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<!DOCTYPE html>
<html>
<head>
<title>Q ⟩ Gate</title>
<meta charset="utf-8">
<meta name="viewport" content="width=device-width,initial-scale=1.0">
<meta name="description" content="Quantum computing in your browser.">
<meta name="copyright" content="Stewart Smith 2019–2020">
<meta name="keywords" content="
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<script src="Q/Q.js"></script>
<script src="Q/Q-ComplexNumber.js"></script>
<script src="Q/Q-Matrix.js"></script>
<script src="Q/Q-Qubit.js"></script>
<script src="Q/Q-Gate.js"></script>
<script src="Q/Q-History.js"></script>
<script src="Q/Q-Circuit.js"></script>
<script src="Q/Q-Circuit-Editor.js"></script>
<script src="assets/navigation.js"></script>
</head>
<body>
<main class="api">
<p>
Source code:
<a href="https://github.com/stewdio/q.js/blob/master/source/Q-Gate.js?ts=4" target="_blank">
<code>Q-Gate.js</code>
</a>
</p>
<hr>
<h3>Gates to walk through</h3>
<p>
A quantum computer is just a collection of
<a href="Q-Qubit.html">qubits</a>.
These qubits hold values like
<code>1</code>, <code>0</code>, or some value inbetween.
A quantum computer
is able to calculate things
by changing the value of its qubits over time.
We tell the computer exactly how it should change
the value of a qubit
by instructing it to
“walk through” a series of
<a href="https://en.wikipedia.org/wiki/Quantum_logic_gate" target="_blank">quantum gates</a>.
As the qubit
<a href="https://en.wikipedia.org/wiki/The_Gates" target="_blank">“walks through” each gate</a>
its value is changed based on the type of gate
it is passing through.
</p>
<p>
Contrary to what pop-science might tell you,
there is nothing random or unpredictable
about this process.
Mathematically, a
<a href="Q-Qubit.html">qubit</a> is just a
<a href="Q-Matrix.html">matrix</a>.
Similarly, a quantum gate is also just a <a href="Q-Matrix.html">matrix</a>.
To apply a gate to a qubit
is to
<a href="Q-Matrix.html#.prototype.multiply">multiply these matrices together</a>.
To demonstrate this,
let’s begin with a
<a href="Q-Qubit.html#.HORIZONTAL">“Horizontal”
qubit</a>—commonly thought of as
representing “off.”
It has the following matrix form:
</p>
<div class="center">
<div class="matrix qubit">
<div class="matrix-bracket-left"></div>
<div class="matrix-bracket-right"></div>
<table>
<tr><td>1</td></tr>
<tr><td>0</td></tr>
</table>
</div>
<a href="Q-Qubit.html#.HORIZONTAL">Horizontal qubit</a>
</div>
<p>
We would like to flip the value of this qubit
from “off” to “on.”
The result will be a
<a href="Q-Qubit.html#.VERTICAL">“Vertical”
qubit</a>
with the following matrix form:
</p>
<div class="center">
<div class="matrix qubit">
<div class="matrix-bracket-left"></div>
<div class="matrix-bracket-right"></div>
<table>
<tr><td>0</td></tr>
<tr><td>1</td></tr>
</table>
</div>
<a href="Q-Qubit.html#.VERTICAL">Vertical qubit</a>
</div>
<p>
In order to achieve this
we must apply a <a href="#.PAULI_X">Pauli X gate</a>
to our
<a href="Q-Qubit.html#.HORIZONTAL">Horizontal qubit</a>.
Pauli X gates have the effect of “flipping” the value of a qubit.
They are often thought of
as the quantum equivalent of
a <a href="https://en.wikipedia.org/wiki/Inverter_(logic_gate)" target="_blank">classical NOT gate</a>.
Pauli X gates have the following matrix form:
</p>
<div class="center">
<div class="matrix">
<div class="matrix-bracket-left"></div>
<div class="matrix-bracket-right"></div>
<table>
<tr>
<td>0</td><td>1</td>
</tr>
<tr>
<td>1</td><td>0</td>
</tr>
</table>
</div>
<a href="#.PAULI_X">Pauli X gate</a>
</div>
<p>
We can now apply the
<a href="#.PAULI_X">Pauli X gate</a>
to the
<a href="Q-Qubit.html#.HORIZONTAL">Horizontal qubit</a>
by multiplying their matrices together.
</p>
<div class="center">
<div class="matrix">
<div class="matrix-bracket-left"></div>
<div class="matrix-bracket-right"></div>
<table>
<tr>
<td>0</td><td>1</td>
</tr>
<tr>
<td>1</td><td>0</td>
</tr>
</table>
</div>
×
<div class="matrix qubit">
<div class="matrix-bracket-left"></div>
<div class="matrix-bracket-right"></div>
<table>
<tr><td>1</td></tr>
<tr><td>0</td></tr>
</table>
</div>
=
<div class="matrix qubit">
<div class="matrix-bracket-left"></div>
<div class="matrix-bracket-right"></div>
<table>
<tr><td>0</td></tr>
<tr><td>1</td></tr>
</table>
</div>
</div>
<p>
As anticipated, the resulting product matrix represents a
<a href="Q-Qubit.html#.VERTICAL">Vertical qubit</a>.
