Query: please explain this
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### 2.2 Quantum computation

In circuit-based quantum computing, a number $q$ of qubits is manipulated. The state $|\psi\rangle \in \mathbb{C}^{2}$ of each qubit is a two-dimensional complex vector that can be expressed as a normalised complex linear combination (superposition) of two computational basis states

$$
\begin{aligned}
& |\psi\rangle=\alpha|0\rangle+\beta|1\rangle, \quad|0\rangle=\left[\begin{array}{l}
1 \\
0
\end{array}\right], \quad|1\rangle=\left[\begin{array}{l}
0 \\
1
\end{array}\right] \\
& |\alpha|^{2}+|\beta|^{2}=1, \quad \alpha, \beta \in \mathbb{C}
\end{aligned}
$$

Note that we use bra-ket notation, where column vectors are represented by kets e.g. $|\psi\rangle$ and row
vectors are represented by bras e.g. $\langle\psi|$. A bra-ket e.g. $\langle\psi \mid \phi\rangle$ corresponds to the dot product resulting from multiplying a row vector by a column vector, while a ket-bra e.g. $|\psi\rangle\langle\phi|$ corresponds to the matrix obtained by multiplying a column vector by a row vector. The joint state of the qubits is an element of the tensor product $\bigotimes_{j=1}^{q} \mathbb{C}^{2}$. This space is mathematically equivalent (i.e. isomorphic) to $\mathbb{C}^{2^{q}}$ as a vector space. This equivalence is evidenced by the reference basis for this space, the so-called computational basis, defined as $\left\{\left|c_{q-1} \ldots c_{0}\right\rangle: c_{k} \in\{0,1\}\right\}$, where $\left|c_{q-1} \ldots c_{0}\right\rangle$ corresponds to the tensor product $\left|c_{q-1}\right\rangle \otimes \ldots \otimes\left|c_{0}\right\rangle$. For a $q$-qubit state and a positive integer $j=\sum_{k=0}^{q-1} c_{k} 2^{k}, c_{k} \in\{0,1\}$, we define $|j\rangle:=\left|c_{q-1} \ldots c_{0}\right\rangle$ (i.e. the $k^{\prime}$ 'th qubit is set to $\left|c_{k}\right\rangle$; for instance, $|6\rangle=|110\rangle)$. Quantum circuits are often drawn in diagrammatic form as in e.g. (8). In these diagrams, wires (represented by horizontal lines) can either represent single qubits, or a group of qubits (i.e. a register) when crossed by an oblique line.

The state of a quantum system evolves through the application of gates, each of which corresponds to a unitary matrix $U$ (i.e. a matrix satisfying $U^{\dagger} U=I$, where $U^{\dagger}:=\bar{U}^{T}$ ). Gates are drawn as boxes in circuit diagrams, and may act on single qubits or be entangling, i.e. act on multiple qubits. An example of a single-qubit gate is the $R_{X}$ gate, defined as $R_{X}(\theta):=e^{-i \theta X}$. The matrices
$X:=\left[\begin{array}{cc}0 & 1 \\ 1 & 0\end{array}\right] \quad Y:=\left[\begin{array}{cc}0 & -i \\ i & 0\end{array}\right] \quad Z:=\left[\begin{array}{cc}1 & 0 \\ 0 & -1\end{array}\right]$
are the so-called Pauli matrices (Nielsen and Chuang, 2011), and the $R_{Y}$ and $R_{Z}$ gates are similarly defined.

The entangling gates we use are controlled gates of the form $C U$, where the action of a gate $U$ on a target register depends on the state of a control register. For example, a $C R_{X}$ gate applies a $R_{X}$ gate to the target qubit when the control register is in some specified state $|\psi\rangle$, and performs the identity operation $I$ if the control register is in any other state; if the control register or the target register is in a superposition, the operation is applied to each basis element by linearity. Diagramatically, a controlled gate is represented as a vertical line connecting a box on the target register to a circle on the control register, above which we indicate the state the gate is controlled on.

Qubits can be measured; measuring a qubit in state $|\psi\rangle=\alpha|0\rangle+\beta|1\rangle$ with respect to the computational basis yields an output of 0 with probability $|\alpha|^{2}=|\langle 0 \mid \psi\rangle|^{2}$ and output 1 with probability $|\beta|^{2}=|\langle 1 \mid \psi\rangle|^{2}$; this is the so-called Born rule. Thus, quantum computation is inherently probabilistic; in general, several runs (often called shots) of a
circuit are needed to obtain the desired result. For instance, some quantum algorithms rely on postselection, a process which discards all executions of a quantum circuit in which a measurement did not result in a value. For instance, postselecting on the $|0\rangle$ state is indicated in a circuit by a $\langle 0|$ placed near the end of a wire, and discards all computations for which measuring all of the corresponding qubits did not yield the value 0 .

An observable is represented by a hermitian matrix $O$, i.e. a matrix satisfying $O^{\dagger}=\bar{O}^{T}=O$. The expectation value associated with an observable $O$ applied to a state $|\psi\rangle$ is given by

$$
\langle O\rangle_{|\psi\rangle}:=\langle\psi| O|\psi\rangle \in \mathbb{R} .
$$

An observable is usually associated to a concrete physical property that can be observed, though this is not always the case. Examples of common observables are the Pauli matrices introduced above, which can be associated to the direction in threedimensional space the spin of a particle is pointing (hence the designation $X, Y, Z$ ). For further details on quantum computation, we refer the reader to Nielsen and Chuang (2011).
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Subject: physics
Intent: Paper Understanding