Phase in Quantum Computing

1
Phase in Quantum Computing
2
Main concepts of computing illustrated with
simple examples
3
Quantum Theory Made Easy
Classical
Quantum
probabilities
amplitudes
p0
a0
0
0
p1
a1
1
1
pi is a real number
ai is a complex number
p0p11
a02a12 1
Prob(i)pi
Prob(i)ai2
qubit
bit
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Quantum Theory Made Easy
Classical Evolution
Quantum Evolution
0
0
0
0
0
0
0
0
0
1
0
1
0
1
0
1
1
0
1
0
1
0
1
0
1
1
1
1
1
1
1
1
transition probabilities
transition amplitudes
unitary matrix
stochastic matrix
5
Interference
0
0
1
1
qubit input
0
50
100
0
1
50
measure
0
50
0
1
50
100
measure
6
Interfering Pathways
1.0 H
100 H
0.707
0.707
50
50
0.707 H
0.707 C
50 H
0.707
50 C
90
0.707
10
0.707
80
-0.707
20
1.0 C
0.0 H
85 C
15 H
Always addition!
Subtraction!
Classical
Quantum
7
Superposition
Qubits
amplitudes
a0
0
a1
1
ai is a complex number
Schrödingers Cat
( ? ?)
1
v2
8

• Classical versus quantum computers

9
Some differences between classical and quantum
computers
superposition
Hidden properties of oracles
10
Randomised Classical Computation versus Quantum
Computation
Deterministic Turing machine
Probabilistic Turing machine
11
Probabilities of reaching states
12
Formulas for reaching states
13
Relative phase, destructive and constructive
inferences
Destructive interference
Constructive interference
14
Most quantum algorithms can be viewed as big
interferometry experiments
Equivalent circuits
15
The eigenvalue kick-back concept
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There are also some other ways to introduce a
relative phase
17
The eigenvalue kick-back concept
Now we know that the eigenvalue is the same as
relative phase
18
The eigenvalue kick-back concept illustrated
for DEUTSCH
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The shift operation as a generalization to
Deutschs Tricks
20
Change of controlled gate in Deutsch with
Controlled-Ushift gate
21
Now we deal with new types of eigenvalues and
eigenvectors
22
The general concept of the answer encoded in phase
23
Shift operator allows to solve Deutschs problem
with certainty
24
Controlling amplitude versus controlling phase
25
Controlling amplitude versus controlling phase
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Exercise for students
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Exercise for students
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Dave BaconLawrence Ioannou

• Sources used