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<!DOCTYPE html>
<html>
<head>
<title>Q ⟩ Qubit</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="
Q, Q.js, Q-js, Qjs, quantum JavaScript,
quantum, quantum physics, quantum mechanics, superposition,
quantum computer, quantum computer programming, quantum computing, QC,
quantum simulator, quantum computer simulator,
qubit, qbit, gate, Hadamard, Bloch, Bloch Sphere,
Web, Web site, website, Web browser, browser, HTML, HTML5, JavaScript, ES6, CSS,
Chrome, Firefox, Safari, Opera, Brave, Edge, WebKit, Blink, Gecko, Mozilla,
Stewart Smith, Stewart, Stew, Stuart, Steven, Steve, Stewdio, stewartsmith, stew_rtsmith, @stew_rtsmith,
Moar, Moar Technologies Corp, MTC,
Google, IBM, Microsoft, Amazon, NASA, DWave, D-Wave,
Quil, OpenQASM,
ProjectQ, Qiskit,
Quantum Development Kit, Cirq, Strawberry Fields, t|ket>,
QCL, Quantum pseudocode, Q#, Q|SI>, Q language, qGCL, QFC, QML, LIQUi|>, Quipper,
Stanford CS 269 Q: Quantum Computer Programming">
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</head>
<body>
<main class="api">
<p>
Source code:
<a href="https://github.com/stewdio/q.js/blob/master/source/Q-Qubit.js?ts=4" target="_blank">
<code>Q-Qubit.js</code>
</a>
</p>
<hr>
<h3>Perfect pairs</h3>
<p>
A <a href="https://en.wikipedia.org/wiki/Qubit" target="_blank">qubit</a>
is just a pair of numbers.
That’s it—two
<a href="https://youtu.be/YYOKMUTTDdA" target="_blank">shiny happy number values,
holding hands</a>.
Let’s call the first number
“<code><a href="#this.alpha">alpha</a></code>”
and the second number
“<code><a href="#this.beta">beta</a></code>.”
Look at this handsome couple:
</p>
<br>
<div class="center">
<code>alpha</code>, <code>beta</code>
</div>
<br>
<p>
We can package
<code><a href="#this.alpha">alpha</a></code>
and
<code><a href="#this.beta">beta</a></code>
together
by storing them in a very small
<a href="Q-Matrix.html">matrix</a>
that is only <strong>one</strong> unit wide
and <strong>two</strong> units tall—and
because this matrix is only one unit wide
it is not merely a matrix, it’s also a
<a href="Q-Matrix.html#Vectors">vector</a>.
(What’s a <a href="Q-Matrix.html">matrix</a>?
What’s a <a href="Q-Matrix.html#Vectors">vector</a>?
See the <a href="Q-Matrix.html"><code>Matrix</code> page</a>
for quick refreshers on both.)
</p>
<p>
So when you think of a qubit you can imagine it as
a 1 × 2 <a href="Q-Matrix.html">matrix</a>
<!-- (a <a href="Q-Matrix.html#Vectors"> column vector</a>) -->
containing its
<code><a href="#this.alpha">alpha</a></code> value on the top
and its
<code><a href="#this.beta">beta</a></code> value on the bottom,
like so:
</p>
<br>
<div class="center">
<div class="matrix">
<div class="matrix-bracket-left"></div>
<div class="matrix-bracket-right"></div>
<table>
<tr><td><code>alpha</code></td></tr>
<tr><td><code>beta</code></td></tr>
</table>
</div>
</div>
<br>
<p>
Now that we know a qubit is a
<a href="Q-Matrix.html">matrix</a>,
we also know that we can perform
<a href="Q-Matrix.html#.add">addition</a> with qubits,
perform <a href="Q-Matrix.html#.multiply">multiplication</a> with qubits,
and so on—just as one can do with any matrix.
(That will become rather important
when we eventually introduce the idea of
<a href="Q-Gate.html">quantum gates</a>,
which are also matrices.)