(<a href="https://en.wiktionary.org/wiki/warm_fuzzy" target="_blank">Warm fuzzies</a> all around.)
</p>
<p>
Note how the gate’s matrix
is the first factor
and the qubit’s matrix is the second factor.
The order matters because matrix multiplication is <strong>not</strong>
<a href="https://en.wikipedia.org/wiki/Commutative_property#Commutative_operations_in_mathematics" target="_blank">commutative</a>.
With plain numbers
the order of the factors
does not change the product outcome.
<span class="maths">a × b = b × a</span>.
But with matrices the order does matter.
<span class="maths">[ a ] × [ b ] ≠ [ b ] × [ a ]</span>.
See <a href="https://en.wikipedia.org/wiki/Matrix_multiplication" target="_blank">matrix multiplication</a>
for an in-depth explanation.
</p>
<h4>Applying gates with code</h4>
<p>
Let’s look at three different ways
to apply a gate to a qubit
using Q’s API.
In this first example we will access
our gate’s matrix directly
and multiply it against our qubit—which is itself a matrix.
(<code><a href="Q.html">Q</a>.<a href="Q-Qubit.html">Qubit</a></code> extends <code><a href="Q.html">Q</a>.<a href="Q-Matrix.html">Matrix</a></code>
and thereby inherits <code><a href="Q.html">Q</a>.<a href="Q-Matrix.html">Matrix</a></code>’s methods.
Meanwhile, <code><a href="Q.html">Q</a>.Gate</code> does not extend <code><a href="Q.html">Q</a>.<a href="Q-Matrix.html">Matrix</a></code>,
as it is sometimes desirable to create gates without matrices.
See <a href="#Gates_without_matrices">Gates without matrices</a> below for more details.)
</p>
<pre><code>
<a href="Q.html">Q</a>.Gate.<a href="#.PAULI_X">PAULI_X</a>
.<a href="#this.matrix">matrix</a>
.<a href="Q-Matrix.html#.prototype.multiply">multiply</a>( <a href="Q.html">Q</a>.<a href="Q-Qubit.html">Qubit</a>.<a href="Q-Qubit.html#.HORIZONTAL">HORIZONTAL</a> )
.<a href="Q-Matrix.html#.prototype.isEqualTo">isEqualTo</a>( <a href="Q.html">Q</a>.<a href="Q-Qubit.html">Qubit</a>.<a href="Q-Qubit.html#.VERTICAL">VERTICAL</a> )</code>
<samp>true</samp></pre>
<p>
Rather than access our gate’s matrix directly,
we can instead call the instance method
<code><a href="#this.applyToQubit">applyToQubit</a></code>.
</p>
<pre><code>
<a href="Q.html">Q</a>.Gate.<a href="#.PAULI_X">PAULI_X</a>
.<a href="#this.applyToQubit">applyToQubit</a>( <a href="Q.html">Q</a>.<a href="Q-Qubit.html">Qubit</a>.<a href="Q-Qubit.html#.HORIZONTAL">HORIZONTAL</a> )
.<a href="Q-Matrix.html#.prototype.isEqualTo">isEqualTo</a>( <a href="Q.html">Q</a>.<a href="Q-Qubit.html">Qubit</a>.<a href="Q-Qubit.html#.VERTICAL">VERTICAL</a> )</code>
<samp>true</samp></pre>
<p>
We can also begin with the qubit
rather than the gate.
</p>
<pre><code>
<a href="Q.html">Q</a>.<a href="Q-Qubit.html">Qubit</a>.<a href="Q-Qubit.html#.HORIZONTAL">HORIZONTAL</a>
.<a href="Q-Qubit.html#.prototype.applyGate">applyGate</a>( <a href="Q.html">Q</a>.Gate.<a href="#.PAULI_X">PAULI_X</a> )
.<a href="Q-Matrix.html#.prototype.isEqualTo">isEqualTo</a>( <a href="Q.html">Q</a>.<a href="Q-Qubit.html">Qubit</a>.<a href="Q-Qubit.html#.VERTICAL">VERTICAL</a> )</code>
<samp>true</samp></pre>
<h4>Gates without matrices</h4>
<p>
In quantum computing theory a quantum gate <strong>is</strong> a matrix.
But Q expands this notion of a “gate”
to the more abstract notion of an “operation”—any function
that is performed upon a qubit.
This allows us to build custom “gates” that
perform data visualizations,
trigger interface changes,
and so on.
</p>
<h4>The punchline</h4>
<p>
While applying gates to qubits in the above fashion is useful
for some debugging and
for building single-qubit state visualizations,
it has no practical use for
full circuit evaluation as
it cannot represent the state of
a multiple-qubit system, let alone
<a href="https://en.wikipedia.org/wiki/Bell_state" target="_blank">quantum entanglement</a>.
For that we will need a proper
<code><a href="Q-Circuit.html">Circuit</a></code> class
for simulating multi-qubit states.
</p>
<hr>
<h3 id="Constructor">Constructor</h3>
<p>
<span class="constructor">Gate</span>
<code class="value-type">Function( params: Object ) => <a href="Q.html">Q</a>.Gate</code>
<br>
Expects a single object whose properties
will be applied directly to this gate instance.
No particular properties are required
and even the object argument itself is optional.
The properties <em>not</em> listed as “optional” below
are ones that will be assigned defaults
upon instantiation by the constructor.
