Quantum Computation and Algorithms

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# Quantum Computation and Algorithms - PowerPoint PPT Presentation

Quantum Computation and Algorithms. Debasis Sadhukhan M.Sc. Physics, IIT Bombay. Basics of Quantum Computation . Quantum Circuits Quantum Fourier Transform and it’s applications. Quantum Search Algorithm. Plan of Talk. WHAT WE NEED TO KNOW Basic Quantum Mechanics &

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## Quantum Computation and Algorithms

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### Quantum Computation and Algorithms

M.Sc. Physics, IIT Bombay

Basics of Quantum Computation.

• Quantum Circuits
• Quantum Fourier Transform and it’s applications.
• Quantum Search Algorithm
Plan of Talk

WHAT WE NEED TO KNOW

• Basic Quantum Mechanics &
• A little Background of Computer Science
BACKGROUND

So, if the state can’t be written in the product state form, then they are Entangled. They are called to be Entangled State.

• Classical Analogy: No classical analog exists. But you can think of : Harry Potter and Voldemort
Quantum EntanglementThe Greatest Love Story Ever Told

Examples: Bell states or EPR pairs

Some of the very important applications are :

• Super-dense coding
• Quantum Teleportation
• Quantum Cryptography
• Quantum Games
Applications

Represent a quantum state as a triangle with attached wires & do operation on quantum states just manipulating this picture

Graphical Tensor Notation

Execution of an classical algorithm require hardware, consist of many electrical circuits containing wires and logic gates.

• These logic gates are the basic building block of a classical computer.
• Similarly, to execute a quantum algorithm we must have a quantum computer where the building blocks are quantum gates.
• So, What are the Quantum Gates…?
• As the name suggests, the gates are quantum, the laws of quantum mechanics must be applicable here.
• So, they must be unitary operator and can be made reversible.
Quantum Gates

Note: The target and control qubit are not basis independent i.e. our target and control qubit may change if we use a different basis .

• In Classical Computation, we have seen NAND and NOR gate as universal quantum gate. A similar universality is true for quantum computation also.
• Every classical gates can be created using unitary quantum gates. In that sense quantum circuits include all the classical circuits.
• So, universality of quantum gates is obvious.
Uni1versal Quantum Gates

An algorithm is a well defined procedure or a set of instructions to perform an information processing task.

• Turing-Church Thesis: Any algorithmic process can be simulated efficiently using a probabilistic Turing machine.
• Complexity Classes: P , NP
• Quantum algorithms are those that uses quantum mechanical principles at the time of it’s execution.

Hard to design !

Quantum Algorithms

The final state of the 1st register:

Now, apply Inverse Fourier Transform on the 1st register.

Final state:

Overall Circuit:

Phase Estimation

The major applications are

• Order finding
• Prime factorizationThese can be used to break the cryptosystem used in classical computer
• Period Finding etc.
Applications of QFT

Examples:

C:\Users\DEBASIS\Desktop\GroversQuantumSearchAlgorithm.cdf

C:\Users\DEBASIS\Desktop\SimulatedQuantumComputerAlgorithmForDatabaseSearching.cdf

• Drawback:
• Still, the problem remains in NP class.
• If we don’t know the exact no of solution, we may not reach to our solution as no of iteration explicitly depends on M.
Examples and Drawback

References:

• [1] Michael A. Nielsen and Isaac I. Chuang, Quantum Computation and Quantum Information, Cambridge University Press(2002).
• [2] Phillip Kaye, Raymond Laflamme and Michele Mosca, An Introduction to Quantum Computing, Oxford University Press(2007).
• [3] Jamie Smith and Michele Mosca, arXiv:1001.0767v2 [quant-ph]
• [4] Lecture notes of John Preskill, California Institute of Technology: http://theory.caltech.edu/~preskill/ph229/
Thank You