</p>
<h4 id="Predictable_couples">Predictable couples</h4>
<p>
The thing that makes this pair of numbers special
is their relationship to each other,
which we can define as follows:
<code><a href="#this.alpha">alpha</a></code>
multiplied by
<code><a href="#this.alpha">alpha</a></code>,
added to
<code><a href="#this.beta">beta</a></code>
multiplied by
<code><a href="#this.beta">beta</a></code>,
must always equal one.
We can express this as an equation:
</p>
<br>
<div class="maths center">
( <code><a href="#this.alpha">alpha</a></code> ×
<code><a href="#this.alpha">alpha</a></code> ) +
( <code><a href="#this.beta">beta</a></code> ×
<code><a href="#this.beta">beta</a></code> ) = 1
</div>
<br>
<p>
The parentheses in the equation above
are not strictly necessary
as the
<a href="https://en.wikipedia.org/wiki/Order_of_operations" target="_blank">“order of operations” rules</a>
dictate that these multiplications
must be carried out prior to
the additions—but emphasis can often be more helpful than brevity when learning something new.
And now that we understand this relationship
we can express it more compactly
using exponents rather than multiplications:
<code><a href="#this.alpha">alpha</a></code> squared
added to
<code><a href="#this.beta">beta</a></code> squared
must always equal one.
</p>
<br>
<div class="maths center">
<code><a href="#this.alpha">alpha</a></code><sup>2</sup> +
<code><a href="#this.beta">beta</a></code><sup>2</sup> = 1
</div>
<br>
<p>
A bit <a href="#Complex_couple">further below</a>
we’ll add one small wrinkle to this equation,
but otherwise this is
what defines a qubit.
That’s it. It’s that easy.
</p>
<h4 title="Brangelina.">Named couples</h4>
<p>
Let’s start plugging in values for
<code><a href="#this.alpha">alpha</a></code>
and
<code><a href="#this.beta">beta</a></code>.
Each different value combination makes a different sort of qubit.
The most frequently used qubit value combinations
have names
and belong to a set of named vectors called
<a href="https://en.wikipedia.org/wiki/Jones_calculus#The_Jones_vector" target="_blank">Jones vectors</a>.
We’ll begin with the two simplest qubits.
This first qubit is named
<a href="#.HORIZONTAL">Horizontal</a>
and the second is named
<a href="#.VERTICAL">Vertical</a>.
Both are composed from a zero and a one
and we can see that this
satisfies our qubit definition above.
</p>
<br>
<div class="center">
<div class="matrix qubit">
<div class="matrix-bracket-left"></div>
<div class="matrix-bracket-right"></div>
<table>
<tr><td class="right"><code>alpha</code> = <strong>1</strong></td></tr>
<tr><td class="right"><code>beta</code> = <strong>0</strong></td></tr>
</table>
</div>
<div class="center middle">
A <a href="#.HORIZONTAL">Horizontal qubit</a> is considered<br>
to be “<code>0</code>” or “<code>OFF</code>.”
</div>
<br>
<br>
<div class="maths">
<table class="equation">
<tr>
<td>
<code><a href="#this.alpha">alpha</a></code><sup>2</sup> + <code><a href="#this.beta">beta</a></code><sup>2</sup>
</td>
<td>1</td>
</tr>
<tr>
<td>
<strong>1</strong><sup>2</sup> + <strong>0</strong><sup>2</sup>
</td>
<td>1</td>
</tr>
<tr>
<td>
<strong>1</strong> + <strong>0</strong>
</td>
<td>1</td>
</tr>
<tr>
<td>
<strong>1</strong>
</td>
<td>1 ✓</td>
</tr>
</table>
</div>
<br>
<div class="matrix qubit">
<div class="matrix-bracket-left"></div>
<div class="matrix-bracket-right"></div>
<table>
<tr><td class="right"><code>alpha</code> = <strong>0</strong></td></tr>
<tr><td class="right"><code>beta</code> = <strong>1</strong></td></tr>
</table>
</div>
<div class="center middle">
A <a href="#.VERTICAL">Vertical qubit</a> is considered<br>
to be “<code>1</code>” or “<code>ON</code>.”
