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

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QUANTUM 101

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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

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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 )

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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 :

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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.

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QUANTUM ALGORITHMS

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Gates

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A SIMPLE EXAMPLE

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The circuit

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Analysis
First step
Second step
Measurement

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

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

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Cited 5300 !!!
1996 : 24 year ago

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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

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SOME VECTORS

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

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SOME VECTORS

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SOME OPERATORS
Reflection around u:

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

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

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ANALYSIS: REFLECTION COMPOSITION

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ANALYSIS: DYNAMIQUE

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ANALYSIS: DYNAMIQUE

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ANALYSIS: STOPPING
But remember
so

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

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QBIT PHYSICAL REALISATION

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CLASSIC COMPUTER MECHANICAL

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CLASSIC COMPUTER VACUUM TUBES

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CLASSIC COMPUTER VLSI TRANSISTORS

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QUANTUM COMPUTER JOSEPHSON JUNCTION

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QUANTUM COMPUTER TRAPPED IONS

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

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QUANTUM ECONOMICS

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SMALL ECOSYSTEM

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EMERGING BUT STILL A NICHE

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CASH FOR QBITS

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GLOBAL RACE

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