The Shifting Paradigm of Quantum Computing

Presented at the Nov 2005Annual Dallas Mensa Gathering

By Douglas Matzke, Ph.D.

doug@QuantumDoug.comhttp://www.QuantumDoug.com

DJM Nov 25, 2005

AbstractQuantum 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

OutlineClassical Bits

Distinguishability, Mutual Exclusion, co-occurrence, co-exclusionReversibility and unitary operators

Quantum Bits – QubitsOrthogonal Phase StatesSuperpositionMeasurement and singular operatorsNoise – Pauli Spin Matrices

Quantum RegistersEntanglement and coherence

Entangled Bits – EbitsBell and Magic StatesBell Operator

Quantum AlgorithmsShor’s algorithmGrover’s Algorithm

Quantum CommunicationQuantum Cryptography

Summary

DJM Nov 25, 2005

Classical Bits

Alternate vector notations for multiple coins!!!

DJM Nov 25, 2005

Classical InformationDistinguishability

Definition: Individual items are identifiableCoins, photons, electrons etc are not distinguishableGroups of objects described using statistics

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

Coin lands on heads or tails but not bothFaces 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!Related to simultaneity and synchronization!

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

Phases & Superposition

-

+

90°

State1

State0

(C0 State0 + C1 State1)

0 112

c c= =θ

21 i ic p= =∑ ∑Unitarity Constraint is

C0

C1

10 0

0state

⎡ ⎤= = ⎢ ⎥

⎣ ⎦0

1 11

state ⎡ ⎤= = ⎢ ⎥

⎣ ⎦

For θ = 45°

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Classical vs. Quantum

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Hilbert Space Notation

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Unitary Qubit OperatorsCircuit

Rotate

Hadamard -Superposition

ExistsMatrixSymbolicGate

Shift (Pauli-Z)

Not (Pauli-X)

Identity

cos sinsin cos

θ θθ θ

−⎡ ⎤⎢ ⎥⎣ ⎦

1

0 11 0

σ⎡ ⎤

= ⎢ ⎥⎣ ⎦

1 111 12

H ⎡ ⎤= ⎢ ⎥−⎣ ⎦

0 *σ ψ

*H ψ

*θ ψ

ψ

ψ

ψ

ψ

3

1 00 1

σ⎡ ⎤

= ⎢ ⎥−⎣ ⎦ψ

H

θ

X

Z

0 1⎡ ⎤⎣ ⎦

1 *σ ψ

3 *σ ψ

0

1 00 1

σ⎡ ⎤

= ⎢ ⎥⎣ ⎦

01 σ

11 σ

31 σ

1 θ

1 H

DJM Nov 25, 2005

Matrices 101a bc d

σ⎡ ⎤

= ⎢ ⎥⎣ ⎦

αβ⎡ ⎤

Ψ = ⎢ ⎥⎣ ⎦

*a b a bc d c d

α α βσ

β α β+⎡ ⎤ ⎡ ⎤ ⎡ ⎤

Ψ = =⎢ ⎥ ⎢ ⎥ ⎢ ⎥+⎣ ⎦ ⎣ ⎦ ⎣ ⎦

1

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

1 0 0 1*1 0*0 1σ

+⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤= = =⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥+⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦

00 0 0

0

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

1 1 0 1*1 1*0 1c

H c c cc

+ ⎡ ⎤⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤= = = = ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥− + −⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦

0 1 20.707

c =

=

DJM Nov 25, 2005

and Trine States

( )( )

3 3120

0 1 1

0 1 0 1 2

0 1 0

10 0 1

0 1 0 1

0 1 0

TrTr c c T

Tr c c c c T

Tr c c T

θ= =

→ + =

+ = − =

− = = 0 1 2c =

1 3 2c =0 1 4p =

1 3 4p =

T0

T1

T2

120°

120°

120°

2 290

45

1

1

Not

Not i

Not i

θ

θ

= = −

= − =

= =

Not

90°0

1

0−

90°

DJM Nov 25, 2005

Quantum NoisePauli Spin Matrices

Both Flips

Phase Flip

Bit Flip

Identity

DescriptionMatrixSymbolicLabel

1

1

0 1

1 0

σ

σ

→

→

3 1 1σ → −

0 0 0σ →

2

2

0 1

1 0

i

i

σ

σ

→

→−

1 *σ ψ 1

0 11 0

σ⎡ ⎤

= ⎢ ⎥⎣ ⎦

3 *σ ψ 3

1 00 1

σ⎡ ⎤

= ⎢ ⎥−⎣ ⎦

0 *σ ψ 0

1 01

0 1σ

⎡ ⎤= =⎢ ⎥⎣ ⎦

2 *σ ψ 2

00i

iσ

−⎡ ⎤= ⎢ ⎥⎣ ⎦

DJM Nov 25, 2005

Quantum Measurement

0 112

c c= =

θ

C0

C1

0 10 1c c+

1

0

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

When

then 0 112

p p= =

or 50/50 random!

