The Shifting Paradigm of Quantum Computing
Presented at the Nov 2005Annual Dallas Mensa Gathering
By Douglas Matzke, Ph.D.
[email protected]://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°
DJM Nov 25, 2005
Classical vs. Quantum
DJM Nov 25, 2005
Hilbert Space Notation
DJM Nov 25, 2005
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
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 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
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 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
Top Related