Quantum Computing
Hacking Nature's Computer
About Presenter James Weaver
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 ...
Concepts we'll address today
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 (1985)
Literally the Hello World of quantum algorithms
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
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}
\vert01\rangle
\vert11\rangle
example: f (0) = 0 and f (1) = 1 balanced
Quantum parallelism
what is it, really?
Double-slit experiment
constructive and destructive interference
Text
Choreographing interference
to increase the chance of getting the right answer
Text
Excerpts from “THE TALK” by Scott Aaronson and Zach Weinersmith
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}
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}
+\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}
Lecture 3: One Qubit, Two Qubit by Dave Bacon, University of Washington (Deutsch slightly modified)
=
\begin{bmatrix}
\frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\
\frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}}
\end{bmatrix}
Deutsch (slightly modified)
Why constant vs. balance require only one query
| 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
Deutsch's algorithm
with oracle having balanced function
Expected result is 0% probability of measuring
\vert0\rangle
Leverages phase-kickback from the bottom wire to choreograph constructive and destructive interference
Deutsch-Jozsa algorithm
exponentially faster than classical
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:
see also: Wikipedia Deutsch-Jozsa Decoherence section
2^{(n-1)}+1
1
(Exponentially faster!)
Deutsch-Jozsa algorithm
with oracle having constant function
Deutsch-Jozsa algorithm
example oracle with constant function
Deutsch-Jozsa algorithm
with oracle having balanced function
Deutsch-Jozsa algorithm
example oracle with balanced function
Deutsch-Jozsa implemented in Quil
Deutsch-Jozsa implemented in Quil
Concepts we'll address today
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
Quantum entanglement
Alice and Bob's long running relationship
Quantum entanglement
basic circuit
Quantum teleportation
and the no cloning theorem
Superdense coding
example circuit
Bell inequality test
(CHSH game)
Concepts we'll address today
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
Bernstein-Vazirani algorithm
basic circuit
Simon's algorithm
the quantum portion
Shor's algorithm
Period finding
using Quantum Fourier Transform
Grover's search
finding a needle in a haystack
Concepts we'll address today
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
Complex numbers
aren't complicated
Complex numbers encode
amplitude and phase
Roots of unity
are useful for Shor
Matrices
think two-dimensional arrays
Vector spaces
a home for vectors and scalars
Classic textbook on Quantum Computing
by "Mike & Ike"
By Source (WP:NFCC#4), Fair use,
Quantum Computing
{\frac{1}{\sqrt2}}
\vert{Q}\rangle
+
\vert{A}\rangle
{\frac{1}{\sqrt2}}
Quantum Computing Exposed: Hacking Nature's Computer
By javafxpert
Quantum Computing Exposed: Hacking Nature's Computer
Part two of a gentle introduction to quantum computing