</div>
<br>
<br>
<div class="maths">
<table class="equation">
<tr>
<td>
<code><a href="#this.alpha">alpha</a></code><sup>2</sup> + <code><a href="#this.beta">beta</a></code><sup>2</sup>
</td>
<td>1</td>
</tr>
<tr>
<td>
<strong>0</strong><sup>2</sup> + <strong>1</strong><sup>2</sup>
</td>
<td>1</td>
</tr>
<tr>
<td>
<strong>0</strong> + <strong>1</strong>
</td>
<td>1</td>
</tr>
<tr>
<td>
<strong>1</strong>
</td>
<td>1 ✓</td>
</tr>
</table>
</div>
</div>
<br>
<br>
<p>
What does it mean to be named
<a href="#.HORIZONTAL">Horizontal</a> or
<a href="#.VERTICAL">Vertical</a>?
Where do these orientation-based names come from?
Are there more orientation-based names?
To understand, let’s plot these two qubit vectors on a graph.
We’ll use the
<code><a href="#this.alpha">alpha</a></code> value
as our <span class="symbol">x</span> coordinate
and the
<code><a href="#this.beta">beta</a></code> value
as our <span class="symbol">y</span> coordinate:
</p>
<br>
<div class="center">
<div class="matrix">
<div class="matrix-bracket-left"></div>
<div class="matrix-bracket-right"></div>
<table>
<tr><td><code>alpha</code></td></tr>
<tr><td><code>beta</code></td></tr>
</table>
</div>
→ ( <code>alpha</code>, <code>beta</code> )
</div>
<br>
<p>
Plotting
<code><a href="#this.alpha">alpha</a></code>
and
<code><a href="#this.beta">beta</a></code>
as <span class="symbol">x</span> and <span class="symbol">y</span>
yields (1, 0) for a
<a href="#.HORIZONTAL">Horizontal qubit</a>
and (0, 1) for a
<a href="#.VERTICAL">Vertical qubit</a>.
</p>
<br>
<div class="unit-circle no-animation">
<!-- <div class="unit-circle-outline"></div> -->
<div class="unit-circle-spoke-half unit-circle-spoke unit-circle-spoke-0 dashed"></div>
<div class="unit-circle-spoke-half unit-circle-spoke unit-circle-spoke-180"></div>
<div class="unit-circle-spoke-half unit-circle-spoke unit-circle-spoke-90"></div>
<div class="unit-circle-spoke-half unit-circle-spoke unit-circle-spoke-270 dashed"></div>
<div class="unit-circle-spoke-label unit-circle-spoke-label-90">Vertical</div>
<div class="unit-circle-spoke-label unit-circle-spoke-label-0">Horizontal</div>
<!-- <div class="unit-circle-label unit-circle-label-angle-0">0˚</div>
<div class="unit-circle-label unit-circle-label-angle-90">90˚</div> -->
<div class="unit-circle-label unit-circle-label-point-origin">0, 0</div>
<div class="unit-circle-label unit-circle-label-point-0">1, 0</div>
<div class="unit-circle-label unit-circle-label-point-90">0, 1</div>
<div class="unit-circle-label unit-circle-label-point-180">-1, 0</div>
<div class="unit-circle-label unit-circle-label-point-270">0, -1</div>
</div>
<p>
We can see that the values from a
<a href="#.HORIZONTAL">Horizontal qubit</a>,
when plotted as <span class="symbol">x</span> and <span class="symbol">y</span>,
form a horizontal line from the origin (0, 0)
out to (1, 0).
