Introduction to Quantum Information and Computation

Introduction to Quantum Information and Computation Anu Venugopalan Guru Gobind Singh Indraprastha Univeristy Delhi _______________________________________________ INTERNATIONAL PROGRAM ON QUANTUM INFORMATION (IPQI-2010) Institute of Physics (IOP), Bhubaneswar January 2010 IPQI-2010-Anu Venugopalan

Introduction Quantum computation and quantum information is the study of the information processing tasks that can be accomplished using quantum mechanical systems Quantum Mechanics Computer Science Information Theory Cryptography Mathematics Quantum information & computation IPQI-2010-Anu Venugopalan

Introduction Real computers are physical systems • Stonehenge • Pebbles and beads • Tokens, abacus • Mechanical computers • Difference Engine I (Charles Babbage) • Analytical engine • Electromechanical, electronic (vacuum tubes) • Electronic (semiconductors) IPQI-2010-Anu Venugopalan

Introduction Real computers are physical systems Computer technology in the last fifty years- dramatic miniaturization Faster and smaller – - the memory capacity of a chip approximately doubles every 18 months – clock speeds and transistor density are rising exponentially...what is their ultimate fate???? IPQI-2010-Anu Venugopalan

Moore’s law [www.intel.com] IPQI-2010-Anu Venugopalan

Miniaturization and computers In spite of the dramatic miniaturization in computer technology in the past five decades, our basic understanding of how a computer functions – or what it can do – has not changed. The tiny components inside all computers today still behave and are understood according to classical physics. IPQI-2010-Anu Venugopalan

Extrapolating Moore’s law If Moore’s law is extrapolated, by the year 2020 the basic memory component of the chip would be of the size of an atom – what will be space, time and energy considerations at these scales (heat dissipation…)? At such scales, the laws of quantum physics would come into play - the laws of quantum physics are very different from the laws of classical physics - everything would change! IPQI-2010-Anu Venugopalan

Quantum computation In anticipation of ultimately hitting atomic scales in computer technology, the field of quantum computation was first envisaged Quantum physics offers something new and spectacular. By exploiting delicate quantum phenomena that have no classical analogues, it is possible to do certain computational tasks much more efficiently than can be done by any classical computer – even a supercomputer IPQI-2010-Anu Venugopalan

Quantum computation Quantum information and computing offers a new paradigm and possibilities for computing and will change the way in which scientists think about fundamental operations in computing and the capabilities and ultimate limits to computing Offers powerful techniques for storage and manipulation of information New phenomena - Quantum teleportation - Quantum cryptography IPQI-2010-Anu Venugopalan

Quantum mechanics • At the turn of the last century, several experimental observations could not be explained by the established laws of classical physics and called for a radically different way of thinking --this led to the development of Quantum Mechanics which is today regarded as the fundamental theory of Nature • The price to be paid for this powerful tool is that some of the predictions that Quantum Mechanics makes are highly counterintuitiveand compel us to reshape our classical (‘common sense’) notions......... IPQI-2010-Anu Venugopalan

Introduction to theoretical computer science • What are the capabilities of a computer? • What are the limits of a computer? • What are the problems that can be solved efficiently on a computer and what are the ones that cannot? • How are these questions related to the actual physical make up of the computer? • Does it matter if the computer is made up of gears and columns, vacuum tubes or integrated chips? IPQI-2010-Anu Venugopalan

Introduction to theoretical computer science Alan Turing (1936) Alonso Church, Kurt Gödel, Emil Post - Developed a ‘classical’ mathematical models for computation which was supposedly ‘free’ of any assumptions pertaining to the actual physical mechanism involved in a computer - on closer examination these models revealed subtle assumptions that might well break down when we encounter a new regime of Nature…the quantum domain. IPQI-2010-Anu Venugopalan

Introduction to theoretical computer science Why should we spend time investigating classical computer science (TCS) if we are to study quantum computation? • TCS has a vast body of concepts and techniques that can be applied to and reused in QI and QC- many of the triumphs of QI and QC have come by combining existing ideas from computer science with novel ideas from quantum mechanics IPQI-2010-Anu Venugopalan

Introduction to theoretical computer science One learns to ‘think like a computer scientist’- Computer scientists think in a very different style than does a physicist – anybody wanting a deeper understanding of QI&QC must learn to think like a computer scientist (at least some times!) - very useful for studying QI and QC IPQI-2010-Anu Venugopalan

Introduction to theoretical computer science Key concept of computer science – Algorithm An algorithm is a precise recipe for performing some task – e.g. adding two numbers The fundamental model for algorithms is the Turing Machine IPQI-2010-Anu Venugopalan

The Turing Machine The most influential computer model due to Alan Turing (1936) – captures in a mathematical definition, what we mean when we use the intuitive concept of an algorithm It is said to have been Turing’s response to David Hilbert’s challenge (‘Entscheidungsproblem’) and is also regarded as a computational analog of Gödel’s Incompleteness Theorem in Logic. IPQI-2010-Anu Venugopalan

The Turing Machine The proof process If one were to look over a mathematician's shoulders during a proof derivation, what would one see in his/her notes? Turing abstracted the process appearing in these notes into four principle ingredients IPQI-2010-Anu Venugopalan

