The Shifting Paradigm of Quantum Computing

Presented at the Nov 2005

Annual Dallas Mensa Gathering

By Douglas Matzke, Ph.D.

doug@QuantumDoug.com

http://www.QuantumDoug.com

DJM Nov 25, 2005

Abstract

Quantum computing has shown to efficiently solve problems that classical computers are unable to solve. Quantum computers represent information using phase states in high dimensional spaces, which produces the two fundamental quantum properties of superposition and entanglement.

This talk introduces these concepts in plain-speak and discusses how this leads to a paradigm shift of thinking "outside the classical computing box".

DJM Nov 25, 2005

Biography

Doug Matzke has been researching the limits of computing for over twenty years. These interests led him to investigating the area of quantum computing and earning a Ph.D. in May 2001. In his thirty year career, he has hosted two workshops on physics and computation (PhysComp92 and (PhysComp94), he has contributed to 15 patent disclosures and over thirty papers/talks (see his papers on QuantumDoug.com). He is an enthusiastic and thought provoking speaker.

DJM Nov 25, 2005

Outline Classical Bits

Distinguishability Mutual Exclusion

Quantum Bits – Qubits Orthogonal Phase States Superposition Measurement Noise – Pauli Spin Matrices

Quantum Registers Entanglement

Entangled Bits – Ebits Bell and Magic States

Quantum Algorithms Shor’s algorithm Grover’s Algorithm

Quantum Communication Quantum Cryptography

Summary

DJM Nov 25, 2005

Classical Bits

Alternate vector notations for multiple coins!!!

DJM Nov 25, 2005

Classical Information

Distinguishability Definition: Individual items are identifiable

Coins, photons, electrons etc are not distinguishable Groups of objects described using statistics

Mutual Exclusion (mutex) Definition: Some state excludes another state

Coin lands on heads or tails but not both Faces point in opposite directions in vector notation

+ -

DJM Nov 25, 2005

Coin Demonstration: Act ISetup:

Person stands with both hands behind back

Act I part A: Person shows hand containing a coin then hides it again

Act I part B: Person again shows a coin (indistinguishable from 1st)

Act I part C: Person asks: “How many coins do I have?”

Represents one bit: either has 1 coin or has >1 coin

DJM Nov 25, 2005

Coin Demonstration (cont)

Act II: Person holds out hand showing two identical coins

Received one bit since ambiguity resolved!

Act III: Asks: “Where did the bit of information come from?”

Answer: Simultaneous presence of the 2 coins!

DJM Nov 25, 2005

Space and Time Ideas

Co-occurrence means states exist exactly simultaneously: Spatial prim. with addition operator

Co-exclusion means a change occurred due to an operator: Temporal with multiply operator

* see definitions in my dissertation but originated with Manthey

DJM Nov 25, 2005

Quantum Bits – Qubits

+

-

-

+

Classical bit states: Mutual Exclusive

Quantum bit states: Orthogonal

180°90°

Qubits states are called spin ½

State1

State0

State1

State0

DJM Nov 25, 2005

Classical vs. Quantum

DJM Nov 25, 2005

Phases & Superposition

-

+

90°

State1

State0

(C0 State0 + C1 State1)

0 1

1

2c c

= 45°

21 i ic p Unitarity Constraint is

C0

C1

10 0

0state

0

1 11

state

DJM Nov 25, 2005

Hilbert Space Notation

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Unitary Qubit Operators

Gate Symbolic Matrix Circuit Exists

Identity

Not (Pauli-X)

Shift (Pauli-Z)

Rotate

Hadamard -Superposition

cos sin

sin cos

1

0 1

1 0

1 11

1 12H

0 *

*H

*

3

1 0

0 1

H

θ

X

Z

0 1

1 *

3 *

0

1 0

0 1

01

11

31

1

1 H

DJM Nov 25, 2005

Matrices 101a b

c d

*a b a b

c d c d

1

0 1 1 0*1 1*0 0* 0

1 0 0 1*1 0*0 1

00 0 0

0

1 1 1 1*1 1*0 1* 0

1 1 0 1*1 1*0 1

cH c c c

c

0 1 2

0.707

c

DJM Nov 25, 2005

Trine States

33120

0 1 1

0 1 0 1 2

0 1 0

1

0 0 1

0 1 0 1

0 1 0

Tr

Tr c c T

Tr c c c c T

Tr c c T

0 1 2c 1 3 2c

0 1 4p 1 3 4p

T0

T1

T2

120°

120°

120°

DJM Nov 25, 2005

Quantum Noise Pauli Spin Matrices

Label Symbolic Matrix Description

Identity

Bit Flip

Phase Flip

Both Flips

1

1

0 1

1 0

3 1 1

0 0 0

2

2

0 1

1 0

i

i

1 * 1

0 1

1 0

3 * 3

1 0

0 1

0 * 0

1 01

0 1

2 * 2

0

0

i

i

DJM Nov 25, 2005

Quantum Measurement

0 1

1

2c c

C0

C1

0 10 1c c

1

0

Probability of state is pi = ci2 and p1 = 1- p0ic i

When

then 0 1

1

2p p

or 50/50 random!

