# Quantum Computing

### Hacking Nature's Computer

Java Champion, JavaOne Rockstar, plays well with others, etc :-)

Author of several Java/JavaFX/RaspPi books

Developer Advocate & International Speaker for Pivotal

Mission: "Transform how the world builds software"

Mission: "Transform how the world builds software"

### You are cordially invited to ...

Introduction to quantum computing

Axioms of quantum mechanics (with cats)

• Superposition principle
• Measurement
• Unitary evolution

Quantum mechanical demo (with photons)

Abstracting cats and photons with qubits

Quantum computing algorithms

Quantum entanglement

More algorithms

Supplementary resources

## Deutsch's algorithm

### Determine if function is constant or balanced

Input  a Constant  f(a) Constant  f(a) Balanced  f(a) Balanced  f(a)
0 0 1 0 1
1 0 1 1 0

## Deutsch's algorithm

### How many queries of the oracle to solve?

Classical:

This oracle requires 2 queries classically

Quantum:

We create a superposition of inputs to the oracle for constructive/destructive interference.

1
$1$
2
$2$

## Querying the oracle classically


\begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \end{bmatrix} \cdot \begin{bmatrix} 0 \\ 1 \\ 0 \\ 0 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \\ 1 \end{bmatrix}
$\begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \end{bmatrix} \cdot \begin{bmatrix} 0 \\ 1 \\ 0 \\ 0 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \\ 1 \end{bmatrix}$
\vert01\rangle
$\vert01\rangle$
\vert11\rangle
$\vert11\rangle$

example: f (0) = 0 and f (1) = 1   balanced

Text

Text

## Querying the oracle quantumly


\begin{bmatrix} +\frac{1}{2} & +\frac{1}{2} & +\frac{1}{2} & +\frac{1}{2} \\ +\frac{1}{2} & -\frac{1}{2} & +\frac{1}{2} & -\frac{1}{2} \\ +\frac{1}{2} & +\frac{1}{2} & -\frac{1}{2} & -\frac{1}{2} \\ +\frac{1}{2} & -\frac{1}{2} & -\frac{1}{2} & +\frac{1}{2} \end{bmatrix} \cdot \begin{bmatrix} 0 \\ 0 \\ 1 \\ 0 \end{bmatrix} = \begin{bmatrix} +\frac{1}{2} \\ +\frac{1}{2} \\ -\frac{1}{2} \\ -\frac{1}{2} \end{bmatrix}
$\begin{bmatrix} +\frac{1}{2} & +\frac{1}{2} & +\frac{1}{2} & +\frac{1}{2} \\ +\frac{1}{2} & -\frac{1}{2} & +\frac{1}{2} & -\frac{1}{2} \\ +\frac{1}{2} & +\frac{1}{2} & -\frac{1}{2} & -\frac{1}{2} \\ +\frac{1}{2} & -\frac{1}{2} & -\frac{1}{2} & +\frac{1}{2} \end{bmatrix} \cdot \begin{bmatrix} 0 \\ 0 \\ 1 \\ 0 \end{bmatrix} = \begin{bmatrix} +\frac{1}{2} \\ +\frac{1}{2} \\ -\frac{1}{2} \\ -\frac{1}{2} \end{bmatrix}$

