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

2. What is a computation La Segrada Familia (Barcelona) Architect: Gaudi
Very different than the turing machine…
La Segrada Familia
(Barcelona)
Architect: Gaudi

3. Computation (Algorithm)
What is a computation?
Computation (Algorithm)
Input
Output
011000
Universal computation models: A Q B C ≈
Very different than the turing machine…
Turing machine, 1936
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
Very different than the turing machine…

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

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.
Polynomial time Quantum
algorithm for factoring
Deutsch
Josza [‘92]
Bernstein
Vazirani[‘93]
Simon
[‘94]
Shor[’94] 6 6
Physics of many particles
(non universal computations)
Philosophy
of Science
Algorithms
Cryptography
technology

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

9. The principles of Quantum Physics
Part I:
The principles of Quantum Physics

10. The two slit experiment Part A: bullets
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11. Two slit experiments Part B: Water waves
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12. Two slit experiment Part C: electrons
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Interference pattern for particles!

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

14. Part II:
The Qubit
Very different than the turing machine…

15. 1st quantum principle: Superposition+ b
A quantum particle
can be in a
Superposition
of all its possible
“classical” states + a b + a b
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16. The elementary quantum information unit:
Qubit = Quantum bit
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 |0
Very different than the turing machine…
On the board:
Dirac notation
Vector notation
Transpose
Inner products
density matrix
We can also
talk about qudits,
Of higher dim. + a b 1 |1

17. Part III:
Measurements
Very different than the turing machine…

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

19. Projective measurements
A projective measurement is described by a Hermitian matrix M.
M has eigenvectors (eigenspaces) with associated real eigenvalues
The classical outcome is eigenvalue 𝛌 with probability =
norm squared of projection on the corresponsing eigenspace
& the state collapses to this projection and renormlized
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
Very different than the turing machine… + a b 1 1

20. Part IV:
Dynamics
Very different than the turing machine…

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

22. Quantum Gates On the board:
Very different than the turing machine…
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|0 |0 |0 |1 |1
Nucleos spin – in a large constant magnetic field, apply a perturbation of an oscilating magnetic field in resonant frequency. Generates rotations of the spin. H H
On the board:
compute weights,
repeat with measurement in the middle

24. Part V:
Two qubits
From time to time I will be using more technical terms, as parts of remarks.
Ignore than if you don’t understand them- they are meant for references for those who do understand, but are not necessary
For the rest of the talk. The few technical terms I will need will be explained in detail.

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

26. The space of two qubits The EPR state: The computational Basis
The computational Basis
for the two qubits space
The EPR state:

27. 1st ex. of entanglement: The CHSH game
They win if:
> 0.85! Pr(success) with EPR