Meanwhile,
when we plot the values of a
<a href="#.VERTICAL">Vertical qubit</a>
as <span class="symbol">x</span> and <span class="symbol">y</span>,
it forms a vertical line from the origin (0, 0)
up to (0, 1).
Before we introduce more named
<a href="#Jones_Vectors">Jones vectors</a>
let’s take what we’ve learned about qubit values
and generalize it for vectors
with more than two elements
so we can better understand what these qubit values truly mean.
</p>
<h4>State vectors</h4>
<p>
Did it seem strange to read that
the qubit we refer to as “<code>0</code>” begins with an
<code><a href="#this.alpha">alpha</a></code> value of 1?
(Or that the qubit we refer to as “<code>1</code>” begins with an
<code><a href="#this.alpha">alpha</a></code> value of 0?)
Does that mean we refer to qubits
by their <code><a href="#this.beta">beta</a></code> values?
Is that some sort of quantum computing convention?
</p>
<br>
<div class="center">
<div class="matrix qubit">
<div class="matrix-bracket-left"></div>
<div class="matrix-bracket-right"></div>
<table>
<tr><td class="right"><code>alpha</code> = <strong>1</strong></td></tr>
<tr><td class="right"><code>beta</code> = <strong>0</strong></td></tr>
</table>
</div>
<div class="center middle">
Recall that a
<a href="#.HORIZONTAL">Horizontal qubit</a><br>
is considered
“<code>0</code>” or “<code>OFF</code>.”
</div>
</div>
<br>
<p>
The short answer is “No.”
To understand why,
we must recognize that
a qubit is the simplest example of a
quantum <em>state vector</em>—a
list of all possible states
for a quantum system to exist in,
with each possible state accompanied by the probability
that the system is indeed in that state.
When a single qubit is measured
there are only two possible states
for it to be in:
<code><strong>0</strong></code> or <code><strong>1</strong></code>.
This is why a qubit is represented by a two-element vector;
one element per possible outcome.
On this very short list of possible outcomes,
<code><strong>0</strong></code> is the first possible outcome
and <code><strong>1</strong></code> is the second possible outcome.
When we say that a <a href="#.HORIZONTAL">Horizontal</a> qubit is “<code>0</code>”
<a href="https://www.youtube.com/watch?v=DfSL-HeIrxA" target="_blank">we’re not referring to the qubit’s
<code>beta</code> value at all</a>.
We’re instead highlighting the fact that
the <code>1</code> in this (1, 0) pair
happens to be in the “zeroth” slot—the
<code><a href="#this.alpha">alpha</a></code> slot
of this
(<code><a href="#this.alpha">alpha</a></code>,
<code><a href="#this.beta">beta</a></code>) pair.
We’re saying that
for the possible outcome “<code>0</code>”
our qubit is voting 1, or <code>TRUE</code>.
At the same time we’re saying that
for the possible outcome “<code>1</code>”,
our qubit is voting 0, or <code>FALSE</code>.
</p>
<br>
<div class="center">
<div class="matrix qubit">
<div class="matrix-bracket-left"></div>
<div class="matrix-bracket-right"></div>
<table class="chart">
<tr>
<td class="font-sans">
Outcome will be <code><strong>0</strong></code>:
<code>TRUE</code> (<code>1</code>)
</td>
</tr>
<tr>
<td class="font-sans">
Outcome will be <code><strong>1</strong></code>:
<code>FALSE</code> (<code>0</code>)
</td>
</tr>
</table>
</div>
<div class="center middle">
<a href="#.HORIZONTAL">Horizontal qubit</a>
</div>
</div>
<br>
<p>
For good measure
let’s look at the converse example.
