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Date: Tue, 30 Mar 2021 17:17:39 +0200
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+---
+title: "Quantum gate"
+firstLetter: "Q"
+publishDate: 2021-03-29
+categories:
+- Quantum information
+
+date: 2021-03-29T21:37:57+02:00
+draft: false
+markup: pandoc
+---
+
+
+# Quantum gate
+
+In quantum computing, **quantum gates** are the equivalent
+of classical binary logic gates such as $\mathrm{NOT}$, $\mathrm{AND}$, etc.
+Because of the continuous nature of qubits,
+the number of possible quantum gates is uncountably infinite,
+so we only consider the most important examples here.
+
+
+## One-qubit gates
+
+As an example, consider the following must general single-qubit state $\ket{\psi}$:
+
+$$\begin{aligned}
+    \ket{\psi}
+    = \alpha \ket{0} + \beta \ket{1}
+    = \begin{bmatrix} \alpha \\ \beta \end{bmatrix}
+\end{aligned}$$
+
+Arguably the most famous and/or most fundamental quantum gates are the **Pauli matrices**:
+
+$$\begin{aligned}
+    \boxed{
+        X =
+        \begin{bmatrix}
+            0 & 1 \\
+            1 & 0
+        \end{bmatrix}
+    }
+    \qquad
+    \boxed{
+        Y =
+        \begin{bmatrix}
+            0 & -i \\
+            i & 0
+        \end{bmatrix}
+    }
+    \qquad
+    \boxed{
+        Z =
+        \begin{bmatrix}
+            1 & 0 \\
+            0 & -1
+        \end{bmatrix}
+    }
+\end{aligned}$$
+
+They have the following effect on $\ket{\psi}$.
+Note that $X$ is equivalent to the classical $\mathrm{NOT}$ gate
+(and is often given that name),
+and $Z$ is sometimes called the **phase-flip gate**:
+
+$$\begin{aligned}
+    X \ket{\psi}
+    = \begin{bmatrix} \beta \\ \alpha \end{bmatrix}
+    \qquad
+    Y \ket{\psi}
+    = \begin{bmatrix} -i \beta \\ i \alpha \end{bmatrix}
+    \qquad
+    Z \ket{\psi}
+    = \begin{bmatrix} \alpha \\ -\beta \end{bmatrix}
+\end{aligned}$$
+
+In fact, $Z$ is a specific case of the **phase shift gate** $R_\phi$,
+which modifies the qubit's phase without changing its amplitudes.
+For an angle $\phi$, it is given by:
+
+$$\begin{aligned}
+    \boxed{
+        R_\phi =
+        \begin{bmatrix}
+            1 & 0 \\
+            0 & e^{i \phi}
+        \end{bmatrix}
+    }
+\end{aligned}$$
+
+For $\phi = \pi$, we recover the Pauli-$Z$ gate.
+In general, the action of $R_\phi$ is as follows:
+
+$$\begin{aligned}
+    R_\phi \ket{\psi}
+    = \begin{bmatrix} \alpha \\ e^{i \phi} \beta \end{bmatrix}
+\end{aligned}$$
+
+Two common special cases of $R_\phi$
+are $\phi = \pi/2$ and $\phi = \pi/4$,
+respectively called $S$ and $T$:
+
+$$\begin{aligned}
+    \boxed{
+        S = R_{\pi/2} =
+        \begin{bmatrix}
+            1 & 0 \\
+            0 & i
+        \end{bmatrix}
+    }
+    \qquad \quad
+    \boxed{
+        T = R_{\pi/4} =
+        \frac{1}{\sqrt{2}}
+        \begin{bmatrix}
+            \sqrt{2} & 0 \\
+            0 & 1 + i
+        \end{bmatrix}
+    }
+\end{aligned}$$
+
+Finally, we have the **Hadamard gate** $H$,
+which is defined as follows:
+
+$$\begin{aligned}
+    \boxed{
+        H = \frac{1}{\sqrt{2}}
+        \begin{bmatrix}
+            1 & 1 \\
+            1 & -1
+        \end{bmatrix}
+    }
+\end{aligned}$$
+
+Its action consists of rotating the qubit
+by $\pi$ around the axis $(X + Z) / \sqrt{2}$ of the Bloch sphere:
+
+$$\begin{aligned}
+    H \ket{\psi}
+    = \frac{1}{\sqrt{2}} \begin{bmatrix} \alpha + \beta \\ \alpha - \beta \end{bmatrix}
+\end{aligned}$$
+
+Notably, it maps the eigenstates of $X$ and $Z$ to each other,
+and is its own inverse (i.e. unitary):
+
+$$\begin{aligned}
+    H \ket{0} = \ket{+}
+    \qquad
+    H \ket{1} = \ket{-}
+    \qquad
+    H \ket{+} = \ket{0}
+    \qquad
+    H \ket{-} = \ket{1}
+\end{aligned}$$
+
+The **Clifford gates** are a set including $X$, $Y$, $Z$, $H$ and $S$,
+or more generally any gates that rotate
+by multiples of $\pi/2$ around the Bloch sphere.
+This set is **not universal**, meaning that if we start from $\ket{0}$,
+we can only reach $\ket{0}$, $\ket{1}$, $\ket{+}$, $\ket{-}$, $\ket{+i}$ $\ket{-i}$ using these gates.
+
+If we add *any* non-Clifford gate, for example $T$,
+then we can reach any point on the Bloch sphere,
+which means that the set is **universal**.
+
+However, there is a problem: a qubit has an uncountable infinity of states,
+but a quantum circuit consists of a countably infinite sequence of gates, at most.
