Quantum Computing
Understanding the Deutsch-Jozsa algorithm
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 ...
Why use a quantum computer?
Feasible on classical computers
Feasible on quantum computers
Solutions to problems
some problems may be solved exponentially faster
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
generalizes Deutsch's algorithm to n qubits
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
Deutsch-Jozsa in web-based circuit simulator
from quantum computing community member @rasasankar
by Ravisankar from Thrissur, India
Computer Science student
TKM College of Engineering
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