</p>
<br>
<div class="center">
<div class="matrix qubit">
<div class="matrix-bracket-left"></div>
<div class="matrix-bracket-right"></div>
<table class="chart">
<tr>
<td class="font-sans">
Outcome will be <code><strong>0</strong></code>:
<code>FALSE</code> (<code>0</code>)
</td>
</tr>
<tr>
<td class="font-sans">
Outcome will be <code><strong>1</strong></code>:
<code>TRUE</code> (<code>1</code>)
</td>
</tr>
</table>
</div>
<div class="center middle">
<a href="#.VERTICAL">Vertical qubit</a>
</div>
</div>
<br>
<p>
To further clarify—and
to hint at how a <a href="Q-Circuit.html">quantum circuit</a>
functions—let’s look at a state vector
for a quantum system composed of <strong>two</strong> qubits.
With one qubit there were two possible outcomes:
<code><strong>0</strong></code> and <code><strong>1</strong></code>.
(And because there are only two elements of a qubit vector we named them
<code><a href="#this.alpha">alpha</a></code> and
<code><a href="#this.beta">beta</a></code>
to make referring to them more convenient.)
For two qubits there are four possible outcomes:
<code><strong>00</strong></code>,
<code><strong>01</strong></code>,
<code><strong>10</strong></code>, and
<code><strong>11</strong></code>.
(We won’t bother to name elements of state vectors larger than two.
It would get unwieldy rather quickly.)
Which of those four possible outcomes
might the following state vector represent?
</p>
<br>
<div class="center">
<div class="matrix">
<div class="matrix-bracket-left"></div>
<div class="matrix-bracket-right"></div>
<table>
<tr><td>0</td></tr>
<tr><td>0</td></tr>
<tr><td>1</td></tr>
<tr><td>0</td></tr>
</table>
</div>
<div class="center middle">
A state vector for<br>
<strong>two</strong> qubits.
</div>
</div>
<br>
<p>
The above vector represents four possible outcomes
and we see that
three out of those four possible outcomes
are <code>FALSE</code> (<code>0</code>).
Meanwhile, the third of those four possible outcomes
is <code>TRUE</code> (<code>1</code>).
Because the result value of the third possible outcome is
<code><strong>10</strong></code>
we see that this two qubit vector state
is telling us it represents a result of
<code><strong>10</strong></code>.
Let’s break this down the same way we did
with
<a href="#.HORIZONTAL">Horizontal</a>
and
<a href="#.VERTICAL">Vertical</a>
qubits.
</p>
<br>
<div class="center">
<div class="matrix qubit">
<div class="matrix-bracket-left"></div>
<div class="matrix-bracket-right"></div>
<table class="chart">
<tr>
<td class="font-sans">
Outcome will be <code><strong>00</strong></code>:
<code>FALSE</code> (<code>0</code>)
</td>
</tr>
<tr>
<td class="font-sans">
Outcome will be <code><strong>01</strong></code>:
<code>FALSE</code> (<code>0</code>)
</td>
</tr>
<tr>
<td class="font-sans">
Outcome will be <code><strong>10</strong></code>:
<code>TRUE</code> (<code>1</code>)
</td>
</tr>
<tr>
<td class="font-sans">
Outcome will be <code><strong>11</strong></code>:
<code>FALSE</code> (<code>0</code>)
</td>
</tr>
</table>
</div>
<div class="center middle">
A state vector for<br>
<strong>two</strong> qubits.
</div>
</div>
<br>
<p>
We’ve learned that
a single qubit is the simplest example of
a quantum state vector.
It is a list of the votes per each possible
outcome—and for a single qubit there are only two possible outcomes.
We’ve also seen that we can represent the state of a multi-qubit system
where there are more than two possible outcomes.
</p>
<h4 id="Braket_notation">Ket notation</h4>
<p>
<a href="https://en.wikipedia.org/wiki/Paul_Dirac" target="_blank">Paul Dirac</a>’s
<a href="https://en.wikipedia.org/wiki/Bra%E2%80%93ket_notation" target="_blank">“bra-ket” notation</a>
offers us a more compact
means of describing <a href="#State_vectors">quantum state vectors</a>—and by extension, qubits.
