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# Quantum Information

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This is the presentation used by Dr Charles Bennet ,Fellow IBM during his Video Conferencing Lecture on Quantum Information for students at NIT Warangal

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### Quantum Information

1. 1. Quantum Information and the Importance of Basic Research Charles H. Bennett IBM Research Yorktown NIT Warangal Videoconference 25 Sep 2009
2. 2. Computational complexity of physical states and evolutions Information carrying capacity of physical interactions Physical (e.g. thermodynamic) resources required for communication and computation
3. 4. Some computations require a great many intermediate steps to get to the answer. Factoring large numbers is thought to require more than polynomial time. Classical Computation Theory shows how to reduce all computations to a sequence of NANDs and Fanouts. It classifies problems into solvable and unsolvable, and among the solvable ones classifies them by the resources (e.g. time, memory, luck) required to solve them. Complexity classes P, NP, PSPACE…
4. 5. a b a a XOR b a a b b c c XOR (a AND b) XOR gate Toffoli gate Conventional computer logic uses irreversible gates, eg NAND, but these can be simulated by reversible gates. Toffoli gate is universal. Reversible gates were used to show that computation is thermodynamically reversible in principle. Now they are used for quantum computation . self-inverse NAND gate a b NOT(a AND b) no inverse a Fanout
5. 6. Logical Reversibility and Thermodynamics <ul><li>Landauer Principle: each erasure of a bit, or other logical 2:1 mapping of the state of a physical computer, increases the entropy of its environment by k log 2. </li></ul><ul><li>Reversible computers, which by their hardware and programming avoid these logically irreversible operations, can in principle operate with arbitrarily little energy dissipation per step. </li></ul>
6. 7. Ordinary irreversible computation can be viewed as an approxi-mation or idealization, often quite justified, in which one considers only the evolution of the computational degrees of freedom and neglects the cost of exporting entropy to the environment.
7. 8. RNA Polymerase may be viewed as a reversible tape-copying Turing machine. The chemical reaction is reversible, but in vivo it is driven forward by removal of PP.
8. 9. In vitro, by adjusting PP vs XTP concentrations, the copying can be made to drift forward or backward while dissipating < kT dissipation per step.
9. 10. Reversible clockwork computer I invented in 1982 (highly impractical)
10. 11. A Bigger change of Mindset: Quantum Information
11. 12. or EPR state
12. 14. Ordinary classical information, such as one finds in a book, can be copied at will and is not disturbed by reading it. <ul><li>Trying to describe your dream </li></ul><ul><li>changes your memory of it, </li></ul><ul><li>so eventually you forget the </li></ul><ul><li>dream and remember only what </li></ul><ul><li>you’ve said about it. </li></ul><ul><li>You cannot prove to someone else </li></ul><ul><li>what you dreamed. </li></ul><ul><li>You can lie about your dream and not get caught. </li></ul>But unlike dreams, quantum information obeys well-known laws. Quantum information is more like the information in a dream
13. 17. <ul><li>Superposition Principle </li></ul><ul><li>Between any two reliably distinguishable </li></ul><ul><li>states of a quantum system </li></ul><ul><ul><li>(for example vertically and horizontally polarized single photons) </li></ul></ul><ul><li>there exists other states that are not reliably distinguishable from either original state </li></ul><ul><ul><li>(for example diagonally polarized photons) </li></ul></ul>
14. 19. Measuring an unknown photon’s polarization exactly is impossible (no measurement can yield more than 1 bit about it). Cloning an unknown photon is impossible. (If either cloning or measuring were possible the other would be also). If you try to amplify an unknown photon by sending it into an ideal laser, the output will be polluted by just enough noise (due to spontaneous emission) to be no more useful than the input in figuring out what the original photon’s polarization was. 28.3 o
15. 20. Cryptography: the One Time Pad allows messages to be transmitted in absolute privacy over public channels, but requires the sender and receiver to have shared secret random data (“key”) beforehand. One key digit is used up for each message digit sent. The key cannot be reused. If it, system becomes insecure. Message Cryptogram Key delivered securely beforehand + Key = Cryptogram transmitted publicly Cryptogram  Key = Message One time pad worksheet used by Che Guevara message key cryptogram
16. 21. In the end, Alice and Bob will either agree on a shared secret key, or else they will detect that there has been too much eavesdropping to do so safely. They will not, except with exponentially low probability, agree on a key that is not secret. Quantum Cryptography avoids the need to hand-deliver the key.