Any remaining properties listed below would be undefined
unless supplied as arguments.
</p>
<p>
Here we will create a <a href="https://en.wikipedia.org/wiki/Guppy" target="_blank">“gup”</a> gate
that borrows its <code><a href="#this.matrix">matrix</a></code> property
from the pre-existing <code><a href="#.PAULI_X">PAULI_X</a></code> gate
<a href="#.constantss">constant</a>.
This will enable our custom gate
to flip
<a href="Q-Qubit.html">qubits</a> from a
<a href="Q-Qubit.html#.HORIZONTAL">horizontal state</a>,
ie. “off” or <span class="complex-vector ket">0</span>,
to a <a href="Q-Qubit.html#.VERTICAL">vertical state</a>,
ie. “on” or <span class="complex-vector ket">1</span>,
and back again—just like a
<code><a href="#.PAULI_X">PAULI_X</a></code> gate.
</p>
<pre><code>
var gup = new <a href="Q.html">Q</a>.Gate({
<a href="#this.symbol">symbol</a>: 'G',
<a href="#this.name">name</a>: 'Gup',
<a href="#this.nameCss">nameCss</a>: 'gup',<!--
<a href="#this.description">description</a>: 'This “gup” gate flips qubits.', -->
<a href="#this.matrix">matrix</a>: <a href="Q.html">Q</a>.Gate.<a href="#.PAULI_X">PAULI_X</a>.<a href="#this.matrix">matrix</a>
})
</code></pre>
<p>
We can now test that our new
<code>gup</code> gate
flips qubit states between <span class="complex-vector ket">0</span>
and <span class="complex-vector ket">1</span>.
</p>
<pre>
<code><span class="comment">// Verify a “Horizontal” qubit has a value of |0⟩
// and a “Vertical” qubit has a value of |1⟩.</span>
<a href="Q.html">Q</a>.<a href="Q-Qubit.html">Qubit</a>.<a href="Q-Qubit.html#.HORIZONTAL">HORIZONTAL</a>.<a href="Q-Qubit.html#this.toStateVectorText">toStateVectorText</a>()</code>
<samp>"|0⟩"</samp>
<code><a href="Q.html">Q</a>.<a href="Q-Qubit.html">Qubit</a>.<a href="Q-Qubit.html#.VERTICAL">VERTICAL</a>.<a href="Q-Qubit.html#this.toStateVectorText">toStateVectorText</a>()</code>
<samp>"|1⟩"</samp>
<code><span class="comment">// Now flip a “Horizontal” |0⟩ to a “Vertical” |1⟩
// and flip a “Vertical” |1⟩ to a “Horizontal” |0⟩.</span>
gup.<a href="#this.applyToQubit">applyToQubit</a>( <a href="Q.html">Q</a>.<a href="Q-Qubit.html">Qubit</a>.<a href="Q-Qubit.html#.HORIZONTAL">HORIZONTAL</a> )
.<a href="Q-Qubit.html#this.toStateVectorText">toStateVectorText</a>()</code>
<samp>"|1⟩"</samp>
<code>gup.<a href="#this.applyToQubit">applyToQubit</a>( <a href="Q.html">Q</a>.<a href="Q-Qubit.html">Qubit</a>.<a href="Q-Qubit.html#.VERTICAL">VERTICAL</a> )
.<a href="Q-Qubit.html#this.toStateVectorText">toStateVectorText</a>()</code>
<samp>"|0⟩"</samp></pre>
<p>
We can similarly use a qubit’s
<code><a href="Q-Qubit.html#.prototype.applyGate">applyGate</a></code> method
which internally calls a gate’s
<code><a href="#this.applyToQubit">applyToQubit</a></code> method:
</p>
<pre><code>
<a href="Q.html">Q</a>.<a href="Q-Qubit.html">Qubit</a>.<a href="Q-Qubit.html#.HORIZONTAL">HORIZONTAL</a> <span class="comment">// Value begins at |0⟩.</span>
.<a href="Q-Qubit.html#.prototype.applyGate">applyGate</a>( gup ) <span class="comment">// Value is now |1⟩.</span>
.<a href="Q-Qubit.html#this.toStateVectorText">toStateVectorText</a>()<span class="comment">// Output that as text.</span></code>
<samp>"|1⟩"</samp>
<code><a href="Q.html">Q</a>.<a href="Q-Qubit.html">Qubit</a>.<a href="Q-Qubit.html#.VERTICAL">VERTICAL</a> <span class="comment">// Value begins at |1⟩.</span>
.<a href="Q-Qubit.html#.prototype.applyGate">applyGate</a>( gup ) <span class="comment">// Value is now |0⟩.</span>
.<a href="Q-Qubit.html#this.toStateVectorText">toStateVectorText</a>()<span class="comment">// Output that as text.</span></code>
<samp>"|0⟩"</samp></pre>
<p>
We can also do this in one go:
</p>
<pre><code>
<a href="Q.html">Q</a>.<a href="Q-Qubit.html">Qubit</a>.<a href="Q-Qubit.html#.HORIZONTAL">HORIZONTAL</a> <span class="comment">// Value begins at |0⟩.</span>
.<a href="Q-Qubit.html#.prototype.applyGate">applyGate</a>( gup ) <span class="comment">// Value is now |1⟩.</span>
.<a href="Q-Qubit.html#.prototype.applyGate">applyGate</a>( gup ) <span class="comment">// Value is back to |0⟩.</span>
.<a href="Q-Qubit.html#this.toStateVectorText">toStateVectorText</a>()<span class="comment">// Output that as text.</span></code>
<samp>"|0⟩"</samp></pre>
<ul class="properties">
<li>
<dt id="this.symbol">symbol</dt>
<dd>
<code class="value-type">String</code>
One or two non-numeric characters
used as a unique identifier among gates,
and as a label in visual representations
such as circuit diagrams.
