From 6ce0bb9a8f9fd7d169cbb414a9537d68c5290aae Mon Sep 17 00:00:00 2001 From: Prefetch Date: Fri, 14 Oct 2022 23:25:28 +0200 Subject: Initial commit after migration from Hugo --- source/know/concept/quantum-gate/index.md | 296 ++++++++++++++++++++++++++++++ 1 file changed, 296 insertions(+) create mode 100644 source/know/concept/quantum-gate/index.md (limited to 'source/know/concept/quantum-gate/index.md') diff --git a/source/know/concept/quantum-gate/index.md b/source/know/concept/quantum-gate/index.md new file mode 100644 index 0000000..e7a83b0 --- /dev/null +++ b/source/know/concept/quantum-gate/index.md @@ -0,0 +1,296 @@ +--- +title: "Quantum gate" +date: 2021-03-29 +categories: +- Quantum information +layout: "concept" +--- + +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, from every one-qubit gate $U$, +we can define a two-qubit **controlled U gate** $\mathrm{CU}$, +which applies $U$ to $\Ket{\psi_2}$ if $\Ket{\psi_1}$ is true: + + + + + +$$\begin{aligned} + \boxed{ + \mathrm{CU} = + \begin{bmatrix} + 1 & 0 & 0 & 0 \\ + 0 & 1 & 0 & 0 \\ + 0 & 0 & u_{00} & u_{01} \\ + 0 & 0 & u_{10} & u_{11} + \end{bmatrix} + } +\end{aligned}$$ + +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. -- cgit v1.2.3