17. 22. First quantum cryptography paper was presented not far away from here in 1984
18. 24. Original Quantum Cryptographic Apparatus built in 1989 transmitted information secretly over a distance of about 30 cm. Senders side produces very faint green light pulses of 4 different polarizations Quantum channel is an empty space about 30 cm long. There is no Eavesdropper, but if there were she would be detected. Calcite separates polarizations. Photomultipliers A and B detect single photons.
19. 25. Modern Quantum Crypto Key Distribution at University of Geneva Also experiments at several other labs, and two commercial QKD systems.
20. 26. So far, all commercial QKD equipment has taken the form of boring rectangular metal boxes. Sales have been less than anticipated. Perhaps some new ideas are needed. ID-Quantique MagiQ Technologies
21. 27. Frogs, if cooled to 5 o C to reduce thermal noise in their retinas, can see single photons. This raises the possibility of an all-natural quantum cryptography system,
22. 28. or EPR state
23. 32. “ Violation of Bell’s Inequality”
24. 35. <ul><li>A “message” backward in time is safe from paradox under two conditions, either of which frustrate your ability to advise your broker what stocks to buy or sell yesterday: </li></ul><ul><li>Sender can’t control it (EPR effect) OR </li></ul><ul><li>Receiver disregards it (Cassandra myth). </li></ul>
25. 36. unknown quantum state teleported state BBCJPW ’93
26. 37. A classical channel is a quantum channel with an eavesdropper . A classical computer is a quantum computer handicapped by having eavesdroppers on all its wires.
27. 38. Fast Quantum Computation (Grover algorithm) (Shor algorithm) (For a quantum computer, factoring is about as easy as multiplication, due to the availability of entangled intermediate states.) (For a classical computer, factoring appears to be exponentially harder than multiplication, by the best known algorithms.)
28. 39. A Computer can be compared to a Stomach Classical Computer Quantum Computer n -bit input n -bit output Because of the superposition principle and the possibility of entanglement, the intermediate state of an n-qubit quantum computer state requires 2 n complex numbers to describe, giving a lot more room for maneuvering a |0000>+ b |0001>+ c |0010>+ d |0011>+… n -bit intermediate state e.g. 0100
29. 40. How Much Information is “contained in” n qubits, compared to n classical bits, or n analog variables? Digital Analog Quantum Information required to specify a state Information extractable from state n bits n bits 2 n complex numbers n bits n real numbers n real numbers Good error correction yes no yes
30. 41. This 7 qubit molecule was used to factor 15 4
31. 42. Physical systems actively considered for quantum computer implementation <ul><li>Liquid-state NMR </li></ul><ul><li>NMR spin lattices </li></ul><ul><li>Linear ion-trap spectroscopy </li></ul><ul><li>Neutral-atom optical lattices </li></ul><ul><li>Cavity QED + atoms </li></ul><ul><li>Linear optics with single photons </li></ul><ul><li>Nitrogen vacancies in diamond </li></ul><ul><li>Topological defects in </li></ul><ul><li>fractional quantum Hall effect systems </li></ul><ul><li>Electrons on liquid He </li></ul><ul><li>Small Josephson junctions </li></ul><ul><ul><li>“ charge” qubits </li></ul></ul><ul><ul><li>“ flux” qubits </li></ul></ul><ul><li>Spin spectroscopies, impurities in semiconductors </li></ul><ul><li>Coupled quantum dots </li></ul><ul><ul><li>Qubits: spin,charge,excitons </li></ul></ul><ul><ul><li>Exchange coupled, cavity coupled </li></ul></ul>
32. 43. The Downside of Entanglement
33. 45. Quantum Fault Tolerant Computation Clean qubits are brought into interaction with the quantum data to siphon off errors, even those that occur during error correction itself.
34. 46. CMOS Device Performance The end of Moore’s law for conventional hardware is widely predicted on fundamental physical grounds. Can quantum computers give Moore’s law a new lease on life? If so, how soon will we have them?