Will default to <code>"?"</code> if undefined at creation.
</dd>
</li>
<li>
<dt id="this.symbolAmazonBraket">symbolAmazonBraket</dt>
<dd>
<code class="value-type">String</code>
Identifier to be used for this gate
when exporting circuit data
as <a href="https://aws.amazon.com/braket/" target="_blank">Amazon Braket</a> code.
If not supplied to the constructor,
<code>Gate</code> will define this value as
<code>this.<a href="#this.symbol">symbol</a>.toLowerCase()</code>.
</dd>
</li>
<li>
<dt id="this.symbolSvg">symbolSvg</dt>
<dd>
<code class="value-type">String [optional]</code>
A visual representation of this gate’s symbol
in <a href="https://en.wikipedia.org/wiki/Scalable_Vector_Graphics" target="_blank">scalable vector graphic</a> format.
</dd>
</li>
<li>
<dt id="this.name">name</dt>
<dd>
<code class="value-type">String</code>
The full name of this gate
to be used as a title or long label
in a visual representations, etc.
Will default to <code>"Unknown"</code> if undefined at creation.
</dd>
</li>
<li>
<dt id="this.nameCss">nameCss</dt>
<dd>
<code class="value-type">String</code>
Identifier to be used in constructing a
<a href="https://en.wikipedia.org/wiki/Cascading_Style_Sheets" target="_blank">Cascading Style Sheet</a> class name
as defined in <code>Q-Circuit-Editor.css</code>.
Will default to <code>"unknown"</code> if undefined at creation.
</dd>
</li>
<!-- li>
<dt id="this.description">description</dt>
<dd>
<code class="value-type">String [optional]</code>
Text describing the intended use of this gate.
</dd>
</li -->
<li>
<dt id="this.matrix">matrix</dt>
<dd>
<code class="value-type"><a href="Q.html">Q</a>.<a href="Q-Matrix.html">Matrix</a> [optional]</code>
The matrix representation of this gate operation’s logic,
if one exists.
(All true quantum gates will have a matrix representation,
but some visualization operations—such as
the creation of a
<a href="Q-Qubit.html#Bloch_sphere">Bloch sphere</a>—will not.)
</dd>
</li>
<li>
<dt id="this.applyToQubit">applyToQubit</dt>
<dd>
<code class="value-type">Function</code>
An action to be performed upon an input
<code><a href="Q-Qubit.html">Qubit</a></code> instance.
If not supplied to the <code>Gate</code> constructor,
<code>Gate</code> will create an operation
that multiplies the input qubit by the gate instance’s
<code><a href="#this.matrix">matrix</a></code>.
If the gate instance’s
<code><a href="#this.matrix">matrix</a></code> is undefined,
<code>Gate</code> will create an operation
that simply returns the input qubit untouched.
While this is useful
for inspection and debugging a single gate’s functionality,
or building a visualization of a single qubit’s changing state,
it has no practical use for circuit evaluation
as it cannot represent the state of a multiple-qubit system,
let alone entanglement.
</dd>
</li>
<li>
<dt id="this.index">index</dt>
<dd>
<code class="value-type">Number</code>
An auto-incrementing identification number assigned to the instance,
used for minding the total number of instances created.
</dd>
</li>
</ul>
<hr>
<h3>Static properties</h3>
<ul class="properties">
<li>
<dt id=".index">index</dt>
<dd>
<code class="value-type">Number</code>
The number of instances created so far.
</dd>
</li>
<li>
<dt id=".findBy">findBy</dt>
<dd>
<code class="value-type">Function( key: String, value: * ) ⇒ Q.Gate</code>
Returns the first object within
<code><a href="Q.html">Q</a>.Gate.<a href="#.constants">constants</a></code>
to satisfy the constraint
<code>object[ key ] === value</code>.
</dd>
</li>
<li>
<dt id=".findBySymbol">findBySymbol</dt>
<dd>
<code class="value-type">Function( symbol: String ) ⇒ Q.Gate</code>
Returns the result of calling
the static method <code><a href="#.findBy">findBy</a></code> with
<code>"symbol"</code> as the <code>key</code> argument
and <code>symbol</code> as the value argument.
</dd>
</li>
<li>
<dt id=".findByName">findByName</dt>
<dd>
<code class="value-type">Function( name: String ) ⇒ Q.Gate</code>
Returns the result of calling
the static method <code><a href="#.findBy">findBy</a></code> with
<code>"name"</code> as the <code>key</code> argument
and <code>name</code> as the value argument.
</dd>
</li>
</ul>
<h4>Constants and constant creation</h4>
<ul class="properties">
<li>
<dt id=".constants">constants</dt>
<dd>
<code class="value-type">Object</code>
Constants are appended <em>directly</em> to the
<code><a href="Q.html">Q</a>.Gate</code> object.
