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### "Exploring Quantum Computing: Harnessing Parallelism and Entanglement for Revolutionary Algorithms" ###

Dive into the realm of quantum computing and unlock the power of parallelism and entanglement in shaping revolutionary algorithms. Understand qubits, tensor products, entangled states, and quantum algorithms like Shor's for factoring large numbers efficiently. Explore the interplay between classical and quantum mechanics, and grasp the potential these technologies hold for transforming computation. ###

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### "Exploring Quantum Computing: Harnessing Parallelism and Entanglement for Revolutionary Algorithms" ###

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  1. Lecture 16 • Quantum computing • Ubiquitous Internet Services • The client server paradigm • DNS • Electronic Mail • World Wide Web

  2. Quantum Computing • In a quantum system the amount of parallelism increases exponentially with the size of the system. • Access to the results disturbs the quantum state through a process called decoherence. • Qubit – a unit vector in a 2-dimensional complex space where a complex base has been chosen.

  3. Classical mechanics • The individual states of particles combine through the cartesian product. • If X and Y are vectors the vector product has dimension dim(X) + dim(Y). • Given a system of n particles the states of the system form a vector space with 2 x n dimensions. • Given n bits we can construct 2n n-tuples and describe a system with 2n states.

  4. Quantum mechanics • The individual states of particles combine through the tensor product. • If X and Y are vectors the vector product has dimension dim(X) x dim(Y). • Given a system of n particles the states of the system form a vector space with 2n dimensions. • The extra states that do not have a classical analogy are called entangled states.

  5. Quantum mechanics (cond’t) • A quantum bit can be in infinitely many superposition states. • When the qubit is measured the measurement changes the state of the particle to one of the two basic states, thus from a qubit we can only extract a clasical bit of information. • Example – the use of polarized light to transmit information.

  6. Polarized light • A photon’s polarized state can be modeled as a unit vector and expresses as a combination of two basis vectors. • There are infinitely many possible orientations of the unit vector  a qubit can be in infinitely many superposition states. • Measuring the polarization is equivalent with projecting the random vector onto one of the two bases.

  7. Quantum algorithms • 1994 Peter Shor found a polynomial time algorithm for factoring n-bit numbers on quantum computers. • The algorithm reduces the factoring problem to finding the period of a function and uses quantum parallelism to find the superposition of all values of the function in one step. • Then the algorithm calculates the quantum Fourier transform of the function in one step.

  8. Ubiquitous Internet Services • The client-server paradigm. • Asynchrounous RPCs • Servers are reactive programs. • Stateless versus statefull servers.

  9. DNS • DNS – Domain Name System, RFC 1034-5 • Distributed database consisting of a hierarchy of name servers. • Each organization has one or more local or authoritative name servers. • Maps host names to IP addresses.

  10. Electronic Mail • Asynchronous communication model • SMTP based upon TCP.

  11. The World Wide Web • Developed at CERN in late 80s. • Stateless servers. • URLs  identify a host and the path to a resource (file) on that host. • HTTP  Hypertext Transfer Protocol • HTML  Hypertext Markup Language

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