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1. 1
For computer
scientists
Prof. Dorit Aharonov
School of computer science and engineering
Hebrew university, Jerusalem, Israel
Quantum computation

2. What is a computation
La Segrada Familia
(Barcelona)
Architect: Gaudi

3. What is a computation?
A
Q
B C A
Turing machine, 1936
≈
Computation
(Algorithm)
Output
011000
Input
0111001
Universal computation models:
Uniform
Circuits
≈
≈

4. Game of life
Rules:
A living site:
stays alive if it has
2 or 3 live neighbors
Otherwise dies
A dead site:
comes to life
if it has exctly 3
living neighbors

5. 5
Quantum computation is the only computational model which
credibly challenges the Extended Church Turing thesis
The Extended Church Turing thesis:
“Any physically realizable computational model can be
simulated efficiently by a randomized Turing machine”
A corner stone thesis in computer science:
≈ ≈
The Extended Church-Turing thesis (ECTT)

6. 6
Bird’s view on Quantum computation
Inherently different from standard
“classical” computers. We believe that
it will be exponentially more powerful
for certain tasks.
6
6
Philosophy
of Science
Physics of many particles
(non universal computations)
Cryptography
Algorithms
technology
Polynomial time
Quantum
algorithm for
factoring
Shor[’94]
Deutsch
Josza [‘92]
Bernstein
Vazirani[‘93]
Simon
[‘94]

7. 7
About this school
Goal: Intro to quantum computation & complexity
Some important notions, results, open questions
Note: We will not cover many important things…
(A partial list will be provided & updated )
Two remarks:
1) The lectures are intertwined, not independent!
2) The TA sessions are mainly exercises.
Do them! We rely on them in the next lecture.

8. 8
Intro Lecture:
Qubits
Part 1: The principles of quantum Physics
Part2: The qubit
Part 3: Measurements
Part 4: Dynamics
Part 5: Two qubits

9. 9
Part I:
The principles of Quantum Physics

10. 10
The two slit experiment
Part A: bullets

11. 11
Two slit experiments
Part B: Water waves

12. 12
Two slit experiment
Part C: electrons
Interference pattern for particles!

13. 13
Explanation: superpositions and
measurements
The particle passes through both
paths simultaneously!
If measured,
it collapses to one of the options
1) The superposition principle
2) Measurement gives one option & changes the state

14. 14
Part II:
The Qubit

15. A quantum particle
can be in a
Superposition
of all its possible
“classical” states
+
+
+
a b
a
b
b
a
1st quantum principle:
Superposition

16. The elementary quantum
information unit:
The qubit can be in either one of the
States: 0,1
As well as in any linear combination!
 a vector in a 2 dim Hilbert space
+
a b
0 1
Qubit = Quantum bit
On the board:
Dirac notation
Vector notation
Transpose
Inner products
density matrix
We can also
talk about qudits,
Of higher dim.
|0
|1

17. 17
Part III:
Measurements

18. 18
The quantum measurement
+
a b
When a quantum particle is measured
the answer is Probabilistic
The Superposition collapses
to one of its possible classical states
|b|2
|a|2
0
0 1
1
Those (weird!) aspects have been
tested in thousands of experiments
2nd principle
measurement

19. Projective measurements
A projective measurement is described
by a Hermitian matrix M.
M has eigenvectors (eigenspaces) with
associated real eigenvalues
+
a b
0 1
1
0 On the board:
Measure with respect to Z
Prob=inner product squared
Measure X: The +/- basis
Probablity for 𝛱 a projection
Expected value of measurement:
Direct expression and as Tr(Mρ)
Uncertainty principle
The classical outcome is eigenvalue 𝛌 with probability =
norm squared of projection on the corresponsing eigenspace
& the state collapses to this projection and renormlized

20. 20
Part IV:
Dynamics

21. 21
Dynamics
On the board:
From the differential equation to unitary evolution
(eigenvalues which are primitive roots of unity)
Unitary as preserving inner product
Schrodinger’s equation:
Discrete time evolution:
The Hamiltonian
(A Hermitian operator)

22. 22
Quantum Gates
On the board:
Applying X,Z on computational basis states of a qubit
Linearity
Applying Hadamard on basis states and measuring (“coin flip”)

23. Interference & path integrals
On the board:
compute weights,
repeat with measurement in the middle
H H
|0 |0
|1
|0
|1

24. 24
Part V:
Two qubits

25. The superposition principle for
more qubits
one two
The state of n quantum bits is a superposition of
all 2n possible configurations,
each with its own weight!
three

26. The space of two qubits
 
0
,
1
|
1
,
0
|
2
1

 
The computational Basis
for the two qubits space
The EPR state:

27. 27
1st ex. of entanglement:
The CHSH game
{ 0 ,1 }
B 
X
a b
0.75
Pr(Win)
0
b
a
1
b
a
1
b
a
1
b
a
1
1
0
1
1
0
0
0









 
0
,
1
|
1
,
0
|
2
1

 
They win if:
>
0.85
! Pr(success) with EPR
{0 ,1 }
X A 
b
a
:
{(1,1)}
X
,
X
b
a
:
(1,0)}
(0,1),
{(0,0),
X
,
X
B
A
B
A