For convenience they are also appended to this
<code><a href="Q.html">Q</a>.Gate</code>.constants</code> object
to make looking up constants in the JavaScript console trivial,
and to make iterating across all constants convenient via functions like
<code><a href="https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Global_Objects/Object/entries" target="_blank">Object.entries</a></code>,
<code><a href="https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Global_Objects/Object/keys" target="_blank">Object.keys</a></code>,
<code><a href="https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Global_Objects/Object/values" target="_blank">Object.values</a></code>,
and so on.
<!-- Configured to be unwritable once appended via
<code><a href="https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Global_Objects/Object/defineProperty" target="_blank">Object.defineProperty</a></code>
and they are labeled in uppercase to signal to us that this is so. -->
The intention that a property act as a constant is signaled
by its labelling in all-uppercase.
</dd>
</li>
<li>
<dt id=".createConstant">createConstant</dt>
<dd>
<code class="value-type">Function( key: String, value: * )<!-- → undefined --></code>
Appends a property named by <code>key</code>
with a value of <code>value</code>
to both the
<code>Gate</code> object
and its <code><a href="#.constants">constants</a></code> property.
</dd>
</li>
<li>
<dt id=".createConstants">createConstants</dt>
<dd>
<code class="value-type">Function( … )</code>
Expects an even number of arguments.
Will use each pair in the sequence of arguments to call
<code><a href="#.createConstant">createConstant</a></code>.
</dd>
</li>
</ul>
<br>
<h5>Gates of small consequence</h5>
<ul class="properties">
<li>
<dt id=".IDENTITY">IDENTITY</dt>
<dd>
<code class="value-type"><a href="Q.html">Q</a>.Gate</code>
An Identity gate has no effect on the value of the qubit it operates on;
equivalent to multiplying a value by one.
(Generally when a circuit is created from text or another source,
any included identity gates are ignored.
It is included here for completeness.)
<pre><code>
new <a href="Q.html">Q</a>.Gate({
<a href="#this.symbol">symbol</a>: 'I',<!-- <a href="#this.symbolAmazonBraket">symbolAmazonBraket</a>: 'i', -->
<a href="#this.name">name</a>: 'Identity',
<a href="#this.nameCss">nameCss</a>: 'identity',
<a href="#this.matrix">matrix</a>: <a href="Q.html">Q</a>.<a href="Q-Matrix.html">Matrix</a>.<a href="Q-Matrix.html#.IDENTITY_2X2">IDENTITY_2X2</a>
})
</code></pre>
Matrix representation as declared in
<code><a href="Q.html">Q</a>.<a href="Q-Matrix.html">Matrix</a>.<a href="Q-Matrix.html#.IDENTITY_2X2">IDENTITY_2X2</a></code>:
<div class="maths">
<div class="matrix">
<div class="matrix-bracket-left"></div>
<div class="matrix-bracket-right"></div>
<table>
<tr>
<td>1</td><td>0</td>
</tr>
<tr>
<td>0</td><td>1</td>
</tr>
</table>
</div>
</div>
</dd>
</li>
<li>
<dt id=".CURSOR">CURSOR</dt>
<dd>
<code class="value-type"><a href="Q.html">Q</a>.Gate</code>
Mathematically the “identity cursor” is equivalent
to the <code><a href="#.IDENTITY">IDENTITY</a></code> gate,
however it is used by the visual circuit editor
as a placeholder when a user is building
<a href="https://en.wikipedia.org/wiki/Quantum_logic_gate#Controlled_(cX_cY_cZ)_gates" target="_blank">controlled gates</a>
or
<a href="https://en.wikipedia.org/wiki/Quantum_logic_gate#Swap_(SWAP)_gate" target="_blank">swap gates</a>
(which operate on multiple qubits)
from individual components.
This is a novel Q invention.
<pre><code>
new <a href="Q.html">Q</a>.Gate({
<a href="#this.symbol">symbol</a>: '*',<!-- <a href="#this.symbolAmazonBraket">symbolAmazonBraket</a>: 'i', -->
<a href="#this.name">name</a>: 'Identity',
<a href="#this.nameCss">nameCss</a>: 'identity',
<a href="#this.matrix">matrix</a>: <a href="Q.html">Q</a>.<a href="Q-Matrix.html">Matrix</a>.<a href="Q-Matrix.html#.IDENTITY_2X2">IDENTITY_2X2</a>
})
</code></pre>
Matrix representation as declared in
<code><a href="Q.html">Q</a>.<a href="Q-Matrix.html">Matrix</a>.<a href="Q-Matrix.html#.IDENTITY_2X2">IDENTITY_2X2</a></code>:
<div class="maths">
<div class="matrix">
<div class="matrix-bracket-left"></div>
<div class="matrix-bracket-right"></div>
<table>
<tr>
<td>1</td><td>0</td>
</tr>
<tr>
<td>0</td><td>1</td>
</tr>
</table>
</div>
</div>
</dd>
</li>
</ul>
<h5>Standard single-qubit gates</h5>
<ul class="properties">
<li>
<dt id=".HADAMARD">HADAMARD</dt>
<dd>
<code class="value-type"><a href="Q.html">Q</a>.Gate</code>
Applies a <a href="https://en.wikipedia.org/wiki/Quantum_logic_gate#Hadamard_(H)_gate" target="_blank">Hadamard transform</a> to a single qubit.