(While “bra-ket” offers us two named elements—“bra” and “ket”—for
our purposes we need only focus on the latter.)
Kets represent the result value
that our quantum vector state represents.
They are expressed as values enclosed between
a vertical bar
and a rightward angle bracket.
The following is pronounced “ket zero.”
</p>
<br>
<div class="center">
<span class="Q-state-vector ket">0</span>
</div>
<br>
<p>
We <a href="#Named_couples">began</a>
by stating that
a <a href="#.HORIZONTAL">Horizontal qubit</a>
represents “<code>0</code>”,
and later <a href="#State_vectors">explained</a> why this was so.
Kets provide us a convenient way
to refer to this result state directly
as in-line text
rather than a clunky matrix.
</p>
<br>
<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>= <span class="Q-state-vector ket">0</span>
</div>
<br>
<p>
Similarly, we <a href="#Named_couples">defined</a> a
<a href="#.VERTICAL">Vertical qubit</a>
as representing “<code>1</code>”
and <a href="#State_vectors">illustrated this</a> as well.
We can now also express this column vector as a ket.
</p>
<br>
<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>= <span class="Q-state-vector ket">1</span>
</div>
<br>
<p>
The convenience of ket notation
becomes more apparent
as we represent state vectors
that are larger than a single qubit.
(For <span class="symbol">n</span> qubits
we must use a state vector that has
2<sup><span class="symbol">n</span></sup>
elements.
Meanwhile our ket values are still just
<span class="symbol">n</span> digits long.)
Here we express four possible states
of a two qubit system
as both state vectors and their
<a href="https://en.wikipedia.org/wiki/Qubit#Standard_representation" target="_blank">equivalent kets</a>.
</p>
<br>
<div class="center">
<div class="no-wrap">
<div class="matrix">
<div class="matrix-bracket-left"></div>
<div class="matrix-bracket-right"></div>
<table>
<tr><td>1</td></tr>
<tr><td>0</td></tr>
<tr><td>0</td></tr>
<tr><td>0</td></tr>
</table>
</div>
= <span class="Q-state-vector ket">00</span>,
</div>
<div class="no-wrap">
<div class="matrix">
<div class="matrix-bracket-left"></div>
<div class="matrix-bracket-right"></div>
<table>
<tr><td>0</td></tr>
<tr><td>1</td></tr>
<tr><td>0</td></tr>
<tr><td>0</td></tr>
</table>
</div>
= <span class="Q-state-vector ket">01</span>,
</div>
<div class="no-wrap">
<div class="matrix">
<div class="matrix-bracket-left"></div>
<div class="matrix-bracket-right"></div>
<table>
<tr><td>0</td></tr>
<tr><td>0</td></tr>
<tr><td>1</td></tr>
<tr><td>0</td></tr>
</table>
</div>
= <span class="Q-state-vector ket">10</span>,
</div>
<div class="no-wrap">
<div class="matrix">
<div class="matrix-bracket-left"></div>
<div class="matrix-bracket-right"></div>
<table>
<tr><td>0</td></tr>
<tr><td>0</td></tr>
<tr><td>0</td></tr>
<tr><td>1</td></tr>
</table>
</div>
= <span class="Q-state-vector ket">11</span>
</div>
</div>
<br>
<h4>Superposition</h4>
<p>
You’ve probably heard the term
“<a href="https://en.wikipedia.org/wiki/Quantum_superposition" target="_blank">superposition</a>”—and along with that
you’ve likely been spoonfed some measure of mysticism;
<a href="https://youtu.be/CMdHDHEuOUE" target="_blank">pizza-bagels</a> and whatnot.
In the real, physical world, superposition is indeed weird magic.
But mathematically it’s dead simple:
Superposition is any qubit state
where the
<code><a href="#this.alpha">alpha</a></code>
and
<code><a href="#this.beta">beta</a></code>
values are anything other than
exactly 0 or exactly 1.
