Quantum Computation and Quantum Information

1. Quantum Computation and Quantum Information Introduction and Overview Li Yunpeng Dept. of Computer Science

2. Science offers the boldest metaphysics of the age. It is a thoroughly human construct, driven by the faith that if we dream, press to discover, explain, and dream again, thereby plunging repeatedly into new terrain, the world will somehow come clearer and we will grasp the true strangeness of the universe. And the strangeness will all prove to be connected, and make sense! -Edward O. Wilson

3. Syllabus • Background & History • Quantum Computation • Quantum Information • Implementation of Quantum Computers • Prospect of QC & QI • Resources

4. A Legend • In the fourth century BC, a young man named Pythias was condemned to death by Dionysius, the tyrant of Syracuse, for plotting against him, but Pythias was granted three days’ leave to go home to settle his family’s affairs after his friend Damon agreed to take his place and be executed should Pythias not return. Pythias encountered many problems but managed to return just in time to save Damon. Dionysius was so struck by this remarkable and honorable friendship that he release them both…………………………………….

5. Friendship between CS and Phy. • Present day experimental and theoretical physicists depend on computing, and have incurred a debt that they have repaid many times over by making fundamental contributions to advances in hardware, software, and systems technologies.

6. Physics and the information revolution • Civilization has advanced as people discovered new ways of exploiting various physical resources such as materials, forces and energies. In the twentieth century information was added to the list when the invention of computers allowed complex information processing to be performed outside human brains. • The history of computer technology has involved a sequence of changes from one type of physical realization to another --- from gears to relays to valves to transistors to integrated circuits and so on. Today’s advanced lithographic techniques can create chips with features only a fraction of micron wide (a micron is a micrometer, i.e. a millionth of a metre).

7. Charles Babbage (1791-1871) conceived of most of the essential elements of a modern computer, though in his day there was not the technology available to implement his ideas. Babbage's Analytical Engine was his invention ,which is the ancestor of modern computer!

8. Miniaturization of the Computers • This increase is essentially due to the continual miniaturization of the computer's most elementary component, the transistor. As transistors became smaller more could be integrated into a single microchip, and so the computational power increased. • However this miniaturization process is now reaching a limit, a quantum threshold below which transistors will cease to function. Present ‘state-of-the-art’ components possess features only a few hundreds of nanometers across (a nanometre is a thousandth of a micron, or a billionth of a metre).

10. Request for Quantum Computers • If these chips were to be miniaturized further to the scale of tens of nanometers then their operation would be disrupted by the emergence of quantum phenomena , such as electrons tunneling through the barriers between wires. In order for the science of computation to progress further an alternative to transistor technology must be found, one whose components will function through quantum effects rather than in despite of them.

11. What is QCQI? • Quantum computation and quantum information is the study of the information processing tasks that can be accomplished using quantum mechanical systems.

12. Fundamental fields • Quantum Mechanics • Computer Science • Information Theory • Cryptography

14. A Series Crises in Physics at Early 20th Century • The problems was that classical physics were predicting absurdities such as the existence of an ‘ultraviolet catastrophe’ involving infinite energies, or electrons spiraling in exorable into the atomic nucleus……………………………………

15. The Creation of Quantum Mechanics • Quantum mechanics has been an indispensable part of science, and has been applied with enormous success to everything under and inside the Sun , including the structure of the atom, nuclear fusion in stars, superconductors, the structure of DNA, and the elementary particles of NATURE!

16. Quantum Mechanic • Quantum mechanics is the mathematical structure which embraces, in principle, the whole of physics.

17. It is very difficult… • The rules of Quantum mechanics are simple but even experts find them counter-intuitive, and the earliest antecedents of QCQI may be found in the long-standing desire of physicists to better under-standing quantum mechanics • The best known critic of quantum mechanics, Albert Einstein, went to his grave unreconciled with the theory he helped invent.

18. QIQC for Quantum Mechanics • Generations of Physicists since have wrestled with quantum mechanics in an effort to make its predictions more palatable. • One of the goals of quantum computation and quantum information is to develop tools which sharpen our intuition about quantum mechanics, and make its predictions more transparent to human minds!

19. A Question! • Often the most profound insights in science come when we develop a method for probing a new regime of Nature. • What else shall we discover as we obtain more complete control over single quantum systems, and extend it to more complex systems.

20. Complete Control Over Single Quantum Systems The invention of radio astronomy in the 1930s and 1940s led to a spectacular sequence of discoveries, including the galactic core of the Milky Way galaxy, pulsar, and quasars. Low temperature physics has achieved its amazing successes by finding ways to lower the temperatures of different systems. In a similar way, by obtaining complete control over single quantum systems, we are exploring untouched regimes of Nature in the hope of discovering new and unexpected phenomena.