For the basis qubit states of
<span class="complex-vector ket">0</span> and
<span class="complex-vector ket">1</span>
this has the effect of putting a qubit into <a href="https://en.wikipedia.org/wiki/Quantum_superposition" target="_blank">superposition</a>.
It represents a rotation on the
<a href="Q-Qubit.html#Bloch_sphere">Bloch sphere</a>
around the Z-axis by π radians,
followed by a rotation
around the Y-axis by π÷2 radians.
<pre><code>
new <a href="Q.html">Q</a>.Gate({
<a href="#this.symbol">symbol</a>: 'H',<!-- <a href="#this.symbolAmazonBraket">symbolAmazonBraket</a>: 'h', -->
<a href="#this.name">name</a>: 'Hadamard',
<a href="#this.nameCss">nameCss</a>: 'hadamard',
<a href="#this.matrix">matrix</a>: <a href="Q.html">Q</a>.<a href="Q-Matrix.html">Matrix</a>(
[ Math.SQRT1_2, Math.SQRT1_2 ],
[ Math.SQRT1_2, -Math.SQRT1_2 ]
)
})
</code></pre>
Matrix representation:
<div class="maths">
<div class="matrix">
<div class="matrix-bracket-left"></div>
<div class="matrix-bracket-right"></div>
<table>
<tr>
<td>1</td><td>1</td>
</tr>
<tr>
<td>1</td><td>-1</td>
</tr>
</table>
</div>
×
<table class="division">
<tr class="dividend"><td>1</td></tr>
<tr class="divisor"><td>√ 2</td></tr>
</table>
</div>
</dd>
</li>
<li>
<dt id=".PAULI_X">PAULI_X</dt>
<dd>
<code class="value-type"><a href="Q.html">Q</a>.Gate</code>
The <a href="https://en.wikipedia.org/wiki/Quantum_logic_gate#Pauli-X_gate" target="_blank">Pauli X gate</a>
represents a rotation on the
<a href="Q-Qubit.html#Bloch_sphere">Bloch sphere</a>
around the <strong>X</strong>-axis by π radians.
It is the quantum equivalent of
the <a href="https://en.wikipedia.org/wiki/Inverter_(logic_gate)" target="_blank">classical NOT gate</a>
in that it maps
<span class="complex-vector ket">0</span> to <span class="complex-vector ket">1</span>
and
<span class="complex-vector ket">1</span> to <span class="complex-vector ket">0</span>.
<pre><code>
new <a href="Q.html">Q</a>.Gate({
<a href="#this.symbol">symbol</a>: 'X',<!-- <a href="#this.symbolAmazonBraket">symbolAmazonBraket</a>: 'x', -->
<a href="#this.name">name</a>: 'Pauli X',
<a href="#this.nameCss">nameCss</a>: 'pauli-x',
<a href="#this.matrix">matrix</a>: <a href="Q.html">Q</a>.<a href="Q-Matrix.html">Matrix</a>(
[ 0, 1 ],
[ 1, 0 ]
)
})
</code></pre>
Matrix representation:
<div class="maths">
<div class="matrix">
<div class="matrix-bracket-left"></div>
<div class="matrix-bracket-right"></div>
<table>
<tr>
<td>0</td><td>1</td>
</tr>
<tr>
<td>1</td><td>0</td>
</tr>
</table>
</div>
</div>
</dd>
</li>
<li>
<dt id=".PAULI_Y">PAULI_Y</dt>
<dd>
<code class="value-type"><a href="Q.html">Q</a>.Gate</code>
The <a href="https://en.wikipedia.org/wiki/Quantum_logic_gate#Pauli-Y_gate" target="_blank">Pauli Y gate</a>
represents a rotation on the
<a href="Q-Qubit.html#Bloch_sphere">Bloch sphere</a>
around the <strong>Y</strong>-axis by π radians.
It maps
<span class="complex-vector ket">0</span> to <span class="symbol">i</span><span class="complex-vector ket">1</span>
and
<span class="complex-vector ket">1</span> to -<span class="symbol">i</span><span class="complex-vector ket">0</span>.
<pre><code>
new <a href="Q.html">Q</a>.Gate({
<a href="#this.symbol">symbol</a>: 'Y',<!-- <a href="#this.symbolAmazonBraket">symbolAmazonBraket</a>: 'y', -->
<a href="#this.name">name</a>: 'Pauli Y',
<a href="#this.nameCss">nameCss</a>: 'pauli-y',
<a href="#this.matrix">matrix</a>: <a href="Q.html">Q</a>.<a href="Q-Matrix.html">Matrix</a>(
[ 0, new <a href="Q.html">Q</a>.<a href="Q-ComplexNumber.html">ComplexNumber</a>( 0, -1 )],
[ new <a href="Q.html">Q</a>.<a href="Q-ComplexNumber.html">ComplexNumber</a>( 0, 1 ), 0 ]
)
})
</code></pre>
Matrix representation:
<div class="maths">
<div class="matrix">
<div class="matrix-bracket-left"></div>
<div class="matrix-bracket-right"></div>
<table>
<tr>
<td>0</td><td class="symbol">-i</td>
</tr>
<tr>
<td class="symbol">i</td><td>0</td>
</tr>
</table>
</div>
</div>
</dd>
</li>
<li>
<dt id=".PAULI_Z">PAULI_Z</dt>
<dd>
<code class="value-type"><a href="Q.html">Q</a>.Gate</code>
The <a href="https://en.wikipedia.org/wiki/Quantum_logic_gate#Pauli-Z_gate" target="_blank">Pauli Z gate</a>
represents a rotation on the
<a href="Q-Qubit.html#Bloch_sphere">Bloch sphere</a>
around the <strong>Z</strong>-axis by π radians.