35. 47. <ul><li>Executive Summary </li></ul><ul><li>A Quantum computer can probably be built eventually, but not right away. Maybe in 20 years. </li></ul><ul><li>It would exponentially speed up a few computations like factoring, thereby breaking currently used digital signatures and public key cryptography. (Shor’s algorithm) </li></ul><ul><li>It would speed up many important optimization problems like traveling salesman, but only quadratically, not exponentially. (Grover’s algorithm) </li></ul><ul><li>There would be no speedup for many other problems. For these computational tasks, Moore’s law would still come to an end, even with quantum computers. </li></ul>
36. 48. BQP NP PSPACE NP Complete (e.g. Traveling Salesman, Frustrated classical ground state) Factoring Simulating quantum many-body dynamics (even Fermions*) QMA-complete (e.g. Frustrated quantum ground state) Problems thought to be hard for a classical computer, but easy for a quantum computer Easy for a classical computer Problems thought to be hard even for a quantum computer Multiplication P Find Stationary State of a Dissipative System, Classical or Quantum *Bravyi & Kitaev quant-ph/003137, Ortiz et al cond-mat/0012334
37. 49. <ul><li>But quantum information is good for many other things besides speeding up computation. </li></ul><ul><li>Quantum cryptography. Practical today and secure even against eventual attack by a quantum computer. Quantum cryptography brings back part of the security that is lost because of quantum computers, but does not fully restore public key infrastructure. </li></ul><ul><li>Speeding up the simulation of quantum physics, with applications to chemistry and materials science. </li></ul><ul><li>Communication and Distributed Computing </li></ul><ul><li>Metrology, precision measurement and time standards. </li></ul><ul><li>New quantum information phenomena are continually being discovered. An exciting area of basic science. </li></ul>
38. 50. Science as a career <ul><li>Science can be a very satisfying career. </li></ul><ul><li>I always wanted to be a scientist, from before age 5, so perhaps I’m not the best person to compare science to other career choices. </li></ul><ul><li>Always something new to do and be curious about: the opposite of housework, which consists of doing the same job repeatedly. </li></ul><ul><li>Ability to appreciate progress in all areas of science, not only your own. Scientists, professional or amateur, can still almost be “renaissance persons,” at least with respect to the natural world. </li></ul><ul><li>Less backsliding than any other area of human endeavor. </li></ul><ul><li>If you are patient, and live long enough, you can see questions answered you’ve been wondering about for decades, maybe even help answer them. </li></ul>
39. 51. It was once generally believed that computation had a minimal energy cost of order k B T ln 2 per step (roughly the mean thermal energy of a molecule at ambient temperature T). Now computation is known to be thermodynamically reversible. Quantum effects in computers and communications systems were once thought of as merely a nuisance, making small devices less reliable than their larger cousins. Now they are known to be a valuable resource for both communication and computation, and indeed for understanding the nature of information itself.
40. 52. Discounting the future Individuals and institutions tend to plan ahead about as far in time as they can reasonably predict. For politicians this is usually until the next election. We scarcely think about the very distant future, and plan ahead less when current conditions are chaotic and unpredictable. Lunch time conversation between two astronomers First Astronomer: “…of course the Sun will only last another 5 billion years or so.” Second Astronomer (quite upset): “ What? ” First Astronomer : “I said the Sun will last 5 billion years.” Second Astronomer (relieved): “Oh! I thought you’d said 5 million years .”
41. 53. Reminiscence of a retired engineer at NASA’s Jet Propulsion Lab whose proudest accomplishment, many years earlier, had been working on the power supplies for the two Voyager spacecraft. As I recall he said roughly the following: We wanted to have the Voyagers visit all the big planets. NASA said, “No. Just Jupiter and Saturn. People don’t care about Uranus and Neptune.” We said, “If we don’t do it now, the planets won’t be properly aligned to do it again for another 200 years.” They said, “Legislators understand 2 years, not 200 years. You are only authorized for Jupiter and Saturn.” So we designed all the systems very conservatively, on the pretext of being conservative, but with the intention of giving the Voyagers enough power, fuel, and maneuvering and communication ability to go on to visit and photograph Uranus and Neptune, which they ultimately did to everyone’s great satisfaction.
42. 54. Voyager Photos of Neptune Bottom Picture Caption August 21, 1989 P-34632C Voyager 2-N32C This image of clouds in Neptune's atmosphere is the first that tests the accuracy of the weather forecast that was made eight days earlier to select targets for the Voyager narrow-angle camera. Three of the four targeted features are visible in this photograph; all three are close to their predicted locations. The Great Dark Spot with its bright white companion is slightly to the left of center. The small bright Scooter is below and to the left, and the second dark spot with its bright core is below the Scooter. Strong eastward winds -- up to 400 mph -- cause the second dark spot to overtake and pass the larger one every five days. The spacecraft was 6.1 million kilometers (3.8 million miles) from the planet at the time of camera shuttering, and the image uses the orange, green and clear filters of the camera. The Voyager Mission is conducted by JPL for NASA's Office of Space Science and Applications.
