James L. Weaver
Developer Advocate


 

jweaver@pivotal.io
JavaFXpert.com

CulturedEar.com

@JavaFXpert

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

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Mission: "Transform how the world builds software"

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Mission: "Transform how the world builds software"

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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

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Deutsch's algorithm (1985)

Literally the Hello World of quantum algorithms

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Lecture 4: Quantum Teleportation; Deutsch’s Algorithm by John Watrous, University of Calgary

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

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Lecture 4: Quantum Teleportation; Deutsch’s Algorithm by John Watrous, University of Calgary

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.

 

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1
11
2
22

Querying the oracle classically

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\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}
[1000000100100100]⋅[0100]=[0001]\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
∣01⟩\vert01\rangle
\vert11\rangle
∣11⟩\vert11\rangle

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

Quantum parallelism

what is it, really?

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Double-slit experiment

constructive and destructive interference

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Text

Wikimedia commons Two-Slit_Experiment_Light

Choreographing interference

to increase the chance of getting the right answer

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Text

Excerpts from “THE TALK” by Scott Aaronson and Zach Weinersmith

Querying the oracle quantumly

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\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}
[+12+12+12+12+12−12+12−12+12+12−12−12+12−12−12+12]⋅[0010]=[+12+12−12−12]\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}
[1000000100100100]⋅[+12+12−12−12]=[+12−12−12+12]\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}
[+12+12+12+12+12−12+12−12+12+12−12−12+12−12−12+12]⋅[+12−12−12+12]=[0001]\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}
=[121212−12]= \begin{bmatrix} \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} \end{bmatrix}

Deutsch (slightly modified)

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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

[open]

[open]

[open]

[open]

Deutsch's algorithm

with oracle having constant function

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

open in Quirk quantum circuit simulator

Expected result is 100% probability of measuring

\vert0\rangle
∣0⟩\vert0\rangle

Deutsch's algorithm

with oracle having balanced function

open in Quirk quantum circuit simulator

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Expected result is 0% probability of measuring

\vert0\rangle
∣0⟩\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

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wikipedia.org/wiki/Deutsch–Jozsa_algorithm

Deutsch-Jozsa algorithm, 1992

Determine if function is constant or balanced

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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

see also: Wikipedia Deutsch-Jozsa Decoherence section

 

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2^{(n-1)}+1
2(n−1)+12^{(n-1)}+1
1
11

(Exponentially faster!)

Deutsch-Jozsa algorithm

with oracle having constant function

open in Quirk quantum circuit simulator

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Deutsch-Jozsa algorithm

example oracle with constant function

open in Quirk quantum circuit simulator

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Deutsch-Jozsa algorithm

with oracle having balanced function

open in Quirk quantum circuit simulator

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Deutsch-Jozsa algorithm

example oracle with balanced function

open in Quirk quantum circuit simulator

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Deutsch-Jozsa implemented in Quil

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Deutsch-Jozsa implemented in Quil

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Deutsch-Jozsa in web-based circuit simulator

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from quantum computing community member @rasasankar

by Ravisankar from Thrissur, India

Computer Science student

TKM College of Engineering

see relevant blog post

James L. Weaver
Developer Advocate


 

jweaver@pivotal.io
JavaFXpert.com

CulturedEar.com

@JavaFXpert

Quantum Computing

Understanding the Deutsch-Jozsa algorithm