example: f (0) = 0 and f (1) = 1   balanced

\begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \end{bmatrix} \cdot \begin{bmatrix} +\frac{1}{2} \\ +\frac{1}{2} \\ -\frac{1}{2} \\ -\frac{1}{2} \end{bmatrix} = \begin{bmatrix} +\frac{1}{2} \\ -\frac{1}{2} \\ -\frac{1}{2} \\ +\frac{1}{2} \end{bmatrix}
$\begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \end{bmatrix} \cdot \begin{bmatrix} +\frac{1}{2} \\ +\frac{1}{2} \\ -\frac{1}{2} \\ -\frac{1}{2} \end{bmatrix} = \begin{bmatrix} +\frac{1}{2} \\ -\frac{1}{2} \\ -\frac{1}{2} \\ +\frac{1}{2} \end{bmatrix}$
\begin{bmatrix} +\frac{1}{2} & +\frac{1}{2} & +\frac{1}{2} & +\frac{1}{2} \\ +\frac{1}{2} & -\frac{1}{2} & +\frac{1}{2} & -\frac{1}{2} \\ +\frac{1}{2} & +\frac{1}{2} & -\frac{1}{2} & -\frac{1}{2} \\ +\frac{1}{2} & -\frac{1}{2} & -\frac{1}{2} & +\frac{1}{2} \end{bmatrix} \cdot \begin{bmatrix} +\frac{1}{2} \\ -\frac{1}{2} \\ -\frac{1}{2} \\ +\frac{1}{2} \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \\ 1 \end{bmatrix}
$\begin{bmatrix} +\frac{1}{2} & +\frac{1}{2} & +\frac{1}{2} & +\frac{1}{2} \\ +\frac{1}{2} & -\frac{1}{2} & +\frac{1}{2} & -\frac{1}{2} \\ +\frac{1}{2} & +\frac{1}{2} & -\frac{1}{2} & -\frac{1}{2} \\ +\frac{1}{2} & -\frac{1}{2} & -\frac{1}{2} & +\frac{1}{2} \end{bmatrix} \cdot \begin{bmatrix} +\frac{1}{2} \\ -\frac{1}{2} \\ -\frac{1}{2} \\ +\frac{1}{2} \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \\ 1 \end{bmatrix}$
= \begin{bmatrix} \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} \end{bmatrix}
$= \begin{bmatrix} \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} \end{bmatrix}$

Inp Con
0 0
1 0
Inp Con
0 1
1 1
Inp Bal
0 0
1 1
Inp Bal
0 1
1 0

## Deutsch's algorithm

### with oracle having constant function

Leverages phase-kickback from the bottom wire to choreograph constructive and destructive interference

Expected result is 100% probability of measuring

\vert0\rangle
$\vert0\rangle$

## Deutsch's algorithm

### with oracle having balanced function

Expected result is 0% probability of measuring

\vert0\rangle
$\vert0\rangle$

Leverages phase-kickback from the bottom wire to choreograph constructive and destructive interference

## Deutsch-Jozsa algorithm, 1992

### Determine if function is constant or balanced

Input Constant Constant Balanced Balanced
000 0 1 0 1
001 0 1 1 0
010 0 1 0 1
011 0 1 1 0
100 0 1 0 1
101 0 1 1 0
110 0 1 0 1
111 0 1 1 0

Results when querying our example oracle

## Deutsch-Jozsa algorithm

### How many invocations of the oracle to solve?

Classical:

Our oracle (black box) requires 5 invocations classically

Quantum:

We create a superposition of inputs to the oracle, and use the phase-kickback trick, for constructive/destructive interference.  See:

Lecture 5: A simple searching algorithm; the Deutsch-Jozsa algorithm by John Watrous, University of Calgary

2^{(n-1)}+1
$2^{(n-1)}+1$
1
$1$

(Exponentially faster!)

## Deutsch-Jozsa implemented in Quil

Introduction to quantum computing

Axioms of quantum mechanics (with cats)

• Superposition principle
• Measurement
• Unitary evolution

Quantum mechanical demo (with photons)

Abstracting cats and photons with qubits

Quantum computing algorithms

Quantum entanglement

More algorithms

Supplementary resources

## Bell inequality test

### (CHSH game)

Introduction to quantum computing

Axioms of quantum mechanics (with cats)

• Superposition principle
• Measurement
• Unitary evolution

Quantum mechanical demo (with photons)

Abstracting cats and photons with qubits

Quantum computing algorithms

Quantum entanglement

More algorithms

Supplementary resources

## Grover's search

### finding a needle in a haystack

Introduction to quantum computing

Axioms of quantum mechanics (with cats)

• Superposition principle
• Measurement
• Unitary evolution

Quantum mechanical demo (with photons)

Abstracting cats and photons with qubits

Quantum computing algorithms

Quantum entanglement

More algorithms

Supplementary resources

## Classic textbook on Quantum Computing

### by "Mike & Ike"

By Source (WP:NFCC#4), Fair use,

# Quantum Computing

{\frac{1}{\sqrt2}}
${\frac{1}{\sqrt2}}$
\vert{Q}\rangle
$\vert{Q}\rangle$
+
$+$
\vert{A}\rangle
$\vert{A}\rangle$
{\frac{1}{\sqrt2}}
${\frac{1}{\sqrt2}}$

By javafxpert

# Quantum Computing Exposed: Hacking Nature's Computer

Part two of a gentle introduction to quantum computing

• 1,967