21. QCQI Fit this Problem • They provide a useful series of challenges at varied levels of difficulty for people devising methods to better manipulate single quantum systems, and stimulate the development of new experimental techniques and provide guidance as to the most interesting directions in which to take experiment. • Conversely, the ability to control single quantum systems is essential if we are to harness the power of quantum mechanics for applications to quantum computation and quantum information.

22. To Build a Quantum Computer • Small Quantum computers, capable of doing dozens of operations on a few qubits represent the state of the art in quantum computation. • Experimental prototype for doing quantum cryptography have been demonstrated and are even at the level where they may be useful for some real-world application • However, it remains great difficulty to physicists and engineers of the future to develop a large-scale quantum computer!

23. Verschränkt • Entanglement is a uniquely quantum mechanical recourse that plays a key role in many of the most interesting applications of quantum computation and quantum information; • Entanglement is iron to the classical world’s bronze age. • In recent years there has been a tremendous effort trying to better understand the prosperities of entanglement considered as a fundamental recourse of Nature, of comparable importance to energy, information entropy, or any other fundamental resource!

24. A series crises in Mathematics • Naïve Set Theory • Russell’s Paradox • Axiom Set Theory Prof. Cantor

25. David Hilbert • Hilbert had emphasized between the 1890s and 1930s the importance of asking fundamental questions about the nature of mathematics. • Instead of asking ``is this mathematical proposition true?'' Hilbert wanted to ask ``is it the case that every mathematical proposition can in principle be proved or disproved?'' This was unknown, but Hilbert's feeling, and that of most mathematicians, was that mathematics was indeed complete, although the logical steps might be as yet undiscovered.

26. Hilbert’s Question on Quantum Mechanics • There is a parallel between Hilbert's questions about mathematics and the questions we seek to pose in quantum information theory. Before Hilbert, almost all mathematical work had been concerned with establishing or refuting particular hypotheses, but Hilbert wanted to ask what general type of hypothesis was even amenable to mathematical proof. • Similarly, most research in quantum physics has been concerned with studying the evolution of specific physical systems, but we want to ask what general type of evolution is even conceivable under quantum mechanical rules.

27. Were another Kurt Gödel in QCQI? Kurt Gödel • Gödel destroyed this hope by establishing the existence of mathematical propositions which were undecidable, meaning that they could be neither proved nor disproved. The next interesting question was whether it would be easy to identify such propositions. • Gödel's Impact on Philosophy

28. About “what is a computation” • Progress in mathematics had always relied on the use of creative imagination, yet with hindsight mathematical proofs appear to be automatic, each step following inevitably from the one before. Hilbert asked whether this `inevitable' quality could be captured by a `mechanical' process. In other words, was there a universal mathematical method, which would establish the truth or otherwise of every mathematical assertion?

29. Alan Turing in 1936 Paper • Turing used his machine as a theoretical construct to show that the assumed existence of a mechanical means to establish decidability leads to a contra-diction. In other words, he was initially concerned with quite abstract mathematics rather than practical computation. However, by seriously establishing the idea of automating abstract mathematical proofs rather than merely arithmetic, Turing greatly stimulated the development of general purpose in-formation processing. This was in the days when a "computer" was a person doing mathematics.

30. Computation----BASIC IDEAS • Computation may be defined as the systematic creation of symbols (output) which, under a given method of interpretation, have abstract properties that were specified in other symbols ( input). • “Symbols” here are physical objects, and computation is a process performed by a physical device called a computer.

31. The Classical Theory of Computation • Pioneers such as Gödel, Turing, Church, and Post managed to capture the correct classical theory by intuition alone. • As a result it is often falsely assumed that the foundations of computation theory are self-evident and purely abstract.

32. Church-Turing Thesis The statement of the Church-Turing Principle is stronger than what is strictly necessitated by the former. Indeed it is so strong that it is not satisfied by Turing Machine in classical physics. • Every ‘function which would naturally be regarded as computable’ can be computed by the universal Turing machine. • Physical version of the Church-Turing Principle: ‘Every finitely realizable physical system can be perfectly simulated by a universal model computing machine operating by finite means’. Refer to Deutsh’s famous paper in 1985

33. A Simple Modification… • Any algorithmic process can be stimulated efficiently using a probabilistic Turing Machine.

34. Quantum Turing Machine • :Q  Q  Left, Right, No movement }C • The QTM model is based on the idea that each computational step is the result of a fixed unitary operation, acting on the current global configuration of the whole machine.

