Additive basis

Additive number theory

Recreational mathematics

Area of mathematics

Ungulate subclade

Ungulate

Ustad

\Video[https://www.youtube.com/watch?v=jPIl2G88w_A]
{title=Raag Puriya Dhanashri performed on the <Sitar> by Shahid Parvez}


Puriya

Evening raga

Raga by time of the day

Raga

Some authors use the convention of:
* filled black circle: conventional <controlled quantum gate>, i.e. operate if control qubit is active
* empty (White) circle: operate if control qubit is inactive


Empty circle control qubit notation

{wiki=Quantum_logic_gate\#Controlled_gates}

<Controlled quantum gates> are gates that have two types of input qubits:
* control <qubits>
* operand <qubits> (terminology made up by <Ciro Santilli> just now)
These gates can be understood as doing a certain <unitary operation> only if the control qubits are enabled or disabled.

The first example to look at is the <CNOT gate>.

\Image[https://upload.wikimedia.org/wikipedia/commons/d/dc/Controlled_gate.svg]
{title=Generic <controlled quantum gate> symbol}
{description=
The black dot means "control qubit", and "U" means an arbitrary <Unitary operation>.

When the operand has a conventional symbol, e.g. the <image Quantum NOT gate symbol> for the <quantum NOT gate> to form the <CNOT gate>, that symbol is used in the operand instead. 
}


Controlled quantum gate

Alps

= Multi-qubit quantum gate
{synonym}

= Two-qubit gate
{synonym}

The most common way to construct <multi-qubit gates> is to use <single-qubit gates> as part of a <controlled quantum gate>.


Multi-qubit gate

{c}
{wiki=Quantum_logic_gate\#Phase_shift_gates}


Phase shift gate

{c}

= Quantum NOT gate
{synonym}
{title2}

The <quantum NOT gate> swaps the state of $\ket{0}$ and $\ket{1}$, i.e. it maps:
$$
x \ket{0} + y \ket{y} \to y \ket{0} + x \ket{y}
$$
As a result, this gate also inverts the probability of measuring 0 or 1, e.g.
* if the old probability of 0 was 0, then it becomes 1
* if the old probability of 0 was 0.2, then it becomes 0.8

$$
\begin{bmatrix}
0 & 1 \\
1 & 0 \\
\end{bmatrix}
$$
{title=<Quantum NOT gate> <matrix>}

\Image[https://upload.wikimedia.org/wikipedia/commons/9/91/Qcircuit_CNOT.svg]
{title=<Quantum NOT gate> symbol}


Pauli-X gate

Pauli gate

{c}
{wiki=Quantum_logic_gate\#Hadamard_gate}

The <Hadamard gate> takes $\ket{0}$ or $\ket{1}$ (<quantum states> with probability 1.0 of measuring either 0 or 1), and produces states that have equal probability of 0 or 1.

$$
H = \frac{1}{\sqrt{2}} \begin{bmatrix} 1 & 1 \\ 1 & -1 \end{bmatrix}
$$
{title=<Hadamard gate> <matrix>}

\Image[https://upload.wikimedia.org/wikipedia/commons/1/1a/Hadamard_gate.svg]
{title=<Hadamard gate> symbol}


Ciro Santilli @cirosantilli  40