Grover's algorithm - Introduction to quantum computing

Feb 27, 2020Download as pptx, pdf1 like533 views

This document provides an introduction to Grover's algorithm, a quantum algorithm that allows for faster searching of an unstructured database. It explains some key concepts in quantum computing like superposition, entanglement, and quantum gates. Grover's algorithm uses amplitude amplification to find a target item in a list of N items using only O(sqrt(N)) operations, providing a speedup over classical algorithms. The document analyzes how Grover's algorithm works and its complexity, and discusses some physical implementations of quantum computers.

1
Introduction to quantum
computing:
The grover’s algorithm
Meir TOLEDANO

3
Superposition
Each QBit state s can be is position 0 or 1 or in mixed state of them.
Mathematically speaking :
In other words s lives in the complex Hilbert space

4
Measure
Reading the QBit:
1. return 0 or 1 randomly with probability
1. Change the state accordingly
In order to be consistent (s lives in )

5
Entanglement
When mixing n-QBit together, there are entangled, they work together. Example for 2-
QBit.
Mathematically speaking s lives in
And now we have :

6
Operations
An operation change the state s to the state s’:
U is a 2*2 matrix that transform s in s’.
But U has to preserve the total probability, U is in the unitary group i.e.

11
Analysis
First step
Second step
Measurement

12
Analysis
In term of probability:
So we have created two random fair coins that are completely correlated !!!

13
GROVER’S ALGORITHM:
Super search (Google killer ?)

15
Goal
● Find element that satisfy a condition in a list of size N.
● Classical algorithm run in for the worst case and in for the average case.
● In quantum computing, there is a speedup

17
SOME VECTORS
We have an orthonormal basis
The uniform vector can be written
in this basis

20
SOME OPERATORS
Reflection around m:
One of the contribution of Grover is
to show that this operator is
feasible for “classical” boolean
function

21
GROVER’S ALGORITHM
1. Start with the vector
1. Apply the Hadamard operator then
1. Calculate
1. Do T times
a. Apply
b. Apply

26
ANALYSIS: COMPLEXITY
since
Conclusion : the complexity of Grover’s algo is !!!!
so

33
BUT ALSO...
Molecular quantum computer (First realisation of Shor algorithm)
Topological computing
Electron spin / Quantum dot

39
THANK YOU
https://www.linkedin.com/in/meirtoledano/

Grover's algorithm is a quantum algorithm that allows finding an item in an unstructured database with fewer queries than classical algorithms. It works by amplifying the probability of measuring the target item. The key steps are: (1) initialize a superposition of all possible inputs, (2) apply an oracle to mark the target item, (3) apply Grover's diffusion operator to amplify the target amplitude, and (4) repeat steps 2-3 until the target can be measured with high probability. An example is using Grover's algorithm to find a phone number in an unsorted database with only sqrt(N) queries, compared to N/2 queries classically.

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Grover's algorithm - Introduction to quantum computing

1. 1 Introduction to quantum computing: The grover’s algorithm Meir TOLEDANO

2. 2 QUANTUM 101

3. 3 Superposition Each QBit state s can be is position 0 or 1 or in mixed state of them. Mathematically speaking : In other words s lives in the complex Hilbert space

4. 4 Measure Reading the QBit: 1. return 0 or 1 randomly with probability 1. Change the state accordingly In order to be consistent (s lives in )

5. 5 Entanglement When mixing n-QBit together, there are entangled, they work together. Example for 2- QBit. Mathematically speaking s lives in And now we have :

6. 6 Operations An operation change the state s to the state s’: U is a 2*2 matrix that transform s in s’. But U has to preserve the total probability, U is in the unitary group i.e.

7. 7 QUANTUM ALGORITHMS

8. 8 Gates

9. 9 A SIMPLE EXAMPLE

10. 10 The circuit

11. 11 Analysis First step Second step Measurement

12. 12 Analysis In term of probability: So we have created two random fair coins that are completely correlated !!!

13. 13 GROVER’S ALGORITHM: Super search (Google killer ?)

14. 14 Cited 5300 !!! 1996 : 24 year ago

15. 15 Goal ● Find element that satisfy a condition in a list of size N. ● Classical algorithm run in for the worst case and in for the average case. ● In quantum computing, there is a speedup

16. 16 SOME VECTORS

17. 17 SOME VECTORS We have an orthonormal basis The uniform vector can be written in this basis

18. 18 SOME VECTORS

19. 19 SOME OPERATORS Reflection around u:

20. 20 SOME OPERATORS Reflection around m: One of the contribution of Grover is to show that this operator is feasible for “classical” boolean function

21. 21 GROVER’S ALGORITHM 1. Start with the vector 1. Apply the Hadamard operator then 1. Calculate 1. Do T times a. Apply b. Apply

22. 22 ANALYSIS: REFLECTION COMPOSITION

23. 23 ANALYSIS: DYNAMIQUE

24. 24 ANALYSIS: DYNAMIQUE

25. 25 ANALYSIS: STOPPING But remember so

26. 26 ANALYSIS: COMPLEXITY since Conclusion : the complexity of Grover’s algo is !!!! so

27. 27 QBIT PHYSICAL REALISATION

28. 28 CLASSIC COMPUTER MECHANICAL

29. 29 CLASSIC COMPUTER VACUUM TUBES

30. 30 CLASSIC COMPUTER VLSI TRANSISTORS

31. 31 QUANTUM COMPUTER JOSEPHSON JUNCTION

32. 32 QUANTUM COMPUTER TRAPPED IONS