1. Quantum computation is inherently different from classical computation and is believed to be exponentially more powerful than classical computation for certain tasks. 2. The basic unit of quantum information is the qubit, which can exist in superpositions of states |0> and |1>. Qubits can undergo unitary evolution and measurements, which cause the qubit state to collapse probabilistically. 3. Measurements on multiple entangled qubits do not factorize, allowing quantum algorithms like Deutsch-Jozsa and Shor's algorithm to offer exponential speedups over classical algorithms for certain problems like factoring.

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