43. 55. Surface of Triton, Neptune’s largest moon
44. 57. Quantum Laws & the Universality of Interaction
45. 58. Classically, there are distinct kinds of interaction that cannot be substituted for one another. For example, if I’m a speaker and you’re a member my audience, no amount of talking by me enables you to ask me a question. Quantumly, interactions are intrinsically bidirectional. Indeed there is only one kind of interaction, in the sense that any interaction between two systems can be used to simulate any other. One way in which quantum laws are simpler than classical is the universality of interaction.
46. 63. How do Quantum Speedups Work?
47. 64. Fast Quantum Computation Shor’s algorithm – exponential speedup of factoring – Depends on fast quantum technique for finding the period of a periodic function Grover’s algorithm – quadratic speedup of search – works by gradually focusing an initially uniform superposition over all candidates into one concentrated on the designated element. Speedup arises from the fact that a linear growth of the amplitude of the desired element in the superposition causes a quadratic growth in the element’s probability.
48. 65. Well-known facts from number theory. Let N be a number we are trying to factor. For each a<N, the function f a ( x ) = a x mod N is periodic with period at most N. Moreover it is easy to calculate. Let its period be r a . All known classical ways of finding r a from a are hard. Any algorithm for calculating r a from a can be converted to an algorithm for factoring N. Quantum mechanics makes this calculation easy.
49. 67. 2 Slits 1 photon Shor algorithm uses interference to find unknown period of periodic function. N Slits 1 photon Photon impact point yields a little information about slit spacing Photon impact point yields a lot of information about slit spacing
50. 68. Grover’s quantum search algorithm uses about √ N steps to find a unique marked item in a list of N elements, where classically N steps would be required. In an optical analog, phase plates with a bump at the marked location alternate with fixed optics to steer an initially uniform beam into a beam wholly concentrated at a location corresponding to the bump on the phase plate. If there are N possible bump locations, about √ N iterations are required. P = phase plate F = fixed optics Same optical setup works even with a single photon, so after about √ N iterations it would be directed to the right location.
51. 69. Optimality of Grover’s Algorithm: Why can’t it work in 1 iteration? Repeat the experiment with the phase bump in a different location. Original optical Grover experiment. Because most of the beam misses the bump in either location, the difference between the two light fields can increase only slowly. About √ N iterations are required to get complete separation. (BBBV quant-ph/9701001) Small difference after 1 iteration No difference initially
52. 70. Mask out all but desired area. Has disadvantage that most of the light is wasted. Like classical trial and error. If only 1 photon used each time, N tries would be needed. Non-iterative ways to aim a light beam. Lens: Concentrates all the light in one pass, but to use a lens is cheating. Unlike a Grover iteration or a phase plate or mask, a lens steers all parts of the beam, not just those passing through the distinguished location.
53. 71. How would a working quantum computer change the world? Exponential speedup of factoring and related problems of mainly cryptographic interest. Known public key cryptosystems would be broken. Digital signatures broken. Secret key systems (e.g. DES) still probably OK but would need double the key size. (Some public key functionality can be replaced by unconditionally secure quantum cryptography.) Exponential speedup for simulating quantum physics. Quadratic speedup for optimization problems like travelling salesman. No good reason to hope for exponential speedup. No speedup for some other problems.
54. 72. Fast quantum protocol for the lunch scheduling problem is a distributed form of Grover’s algorithm. A register of log N qubits, initially containing a uniform superposition of all dates, is passed back and forth between the two parties about √ N times, gradually building up amplitude on a conflict-free date. 1. Quantum Savings in Communication Complexity What else is quantum information good for?
55. 74. Prior shared entanglement helps a good deal if Alice and Bob are trying to hold a quiet conversation in a room full of noisy strangers (Gaussian channel in low signal, high noise, low-attenuation limit) ..seulement avec un … ..blah…blah…blah... wahrscheinlich .then suddenly she swerved.. .because they were on their first date .for example, if you want it sweeter,… I love you ..but without his other kidney.. forever. Don’t ask.
56. 75. I love you But it doesn’t help much if they are far apart in an empty room (attenuation) What?