Up until now we’ve thought of
<code><a href="#this.alpha">alpha</a></code>
and
<code><a href="#this.beta">beta</a></code>
values as being either
<code>TRUE</code> (1) or <code>FALSE</code> (0)
but each is actually capable of expressing an entire spectrum between
<code>TRUE</code> (1) and <code>FALSE</code> (0).
Let’s investigate that idea by building on what we’ve already learned.
</p>
<p>
Given the constraint
<code><a href="#this.alpha">alpha</a></code><sup>2</sup> +
<code><a href="#this.beta">beta</a></code><sup>2</sup> = 1, if we plot all of the possible
<code><a href="#this.alpha">alpha</a></code>
and
<code><a href="#this.beta">beta</a></code>
values on a graph
as <span class="symbol">x</span> and <span class="symbol">y</span> respectively,
the outcome is a circle with a radius of 1 centered at the origin (0, 0);
ie. a <a href="https://en.wikipedia.org/wiki/Unit_circle)" target="_blank">unit circle</a>.
All possible combinations of
<code><a href="#this.alpha">alpha</a></code>
and
<code><a href="#this.beta">beta</a></code>
lay on the perimeter of this circle.
To illustrate this, here’s a plot of named
<a href="#Jones_vectors">Jones vectors</a>
as well as their conjugates.
</p>
<div class="unit-circle" id="Unit_circle">
<div class="unit-circle-outline"></div>
<div class="unit-circle-spoke unit-circle-spoke-0"></div>
<div class="unit-circle-spoke unit-circle-spoke-45"></div>
<div class="unit-circle-spoke unit-circle-spoke-90"></div>
<div class="unit-circle-spoke unit-circle-spoke-135"></div>
<div class="unit-circle-spoke-label unit-circle-spoke-label-90">Vertical</div>
<div class="unit-circle-spoke-label unit-circle-spoke-label-45">Diagonal</div>
<div class="unit-circle-spoke-label unit-circle-spoke-label-0">Horizontal</div>
<div class="unit-circle-spoke-label unit-circle-spoke-label-315">Anti-diagonal</div>
<div class="unit-circle-label unit-circle-label-angle-0">0˚</div>
<div class="unit-circle-label unit-circle-label-angle-45">45˚</div>
<div class="unit-circle-label unit-circle-label-angle-90">90˚</div>
<div class="unit-circle-label unit-circle-label-angle-135">135˚</div>
<div class="unit-circle-label unit-circle-label-angle-180">180˚</div>
<div class="unit-circle-label unit-circle-label-angle-225">225˚</div>
<div class="unit-circle-label unit-circle-label-angle-270">270˚</div>
<div class="unit-circle-label unit-circle-label-angle-315">315˚</div>
<div class="unit-circle-label unit-circle-label-point-0">1, 0</div>
<div class="unit-circle-label unit-circle-label-point-45">
<table class="division">
<tr class="dividend"><td>1</td></tr>
<tr class="divisor"><td>√ 2</td></tr>
</table>,
<table class="division">
<tr class="dividend"><td>1</td></tr>
<tr class="divisor"><td>√ 2</td></tr>
</table>
</div>
<div class="unit-circle-label unit-circle-label-point-90">0, 1</div>
<div class="unit-circle-label unit-circle-label-point-135">
<table class="division">
<tr class="dividend"><td>-1</td></tr>
<tr class="divisor"><td>√ 2</td></tr>
</table>,
<table class="division">
<tr class="dividend"><td>1</td></tr>
<tr class="divisor"><td>√ 2</td></tr>
</table>
</div>
<div class="unit-circle-label unit-circle-label-point-180">-1, 0</div>
<div class="unit-circle-label unit-circle-label-point-225">
<table class="division">
<tr class="dividend"><td>-1</td></tr>
<tr class="divisor"><td>√ 2</td></tr>
</table>,
<table class="division">
<tr class="dividend"><td>-1</td></tr>
<tr class="divisor"><td>√ 2</td></tr>
</table>
</div>
<div class="unit-circle-label unit-circle-label-point-270">0, -1</div>
<div class="unit-circle-label unit-circle-label-point-315">
<table class="division">
<tr class="dividend"><td>1</td></tr>
<tr class="divisor"><td>√ 2</td></tr>
</table>,
<table class="division">
<tr class="dividend"><td>-1</td></tr>
<tr class="divisor"><td>√ 2</td></tr>
</table>
</div>
</div>
<p>
What the
<code><a href="#this.alpha">alpha</a></code>
and
<code><a href="#this.beta">beta</a></code>
values represent are the individual
<a href="https://en.wikipedia.org/wiki/Probability_amplitude" target="_blank">probability amplitudes</a>
for each outcome; that a qubit
when measured will be in either a
<span class="Q-state-vector ket">0</span>
or a
<span class="Q-state-vector ket">1</span>
state.