+Therefore, technically, we can never reach the whole Bloch sphere,
+but we *can* come up with circuits that approximate a target state to some degree $\varepsilon$.
+This is the definition of universality:
+any state can be approximated.
+
+
+## Two-qubit gates
+
+As an example, let us consider
+the following two pure one-qubit states $\ket{\psi_1}$ and $\ket{\psi_2}$:
+
+$$\begin{aligned}
+    \ket{\psi_1}
+    = \alpha_1 \ket{0} + \beta_1 \ket{1}
+    = \begin{bmatrix} \alpha_1 \\ \beta_1 \end{bmatrix}
+    \qquad \quad
+    \ket{\psi_2}
+    = \alpha_2 \ket{0} + \beta_2 \ket{1}
+    = \begin{bmatrix} \alpha_2 \\ \beta_2 \end{bmatrix}
+\end{aligned}$$
+
+The composite state of both qubits, assuming they are pure,
+is then their tensor product $\otimes$:
+
+$$\begin{aligned}
+    \ket{\psi_1 \psi_2}
+    = \ket{\psi_1} \otimes \ket{\psi_2}
+    &= \alpha_1 \alpha_2 \ket{00} + \alpha_1 \beta_2 \ket{01} + \beta_1 \alpha_2 \ket{10} + \beta_1 \beta_2 \ket{11}
+    \\
+    &= c_{00} \ket{00} + c_{01} \ket{01} + c_{10} \ket{10} + c_{11} \ket{11}
+\end{aligned}$$
+
+Note that a two-qubit system may be [entangled](/know/concept/quantum-entanglement/),
+in which case the coefficients $c_{00}$ etc. cannot be written as products,
+i.e. $\ket{\psi_2}$ cannot be expressed separately from $\ket{\psi_1}$, and vice versa.
+
+In other words, the general action of a two-qubit quantum gate
+can be expressed in the basis of $\ket{00}$, $\ket{01}$, $\ket{10}$ and $\ket{11}$,
+but not always in the basis of $\ket{0}_1$, $\ket{1}_1$, $\ket{0}_2$ and $\ket{1}_2$.
+
+With that said, the first two-qubit gate is $\mathrm{SWAP}$,
+which simply swaps $\ket{\psi_1}$ and $\ket{\psi_2}$:
+
+$$\begin{aligned}
+    \boxed{
+        \mathrm{SWAP} =
+        \begin{bmatrix}
+            1 & 0 & 0 & 0 \\
+            0 & 0 & 1 & 0 \\
+            0 & 1 & 0 & 0 \\
+            0 & 0 & 0 & 1
+        \end{bmatrix}
+    }
+\end{aligned}$$
+
+This matrix is given in the basis of $\ket{00}$, $\ket{01}$, $\ket{10}$ and $\ket{11}$.
+Note that $\mathrm{SWAP}$ cannot generate entanglement,
+so if its input is separable, its output is too.
+In any case, its effect is clear:
+
+$$\begin{aligned}
+    \mathrm{SWAP} \ket{\psi_1 \psi_2}
+    &= c_{00} \ket{00} + c_{10} \ket{01} + c_{01} \ket{10} + c_{11} \ket{11}
+\end{aligned}$$
+
+Next, there is the **controlled NOT gate** $\mathrm{CNOT}$,
+which "flips" (applies $X$ to) $\ket{\psi_2}$ if $\ket{\psi_1}$ is true:
+
+$$\begin{aligned}
+    \boxed{
+        \mathrm{CNOT} =
+        \begin{bmatrix}
+            1 & 0 & 0 & 0 \\
+            0 & 1 & 0 & 0 \\
+            0 & 0 & 0 & 1 \\
+            0 & 0 & 1 & 0
+        \end{bmatrix}
+    }
+\end{aligned}$$
+
+That is, it swaps the last two coefficients $c_{10}$ and $c_{11}$ in the composite state vector:
+
+$$\begin{aligned}
+    \mathrm{CNOT} \ket{\psi_1 \psi_2}
+    &= c_{00} \ket{00} + c_{01} \ket{01} + c_{11} \ket{10} + c_{10} \ket{11}
+\end{aligned}$$
+
+More generally, each one-qubit gate $U$ can be turned into a **controlled** $U$ **gate**:
+
+$$\begin{aligned}
+    \boxed{
+        \mathrm{CU} =
+        \begin{bmatrix}
+            1 & 0 & 0 & 0 \\
+            0 & 1 & 0 & 0 \\
+            0 & 0 & u_{00} & u_{01} \\
+            0 & 0 & u_{10} & u_{11}
+        \end{bmatrix}
+    }
+\end{aligned}$$
+
+Where the lower-right 2x2 block is simply $U$.
+The general action of this gate is given by:
+
+$$\begin{aligned}
+    \mathrm{CU} \ket{\psi_1 \psi_2}
+    &= c_{00} \ket{00} + c_{01} \ket{01} + (c_{10} u_{00} + c_{11} u_{01}) \ket{10} + (c_{10} u_{10} + c_{11} u_{11}) \ket{11}
+\end{aligned}$$
+
+A set of gates is **universal** if all possible mappings
+from $n$ to $n$ qubits can be approximated using only these gates.
+A minimal universal set is $\{\mathrm{CNOT}, T, S\}$,
+and there exist many others.
+
+
+## References
+1.  J.S. Neergaard-Nielsen,
+    *Quantum information: lectures notes*,
+    2021, unpublished.
+2.  S. Aaronson,
+    *Introduction to quantum information science: lecture notes*,
+    2018, unpublished.
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