#### Transcript Quantum Computation

Debasis Sadhukhan M.Sc. Physics, IIT Bombay 1. Basics of Quantum Computation. 2. Quantum Circuits 3. Quantum Fourier Transform and it’s applications. 4. Quantum Search Algorithm WHAT WE NEED TO KNOW 1. Basic Quantum Mechanics & 2. A little Background of Computer Science Quantum Mechanics Computer Science • Church-Turing Thesis • Computational Complexity Classes (P, NP) • Satisfiability (3-SAT & KSAT) • Reduction • Reversible Computation * * * *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: think of : No classical analog exists. But you can Harry Potter and Voldemort Examples: Bell states or EPR pairs Some of the very important applications are : 1. Super-dense coding 2. Quantum Teleportation 3. Quantum Cryptography 4. Quantum Games * * Graphical Tensor Notation A completely graphical way to represent equations and physical processes * Represent a quantum state as a triangle with attached wires & do operation on quantum states just manipulating this picture Standard Quantum Theory Tensor Network States *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. * * *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. *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 ! * * * * * *The final state of the 1st register: Now, apply Inverse Fourier Transform on the 1st register. Final state: Overall Circuit: * *The major applications are 1. Order finding 2. Prime factorization These can be used to break the cryptosystem used in classical computer 3. Period Finding etc. * * * * * * * * * * * *Examples: C:\Users\DEBASIS\Desktop\GroversQuantumSear chAlgorithm.cdf C:\Users\DEBASIS\Desktop\SimulatedQuantumCo mputerAlgorithmForDatabaseSearching.cdf *Drawback: 1. Still, the problem remains in NP class. 2. 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. * * * 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/