Phase in Quantum Computing. Main concepts of computing illustrated with simple examples.

Phase in Phase in Quantum Quantum

ComputingComputing

Main concepts of Main concepts of computing computing

illustrated with illustrated with simple examplessimple examples

Quantum Theory Made Easy

0

1

Classical

p0

p1

probabilities

Quantum

a0

a1

0

1

amplitudes

p0+p1=1 |a0|2+|a1|2 =1

bit qubit

pi is a real number ai is a complex number

Prob(i)=pi Prob(i)=|ai|2

Quantum Theory Made EasyClassical Evolution

0 0

0 1

1 0

1 1

0 0

0 1

1 0

1 1

Quantum Evolution

stochastic matrix

0 0

0 1

1 0

1 1

0 0

0 1

1 0

1 1

transition probabilities transition amplitudes

unitary matrix

Interference

0

1

0

1

measure

0 50%

1 50%

measure

0 50%

1 50%

100%

0%

100%

0%

qubitinput

Interfering Pathways100% H

50% 50%

50% C50% H

10%

90%

20%

80%

15% H 85% C

1.0 H

0.707 0.707

0.707 C0.707 H

0.707

0.707

-0.707

0.707

0.0 H 1.0 C

Always addition! Subtraction!

Classical Quantum

SuperpositionQubits

a0

a1

0

1

amplitudes

ai is a complex number

1

√2 (| + | )Schrödinger’s Cat

Classical versus Classical versus quantum quantum

computerscomputers

Some differences between classical and quantum computers

superposition

Hidden properties of oracles

Randomised Classical Computation Randomised Classical Computation versus Quantum Computationversus Quantum Computation

Deterministic Turing machine

Probabilistic Turing machine

Probabilities of reaching states

Formulas for reaching statesFormulas for reaching states

Relative phase, destructive and Relative phase, destructive and constructive inferencesconstructive inferences

Destructive interference

Constructive interference

Most quantum algorithms can be viewed Most quantum algorithms can be viewed as big interferometry experimentsas big interferometry experiments

Equivalent circuits

The “eigenvalue The “eigenvalue kick-back” kick-back”

conceptconcept

There are also some other ways to There are also some other ways to introduce a relative phaseintroduce a relative phase

The “eigenvalue kick-back” conceptThe “eigenvalue kick-back” concept

Now we know that the eigenvalue is the same as relative phase

The “eigenvalue kick-The “eigenvalue kick-back” concept back” concept

illustrated for illustrated for DEUTSCHDEUTSCH

The “The “shift operationshift operation” as a generalization to ” as a generalization to Deutsch’s TricksDeutsch’s Tricks

Change of controlled gate in Deutsch with Change of controlled gate in Deutsch with Controlled-Ushift gateControlled-Ushift gate

Now we deal with new types of Now we deal with new types of eigenvalues and eigenvectorseigenvalues and eigenvectors

The The general concept general concept of the answer of the answer encoded in phaseencoded in phase

Shift operator allows to solve Shift operator allows to solve Deutsch’s problem with certaintyDeutsch’s problem with certainty

Controlling amplitude versus Controlling amplitude versus controlling phasecontrolling phase

Controlling amplitude versus Controlling amplitude versus controlling phasecontrolling phase

Exercise for Exercise for studentsstudents

Exercise for students

Dave BaconLawrence Ioannou

Sources used