Measurement itself causes a qubit’s
<a href="https://en.wikipedia.org/wiki/Wave_function_collapse" target="_blank">probability wave to collapse</a>,
bringing an end to its superposition.
The probability that upon measurement a qubit’s
<a href="https://en.wikipedia.org/wiki/Probability_amplitude" target="_blank">probability amplitude</a>
will
<a href="#.prototype.collapse">collapse</a> to
<span class="Q-state-vector ket">0</span>
is <code><a href="#this.alpha">alpha</a></code><sup>2</sup>,
while the probability that it will
<a href="#.prototype.collapse">collapse</a> to
<span class="Q-state-vector ket">1</span>
is <code><a href="#this.beta">beta</a></code><sup>2</sup>.
</p>
<p>
We already know that a
<a href="#.HORIZONTAL">Horizontal qubit</a>
exists in a state of
<span class="Q-state-vector ket">0</span>
(“ket zero”)
and therefore has a 100% chance of being measured as
<span class="Q-state-vector ket">0</span>.
</p>
<br>
<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>
</div>
<br>
<p>
Similarly, we also know that a
<a href="#.VERTICAL">Vertical qubit</a>
exists in a state of
<span class="Q-state-vector ket">1</span>
(“ket one”)
and therefore has a 100% chance of being measured as
<span class="Q-state-vector ket">1</span>.
</p>
<br>
<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>
</div>
<br>
<p>
Meanwhile, a
<a href="#.DIAGONAL">Diagonal qubit</a>
exists in a state of superposition as
<span class="Q-state-vector ket">+</span>
(“ket plus”).
It is a state which does not have a definite result value prior to measurement
but it does of course have a definite state vector
and that state vector has a positive orientation.
(Recall our <a href="#Unit_circle">unit circle</a> diagram above
to see how this value lays in a positive quadrant of the graph.)
There is a 50% chance of it being measured as
<span class="Q-state-vector ket">0</span>
(“ket zero”)
and a 50% chance of it being measured as
<span class="Q-state-vector ket">1</span>
(“ket one”).
</p>
<br>
<div class="center maths">
<div class="matrix qubit">
<div class="matrix-bracket-left"></div>
<div class="matrix-bracket-right"></div>
<table>
<tr><td>1</td></tr>
<tr><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 class="matrix qubit">
<div class="matrix-bracket-left"></div>
<div class="matrix-bracket-right"></div>
<table>
<tr><td><strong>1 ÷ √ 2</strong></td></tr>
<tr><td><strong>1 ÷ √ 2</strong></td></tr>
</table>
</div>
<br>
<br>
<table class="equation">
<tr>
<td>
<code><a href="#this.alpha">alpha</a></code><sup>2</sup> + <code><a href="#this.beta">beta</a></code><sup>2</sup>
</td>
<td>1</td>
</tr>
<tr>
<td>