Destructive and Probabilistic!!

Concepts of projection and singular operators

DJM Nov 25, 2005

Quantum Measurement

DJM Nov 25, 2005

Qubit ModelingQubit Operators: not, Hadamard, rotate & measure gates

Our library in Block Diagram tool by Hyperception

DJM Nov 25, 2005

Quantum Registers Entanglement

Tensor Product is mathematical operator Creates 2q orthogonal dimensions from q qubits: q0 q1 … Unitarity constraint for entire qureg

Separable states Can be created by tensor product Maintained by “coherence” and no noise.

Inseparable states Can’t be directly created by tensor product Concept of Ebit (pieces act as whole) EPR and Bell/Magic states (spooky action at distance) Non-locality/a-temporal quantum phenomena proven as valid

DJM Nov 25, 2005

Qureg Dimensions

0

10 0

0state

0

01 1

1state

1

10 0

0state

1

01 1

1state

1

00 00

0

0

state

0

11 01

0

0

state

0

02 10

1

0

state

0

03 11

0

1

state

= q0 q1

q0

q1

Special kind of linear transformations

DJM Nov 25, 2005

Unitary QuReg Operators

Gate Symbolic Matrix Circuit

Cnot = XOR

Control-not

Cnot2

swap=

cnot*cnot2*cnot

1 0 0 0

0 1 0 0

0 0 0 1

0 0 1 0

*swap

*cnot

2*cnot

1 0 0 0

0 0 0 1

0 0 1 0

0 1 0 0

1 0 0 0

0 0 1 0

0 1 0 0

0 0 0 1

00 01 10 11

x

x=

DJM Nov 25, 2005

Quantum Register Modeling

Qureg Operators: tensor product, CNOT, SWAP & qu-ops

DJM Nov 25, 2005

Reversible ComputingABC

abc

ABC

abc

F T

2 gates back-to-back gives unity gate: T*T = 1 and F*F = 1

3 in & 3 out

DJM Nov 25, 2005

Reversible Quantum CircuitsGate Symbolic Matrix Circuit

Toffoli = control-control-not

Fredkin=control-swap

Deutsch

1

1

1

1

1

1

0 1

1 0

0

0

*D

*T

*F

1

000 001 010 011 100 101 110 111

1

1

1

1

1

0 1

1 0

1

0

0

1

1

1

1

1

1

cos sin

sin cos

0

0 i

i

23

123 x

x

23

1

D

DJM Nov 25, 2005

Toffoli and Fredkin Gates

DJM Nov 25, 2005

Ebits – Entangled Bits EPR (Einstein, Podolsky, Rosen) operator

Bell States

Magic States

0 0 1 0

2 0 3 0

00 11 , 00 11

01 10 , 01 10

B c B c

B c B c

0 0 1 1

2 1 3 0

00 11 , 00 11

01 10 , 01 10

M c M c

M c M c

0 1 2c

1 2c i

H

1B

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

c c

c c

c c

c c

DJM Nov 25, 2005

Step1: Two qubits

Step2: Entangle Ebit

Step3: Separate

Step4: Measure a qubit Other is same if Other is opposite if

EPR – Action at a distance

0 10 , 0

00 11

01 10

? , ?

1, 1

1, 0

answer other

answer other

Linked coins analogy

DJM Nov 25, 2005

Quantum Algorithms Speedup over classical algorithms

Complexity Class: Quantum Polynomial Time Reversible logic gates just mimics classical logic

Requires quantum computer with q>100 qubits Largest quantum computer to date has 7 qubits Problems with decoherence and scalability

Known Quantum Algorithms Shor’s Algorithm – prime factors using QFT Grover’s Algorithm – Search that scales as sqrt(N) No other algorithms found to date after much research

DJM Nov 25, 2005

Quantum Communication

Quantum Encryption Uses fact that measuring qubit destroys state Can be setup to detect intrusion

Quantum Key Distribution Uses quantum encryption to distribute fresh keys Can be setup to detect intrusion

Fastest growing quantum product area Many companies and products In enclosed fiber networks and also open air

DJM Nov 25, 2005

Quantum Mind?

Research with random phase ensembles Ensemble states survive random measurements See paper “Math over Mind and Matter”

Did biology to tap into Quantum Computing? Survival value using fast search We might be extinct if not for quantum mind

Relationship to quantum and consciousness? Movie: “What the Bleep do we know anyhow? Conferences and books

DJM Nov 25, 2005

Summary and Conclusions Quantum concepts extend classical ways of thinking

High dimensional spaces and simultaneity Distinguishability, mutual exclusion, co-occurrence and co-exclusion Reversible computing and unitary transforms Qubits superposition, phase states, probabilities & unitarity constraint Measurement and singular operators Entanglement, coherence and noise Ebits, EPR, Non-locality and Bell/Magic States Quantum speedup for algorithms Quantum ensembles have most properties of qubits

Quantum systems are ubiquitous Quantum computing may also be ubiquitous Biology may have tapped into quantum computing Quantum computing and consciousness may be related