It is a special case of a <a href="#.PHASE">Phase shift gate</a>
where ϕ = π, and is therefore sometimes referred to as a “phase-flip” gate.
It leaves the basis state <span class="complex-vector ket">0</span> unchanged
and maps <span class="complex-vector ket">1</span> to -<span class="complex-vector ket">1</span>.
<pre><code>
new <a href="Q.html">Q</a>.Gate({
<a href="#this.symbol">symbol</a>: 'Z',<!-- <a href="#this.symbolAmazonBraket">symbolAmazonBraket</a>: 'z', -->
<a href="#this.name">name</a>: 'Pauli Z',
<a href="#this.nameCss">nameCss</a>: 'pauli-z',
<a href="#this.matrix">matrix</a>: <a href="Q.html">Q</a>.<a href="Q-Matrix.html">Matrix</a>(
[ 1, 0 ],
[ 0, -1 ]
)
})
</code></pre>
Matrix representation:
<div class="maths">
<div class="matrix">
<div class="matrix-bracket-left"></div>
<div class="matrix-bracket-right"></div>
<table>
<tr>
<td>1</td><td>0</td>
</tr>
<tr>
<td>0</td><td>-1</td>
</tr>
</table>
</div>
</div>
</dd>
</li>
<li>
<dt id=".PHASE">PHASE</dt>
<dd>
<code class="value-type"><a href="Q.html">Q</a>.Gate</code>
<a href="https://en.wikipedia.org/wiki/Quantum_logic_gate#Phase_shift_(%20'%22%60UNIQ--postMath-00000038-QINU%60%22'%20)_gates" target="_blank">Phase gates</a>
are a family of quantum gates
that employ the variable ϕ (phi) to
represent tracing a horizontal arc (a line of latitude)
of ϕ radians
around the
<a href="Q-Qubit.html#Bloch_sphere">Bloch sphere</a>.
They leave the basis state <span class="complex-vector ket">0</span> unchanged
and map <span class="complex-vector ket">1</span> to
<span class="symbol">e</span><sup><span class="symbol">i</span>ϕ</sup>
<span class="complex-vector ket">1</span>.
The probability of measuring a
<span class="complex-vector ket">0</span>
or
<span class="complex-vector ket">1</span>
is unchanged after applying a phase shift,
however it modifies the phase of the quantum state.
This particular form
used for our <code>PHASE</code> constant
represents a rotation on the
<a href="Q-Qubit.html#Bloch_sphere">Bloch sphere</a>
around the Z-axis of π ÷ 2 radians.
<pre><code>
new <a href="Q.html">Q</a>.Gate({
<a href="#this.symbol">symbol</a>: 'P',<!-- <a href="#this.symbolAmazonBraket">symbolAmazonBraket</a>: 'p', -->
<a href="#this.name">name</a>: 'Phase',
<a href="#this.nameCss">nameCss</a>: 'phase',
updateMatrix$: function( phi ){
if( <a href="Q.html">Q</a>.<a href="Q.html#.isUsefulNumber">isUsefulNumber</a>( phi ) === true ) this.phi = phi
this.matrix = new <a href="Q.html">Q</a>.<a href="Q-Matrix.html">Matrix</a>(
[ 1, 0 ],
[ 0, <a href="Q.html">Q</a>.<a href="Q-ComplexNumber.html">ComplexNumber</a>.<a href="Q-ComplexNumber.html#.E">E</a>.<a href="Q-ComplexNumber.html#.prototype.power">power</a>( new <a href="Q.html">Q</a>.<a href="Q-ComplexNumber.html">ComplexNumber</a>( 0, this.phi ))]
return this
},
<a href="#this.applyToQubit">applyToQubit</a>: function( qubit, phi ){
if( <a href="Q.html">Q</a>.<a href="Q.html#.isUsefulNumber">isUsefulNumber</a>( phi ) !== true ) phi = this.phi
const matrix = new <a href="Q.html">Q</a>.<a href="Q-Matrix.html">Matrix</a>(
[ 1, 0 ],
[ 0, <a href="Q.html">Q</a>.<a href="Q-ComplexNumber.html">ComplexNumber</a>.<a href="Q-ComplexNumber.html#.E">E</a>.<a href="Q-ComplexNumber.html#.prototype.power">power</a>( new <a href="Q.html">Q</a>.<a href="Q-ComplexNumber.html">ComplexNumber</a>( 0, phi ))]
)
return new <a href="Q.html">Q</a>.<a href="Q-Qubit.html">Qubit</a>( matrix.<a href="Q-Matrix.html#.prototype.multiply">multiply</a>( qubit ))
}
})
</code></pre>
Note that there is no explicitly defined <code>matrix</code> property.