35. How QTM work? • If a particular final configuration can be reached via two different paths with aptitudes  and -, then the probability of reaching that configuration is |-|2=0,despite the fact that the two paths separately is ||2 in both cases. • Furthermore, s single quantum computer can follow many distinct computational paths in superposition and produce a final output depending on the interference of all of them.

36. QTM ‘s Ability • Quantum computers cannot exceed classical computers in what may be computed, but may possibly exceed them in the efficiency of computation.

37. Connection between Phy. And CS • The fundamental connection between the laws of physics and what is computable was emphasized by Feymann(1982) and Deutsch(1985). Feynman (1982) went one step clos to a true quantum computer with his ‘universal quantum stimulator’. This consists of a lattice of spin systems with nearest-neighbor interactions that are freely specifiable.

38. Two Key Persons

39. Let’s strolling in the river of history!

40. History of QCQI(1) • The first deep insight into quantum information theory came with Bell's 1964 analysis of the paradoxical thought-experiment proposed by Einstein, Podolsky and Rosen (EPR) in 1935. Bell's inequality draws attention to the importance of correlations between separated quantum systems which have interacted (directly or indirectly) in the past, but which no longer influence one another. • In essence his argument shows that the degree of correlation which can be present in such systems exceeds that which could be predicted on the basis of any law of physics which describes particles in terms of classical variables rather than quantum states.

41. History of QCQI(2) • The next link between quantum mechanics and information theory came about when it was realized that simple properties of quantum systems, such as the unavoidable disturbance involved in measurement, could be put to practical use, in quantum cryptography (Wiesner 1983, Bennett et. al. 1982, Bennett and Brassard 1984; for a recent review see Brassard and Crepeau 1996)..

42. History of QCQI(3) • Quantum cryptography covers several ideas, of which the most firmly established is quantum key distribution. This is an ingenious method in which transmitted quantum states are used to perform a very particular communication task: to establish at two separated locations a pair of identical, but otherwise random, sequences of binary digits, without allowing any third party to learn the sequence.

43. History of QCQI(4) • In order to think about computation from a quantum-mechanical point of view, the first ideas involved converting the action of a Turing machine into an equivalent reversible process, and then inventing a Hamiltonian which would cause a quantum system to evolve in a way which mimicked a reversible Turing machine. This depended on the work of Bennett (1973; see also Lecerf 1963) who had shown that a universal classical computing machine (such as Turing's) could be made reversible while retaining its simplicity.

44. History of QCQI(5) • A different approach was taken by Feynman (1982, 1986) who considered the possibility not of universal computation, but of universal simulation---i.e. a purpose-built quantum system which could simulate the physical behaviour of any other. Clearly, such a simulator would be a universal computer too, since any computer must be a physical system. • But his device was not sufficiently specified to be called a computer, since he assumed that any interaction between adjacent two-state systems could be `ordered', without saying how.

45. History of QCQI(6) • In 1985 an important step forward was taken by Deutsch. Deutsch's proposal is widely considered to represent the first blueprint for a quantum computer, in that it is sufficiently specific and simple to allow real machines to be contemplated, but sufficiently versatile to be a universal quantum simulator, though both points are debatable. Deutsch's system is essentially a line of two-state systems, and looks more like a register machine than a Turing machine (both are universal classical computing machines).

46. History of QCQI(7) • Deutsch proved that if the two-state systems could be made to evolve by means of a specific small set of simple operations, then any unitary evolution could be produced, and therefore the evolution could be made to simulate that of any physical system. He also discussed how to produce Turing-like behavior using the same ideas. • Deutsch's simple operations are now called quantum `gates', since they play a role analogous to that of binary logic gates in classical computers. Various authors have investigated the minimal class of gates which are sufficient for quantum computation.

47. History of QCQI(8) • In the early 1990's several authors (Deutsch and Jozsa 1992, Berthiaume and Brassard 1992, Bernstein and Vazirani 1993) sought computational tasks which could be solved by a quantum computer more efficiently than any classical computer. Such a quantum algorithm would play a conceptual role similar to that of Bell's inequality, in defining something of the essential nature of quantum mechanics.

48. History of QCQI(9) • Shor (1994) who astonished the community by describing an algorithm which was not only efficient on a quantum computer, but also addressed a central problem in computer science: that of factorising large integers. • Shor discussed both factorisation and discrete logarithms, making use of a quantum Fourier transform method discovered by Coppersmith (1994) and Deutsch. Further important quantum algorithms were discovered by Grover (1997) and Kitaev (1995).

49. Peter W. Shor • AT&T Labs’ Member • Contact information: Email: shor@research.att.com Address: 180 Park Ave, Florham Park, NJ 07932-0971

50. Prof. Andrew Chi-Chih Yao