The Turing Machine • A set of transformational rules • A method for recording each step in the proof • A method to go back and forth • A mechanism for deciding which rule to apply at a given moment IPQI-2010-Anu Venugopalan

The Turing Machine - the four steps listed are simplified so that a machine could be made to implement them – - translating these steps in terms of symbols (0,1) on a one-dimensional tape with the read/write concept - In this way Turing translated the mechanistic analogues of the human thought process into a mathematical form The deterministic Turing machine IPQI-2010-Anu Venugopalan

The Turing Machine Infinite tape -------- Read/write head Internal states/program IPQI-2010-Anu Venugopalan

The deterministic Turing Machine - the four main elements of the DTM are • Finite State Control • Tape • A read/write tape head • Program IPQI-2010-Anu Venugopalan

DTM- Finite State Control (FSC) - Finite State Control The FSC for a TM can be visualized as a stripped down microprocessor which coordinates the other operations of the machines A finite set of minternal states: q1 , q2, ……….. qm qs : starting state qh: halting state IPQI-2010-Anu Venugopalan

The Tape - the tape is a one dimensional strip which stretches off to infinity in one direction – the tape squares are labelled and each contains one symbol drawn from some alphabet e.g., 0,1 and b (blank) marks the left hand edge of the tape IPQI-2010-Anu Venugopalan

The read/write head - the read/write tape head identifies a single square on the DTM tape as the square that is being currently accessed by the machine IPQI-2010-Anu Venugopalan

Successor Program[adapted from www1.cs.columbi.edu/~zeph/3261/L14/L14.ppt] Sample Rules: If read 1, write 0, go right, repeat If read 0, write 1, HALT! If read , write 1, HALT! Let’s see how they are carried out on a piece of paper that contains the reverse binary representation of 47: IPQI-2010-Anu Venugopalan

Successor Program[adapted from www1.cs.columbi.edu/~zeph/3261/L14/L14.ppt] Program IPQI-2010-Anu Venugopalan

Successor Program[adapted from www1.cs.columbi.edu/~zeph/3261/L14/L14.ppt] IPQI-2010-Anu Venugopalan

Successor Program[adapted from www1.cs.columbi.edu/~zeph/3261/L14/L14.ppt] So the successor’s output on 111101 was 000011 which is the reverse binary representation of 48. IPQI-2010-Anu Venugopalan

Successor Program[adapted from www1.cs.columbi.edu/~zeph/3261/L14/L14.ppt] Similarly, the successor of 127 should be 128, as one can see in the following: IPQI-2010-Anu Venugopalan

Successor Program[adapted from www1.cs.columbi.edu/~zeph/3261/L14/L14.ppt] IPQI-2010-Anu Venugopalan

The Church-Turing Thesis . Church-Turing thesis: Any algorithmic process can be simulated on a Turing machine A Turing Machine – an idealized and rigorously defined mathematical model of a computing device. Many different models of computation are equivalent to the Turing machine (TM). IPQI-2010-Anu Venugopalan

The Church-Turing Thesis . The class of functions computable by a Turing Machine corresponds exactly to the class of functions which we would naturally regard as being computable by an algorithm IPQI-2010-Anu Venugopalan

The Church-Turing Thesis . Despite its lack of adornments, the TM model has proved to be remarkably durable in all the 70 years of its existence. Though computer technology has advanced dramatically, our qualitative understanding of the computation process remains the same - in the strict theoretical sense, all computers are the same. IPQI-2010-Anu Venugopalan

Computation Real computers are finite devices, not infinite, like the Turing machine – they can be understood by a circuit model of computation IPQI-2010-Anu Venugopalan

Circuits NAND NOT AND NOT NAND controlled-not gate (CNOT) IPQI-2010-Anu Venugopalan

Universality The NAND gate can be used to simulate the AND, XOR and NOT gates, provided wires, fanout, and ancilla are available. wire – “memory” Fanout – e.g Ancilla- bits in pre-prepared states. The NAND gate, wires, fanout and ancilla form a universal set of operations for computation. x x x x IPQI-2010-Anu Venugopalan

Irreversibility NAND NAND From the output of the NAND gate it is impossible to determine if the input was (0,1), (1,0), or (0,0) The NAND gate is irreversible - there is no logic gate capable of inverting the NAND. IPQI-2010-Anu Venugopalan

Irreversibility Computing machines inevitably involve devices which perform logical functions that do not have a single-valued inverse. This logical irreversibility is associated with physical irreversibility and requires a minimal heat generation, per machine cycle ~ order of kT for each irreversible function. Landauer’s principle: Any irreversible operation in a circuit is necessarily accompanied by the dissipation of heat. IPQI-2010-Anu Venugopalan

Irreversibility • As the densities and switching speeds of our computational devices continue to increase exponentially, the amount of energy dissipated by these devices must remain at a certain level, otherwise economically impractical cooling apparatus is required. • Conventional computers perform thermodynamically irreversible logic operations. • Information, in the form of bits, is erased. • This bit erasure represents entropy, which is correlated to heat dissipation. IPQI-2010-Anu Venugopalan

Introduction to Quantum Information and Computation