟩\end{array}$$

You may also notice that the Z-gate appears to have no effect on our qubit when it is in either of these two states. This is because the states $|0⟩$ and $|1⟩$ are the two eigenstates of the Z-gate. In fact, the computational basis (the basis formed by the states $|0⟩$ and $|1⟩$ ) is often called the Z-basis. This is not the only basis we can use, a popular basis is the X-basis, formed by the eigenstates of the X-gate. We call these two vectors $|+⟩$ and $|-⟩$ :
$$\begin{array}{l}|+⟩=\frac{1}{\sqrt{2}}(|0⟩+|1⟩)=\frac{1}{\sqrt{2}}\left[\begin{array}{l}1\\ 1\end{array}\right]\\ |-⟩=\frac{1}{\sqrt{2}}(|0⟩-|1⟩)=\frac{1}{\sqrt{2}}\left[\begin{array}{c}1\\ -1\end{array}\right]\end{array}$$
Another less commonly used basis is that formed by the eigenstates of the Y-gate. These are called:
$$|\circlearrowleft ⟩,|\circlearrowright ⟩$$

4. The Hadamard Gate

The Hadamard gate (H-gate) is a fundamental quantum gate. It allows us to move away from the poles of the Bloch sphere and create a superposition of $|0⟩$ and $|1⟩$. It has the matrix:
$$我为Bill\; Gates熬夜加班的那个晚上$$

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