It will be created upon instantiation
when the <a href="#Constructor">constructor</a>
calls the argument object’s <code>updateMatrix$</code> function.
Matrix representation (when ϕ = 1):
<div class="maths">
<div class="matrix">
<div class="matrix-bracket-left"></div>
<div class="matrix-bracket-right"></div>
<table>
<tr>
<td>1</td><td>0</td>
</tr>
<tr>
<td>0</td><td class="symbol">e<sup>i</sup></td>
</tr>
</table>
</div>
</div>
<br>
Using varying values for ϕ is easy.
We can clone this <a href="#.PHASE">PHASE</a> constant,
provide it a new <code><a href="#this.symbol">symbol</a></code>
(so it has a unique identifier),
and a new value for ϕ.
In this example we create a gate called <code>fox</code>
with a ϕ value of π.
<pre><code>
var fox = <a href="Q.html">Q</a>.Gate.<a href="#.PHASE">PHASE</a>.<a href="#.prototype.clone">clone</a>({ symbol: 'F', phi: Math.PI })
</code></pre>
Our value of ϕ is now set equal to π
and we can apply this gate to any qubit.
<pre><code>
fox.<a href="#this.applyToQubit">applyToQubit</a>( <a href="Q.html">Q</a>.<a href="Q-Qubit.html">Qubit</a>.<a href="Q-Qubit.html#.DIAGONAL">DIAGONAL</a> ).<a href="Q-Matrix.html#.prototype.toTsv">toTsv</a>()</code>
<samp>"
0.707
-0.707"
</samp></pre>
But we can also send a
<em>temporary</em> value for ϕ
that will be used only during this function execution;
it will not change <code>fox</code>’s set value of ϕ.
Let’s try temporarily setting ϕ to π ÷ 4.
<pre><code>
fox.<a href="#this.applyToQubit">applyToQubit</a>( <a href="Q.html">Q</a>.<a href="Q-Qubit.html">Qubit</a>.<a href="Q-Qubit.html#.DIAGONAL">DIAGONAL</a>, Math.PI / 4 ).<a href="Q-Matrix.html#.prototype.toTsv">toTsv</a>()</code>
<samp>"
0.707
0.5 + 0.5i"
</samp></pre>
Rather than begin our expression
with a gate,
we can instead begin with a qubit,
and then apply our <code>fox</code> gate to it.
<pre><code>
<a href="Q.html">Q</a>.<a href="Q-Qubit.html">Qubit</a>.<a href="Q-Qubit.html#.DIAGONAL">DIAGONAL</a>
.<a href="Q-Qubit.html#.prototype.applyGate">applyGate</a>( fox )
.<a href="Q-Matrix.html#.prototype.toTsv">toTsv</a>()</code>
<samp>"
0.707
-0.707"
</samp></pre>
But what’s more clever is that additional arguments
will be passed directly to the gate’s
<code><a href="#this.applyToQubit">applyToQubit</a></code> method,
so we can still use temporary values for ϕ.
<pre><code>
<a href="Q.html">Q</a>.<a href="Q-Qubit.html">Qubit</a>.<a href="Q-Qubit.html#.DIAGONAL">DIAGONAL</a>
.<a href="Q-Qubit.html#.prototype.applyGate">applyGate</a>( fox, Math.PI / 4 )
.<a href="Q-Matrix.html#.prototype.toTsv">toTsv</a>()</code>
<samp>"
0.707
0.5 + 0.5i"
</samp></pre>
</dd>
</li>
<li>
<dt id=".PI_8">PI_8</dt>
<dd>
<code class="value-type"><a href="Q.html">Q</a>.Gate</code>
The π ÷ 8 gate
is a special case of a <a href="#.PHASE">Phase shift gate</a>
where the ϕ (phi) variable is set to π ÷ 4.
(But <a href="https://www.quora.com/Why-is-the-quantum-T-gate-called-pi-8-gate-as-it-only-adds-a-phase-difference-of-pi-4-instead-of-pi-8-to-the-state-vector-1" target="_blank">why is it called “π ÷ 8” when it actually divides π by 4?</a>)
Like all phase shift gates,
it represents a rotation on the
<a href="Q-Qubit.html#Bloch_sphere">Bloch sphere</a>
around the Z-axis.
<pre><code>
new <a href="Q.html">Q</a>.Gate({
<a href="#this.symbol">symbol</a>: 'T',<!-- <a href="#this.symbolAmazonBraket">symbolAmazonBraket</a>: 't', -->
<a href="#this.name">name</a>: 'π ÷ 8',
<a href="#this.nameCss">nameCss</a>: 't',
<a href="#this.matrix">matrix</a>: <a href="Q.html">Q</a>.<a href="Q-Matrix.html">Matrix</a>(
[ 1, 0 ],
[ 0, <a href="Q.html">Q</a>.<a href="Q-ComplexNumber.html">ComplexNumber</a>.<a href="Q-ComplexNumber.html#.E">E</a>.<a href="Q-ComplexNumber.html#.prototype.power">power</a>( new <a href="Q.html">Q</a>.<a href="Q-ComplexNumber.html">ComplexNumber</a>( 0, Math.PI / 4 )) ]
)
})
</code></pre>
Matrix representation:
<div class="maths">
<div class="matrix">
<div class="matrix-bracket-left"></div>