Destructive and Probabilistic!!

Concepts of projection and singular operators

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Quantum Measurement

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

Our library in Block Diagram tool by Hyperception

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Quantum RegistersEntanglement

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

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

Inseparable statesCan’t be directly created by tensor productConcept 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

⎡ ⎤= = ⎢ ⎥

⎣ ⎦

10

0 0000

state

⎡ ⎤⎢ ⎥⎢ ⎥= =⎢ ⎥⎢ ⎥⎣ ⎦

01

1 0100

state

⎡ ⎤⎢ ⎥⎢ ⎥= =⎢ ⎥⎢ ⎥⎣ ⎦

00

2 1010

state

⎡ ⎤⎢ ⎥⎢ ⎥= =⎢ ⎥⎢ ⎥⎣ ⎦

00

3 1101

state

⎡ ⎤⎢ ⎥⎢ ⎥= =⎢ ⎥⎢ ⎥⎣ ⎦

⊗ = q0 ⊗ q1

q0

q1

Special kind of linear transformations

DJM Nov 25, 2005

Unitary QuReg Operators

cnot2

CircuitMatrixSymbolicGate

swap =cnot*cnot2*cnot

cnot = XOR Control-not

1 0 0 00 1 0 00 0 0 10 0 1 0

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

*swap ψ

*cnot ψ

2*cnot ψ

ψ

ψ

Φ1 0 0 00 0 0 10 0 1 00 1 0 0

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

ψΦ

Φ

1 0 0 00 0 1 00 1 0 00 0 0 1

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

00 01 10 11⎡ ⎤⎣ ⎦

x

x=

DJM Nov 25, 2005

Quantum Register Modeling

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

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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

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Reversible Quantum Circuits

Fredkin=control-swap

CircuitMatrixSymbolicGate

Deutsch

Toffoli = control-control-not

11

11

11

0 11 0

0

0

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

*D ψ

*T ψ

*F ψ

1ψ

000 001 010 011 100 101 110 111⎡ ⎤⎣ ⎦

11

11

10 11 0

1

0

0

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

11

11

11

cos sinsin cos

0

0 ii

θ θθ

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

2ψ3ψ

1ψ2ψ3ψ x

x

2ψ3ψ

1ψ

D

DJM Nov 25, 2005

Toffoli and Fredkin Gates

DJM Nov 25, 2005

Ebits – Entangled BitsEPR (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

+ −

+ −

=Φ = + =Φ = −

=Ψ = + =Ψ = −

( ) ( )( ) ( )

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

1=B

0 0

0 0

0 0

0 0

0 00 0

0 00 0

c cc c

c cc c

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥−⎢ ⎥−⎢ ⎥⎣ ⎦

=

DJM Nov 25, 2005

Step1: Two qubits

Step2: Entangle Ebit

Step3: Separate

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

EPR: Non-local connection

≈0 10 , 0

00 11

01 10

±

±

Φ = ±

Ψ = ±

? , ?

1, 11, 0

answ er otheransw er other

= == =

±Φ±Ψ

Linked coins analogy

~ ~

~ ~

DJM Nov 25, 2005

Quantum AlgorithmsSpeedup over classical algorithms

Complexity Class: Quantum Polynomial TimeReversible logic gates just mimics classical logic

Requires quantum computer with q>100 qubitsLargest quantum computer to date has 7 qubitsProblems with decoherence and scalability

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

DJM Nov 25, 2005

Quantum CommunicationQuantum Encryption

Uses fact that measuring qubit destroys stateCan be setup to detect intrusion

Quantum Key DistributionUses quantum encryption to distribute fresh keysCan be setup to detect intrusion

Fastest growing quantum product areaMany companies and productsIn enclosed fiber networks and also open air

DJM Nov 25, 2005

Quantum Mind?Did biology to tap into Quantum Computing?

Survival value using fast searchWe might be extinct if not for quantum mind

Research with random phase ensemblesEnsemble states survive random measurementsSee paper “Math over Mind and Matter”

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

DJM Nov 25, 2005

Summary and ConclusionsQuantum concepts extend classical ways of thinking

High dimensional spaces and simultaneityDistinguishability, mutual exclusion, co-occurrence and co-exclusionReversible computing and unitary transformsQubits superposition, phase states, probabilities & unitarity constraintMeasurement and singular operatorsEntanglement, coherence and noiseEbits, EPR, Non-locality and Bell/Magic StatesQuantum speedup for algorithmsQuantum ensembles have most properties of qubits

Quantum systems are ubiquitous Quantum computing may also be ubiquitousBiology may have tapped into quantum ensemble computingQuantum computing and consciousness may be related