1. Introduction to Quantum Informatics Vladimir P.Gerdt Head of research group (sector) on algebraic and quantum computation Laboratory of Information Technologies Joint Institute for Nuclear Research 141980, Dubna, Russia

2. Quantum Computation & Quantum Information Computer Science Information Theory Cryptography Quantum Mechanics Study of information processing tasks that can be accomplished using quantum mechanical systems Digital Design

3. Information and Physics Realisations are getting smaller and faster Evolution of physical realization of classical computers: gears → relays → valves → transistors → integrated circuits → ? Today’s advanced lithographic techniques can create chips with features only a fraction of micron wide (22 nm technology, 2012).

4. Source: Intel Gordon Earle Moore (born January 3, 1929) is an American businessman and co-founder and Chairman Emeritus of Intel Corporation and the author of Moore's Law

5. Inevitable quantum physics With extrapolation of Moore’s law, by the year 2020 the basic memory component (elementary transistor) of the chip would be of the size of an atom (10 -8 cm). 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. In anticipation of ultimately hitting atomic scales in modern technology, the field of quantum computation and quantum information comes into play.

6. Counterintuitive Quantum Mechanics

7. “I think I can safely say that nobody understands quantum mechanics” Richard Feynman “Quantum mechanics; Real Black Magic Calculus” Albert Einstein “Young man, in mathematics you don’t understand things, you just get used to them.” John von Neumann

8. Postulates of Quantum Mechanics Postulate 1: A closed quantum system is described by a unit vector in a complex inner product (Hilbert) space known as state space. Postulate 2: The evolution of a closed quantum system is described by a unitary transformation. Postulate 4: The state space of a composite physical system is the tensor product of the state spaces of the component systems. is often written as or as.

9. Some conventions implicit in Postulate 4 AliceBob

10. Quantum computers and probability When the quantum computer gives you the result of computation, this result is correct only with certain probability Quantum algorithms are designed to "shift" the probability towards correct result Running the same algorithm sufficiently many times you get the correct result with high probability, assuming that we can verify whether the result is correct or not The number of repetitions is much smaller then for usual computers

11. Short History 1970-е: the beginning of quantum information theory 1980: Yuri Manin set forward the idea of quantum computations 1981: Richard Feynman proposed to use quantum computing to model quantum systems. He also described theoretical model of quantum computer 1985: David Deutsch described first universal quantum computer 1994: Peter Shor developed the most important for quantum algorithm (factorization into primes) 1996: Lov Grover developed an algorithm for search in unsorted database with complexity

12. Short History (cont.) 1998: the first quantum computers on two qubits, based on NMR (Oxford; IBM, MIT, Stanford) 2000: quantum computer on 7 qubits, based on NMR (Los-Alamos) 2001: 15 = 3 x 5 on 7- qubit quantum comp. by IBM 2005 - at present: experiments with photons; quantum dots; trapped ions, nuclear spins, etc. 2007: Canadian company D-Wave System announced a prototype quantum computer (Orion) on 16 qubits. 2011: D-Wave System announced D-Wave One, "the world's first commercially available quantum computer," operating on an 128 qubit chip-set. 2012: D-Wave System revealed D-Wave Two, operating on an 512 qubit chip-set.

13. D-Wave’ products commercial quantum computers? D Processor Rainier from D-Wave One (128 qubits) Lockheed Martin bought a version of D-Wave One and upgraded it to D-Wave Two. The New York Times, March 21, 2013 Processor Vesuvius from D-Wave Two (512 qubits) Nasa purchased a 512 qubit D-Wave Two quantum computer for $15 millions. BBC News, May 16, 2013 http://www.bbc.com/news/science-environment-22554494

14. A quantum computer hosts quantum bits (qubits) which can store superpositions of 0 and 1 classical bit: 0 or 1 quantum bit:  |0  +  |1  (superposition!) “qubit” = two-level system |0  |1  |0  Qubits as long as you don’t look ! quantum measurement  flipping a coin |1 |0 |0 + |1 or

15. The routine –Initialization (e.g. all qubits are in state |0 〉 –Quantum computations –Reading of the result (measurement) How quantum computer works Quantum Memory Quantum Logic Unit Classical Computer

16. 0 0 0 0 0 0 0... 0 01 01 01 01 01 01 01... 01 1 0 0 0 0 0 0... 0... 1 1 1 1 1 1 1... 1 state 1 state 2 state 2 n n qubits Quantum parallelism Due to quantum superposition, a quantum computer with n-qubit register can simultaneously work with 2 n classical n-bit strings

17. Quantum circuit model Classical Quantum Unit: bit Unit: qubit 1. Prepare n-bit input 1.Prepare n-qubit input in the computational basis. 2. 1- and 2-bit logic gates 2.Unitary 1- and 2(3)-qubit quantum logic gates 3. Readout value of bits 3. Readout partial information about qubits External control by a classical computer.

18. Criteria for a Quantum Computer 1.Scalable system with well-defined qubits 2.Initializable to a simple fiducial state 3.Long decoherence time 4.Universal set of quantum gates 5.Permit efficient, qubit-specific measurements David DiVincenzo The Physical Implementation of Quantum Computation arXiv:quant-ph/0002077v3

19. Classical logical gates Universal gate sets SWAP gate FANOUT gate AND OR NOT FANOUT Two-gate set: NAND and FANOUT redundant

20. Quantum not gate: Input qubitOutput qubit Matrix representation: One-qubit quantum logic gates

21. Some other single-qubit quantum gates Hadamard gate Phase gate gate

22. Two-and three-qubit gates Control qubit 1 Target qubit Control qubit 2 Control NOT (CNOT) gate CNOT gate Control Target

23. How to clone bits? The quantum fanout Quantum cloning machine?

24. Can we copy a qubit in the unknown state ? The quantum fanout No-cloning theorem

25. Quantum Entanglement: Einstein’s “Spooky action-at-a-distance” or “superposition” “entangled superposition” Entanglement is quantum information resource (teleportation, superdense coding, quantum key distribution,…) Einstein-Podolsky-Rosen (EPR) or Bell pair

26. Example: circuit generating EPR states

27. Quantum entanglement AliceBob Schroedinger (1935): “I would not call [entanglement] one but rather the characteristic trait of quantum mechanics, the one that enforces its entire departure from classical lines of thought.”

28. Alice Bob ab Superdense coding ab Theorist’s impression of a measuring device

29. Alice Bob ab Superdense coding ab

30. Alice Bob ab Superdense coding ab

31. Superdense coding Goal: transmit some classical information from Alice to Bob Condition: Alice wishes to send two classical bits of information to Bob but allowed to send only single qubit Solution: Alice and Bob share a pair of qubits in the entangled qubit Alice applies I, X, Z, or iY gate to her qubit depending on the classical information she wishes to send and then sends her qubit to Bob. Then Bob does the measurement on the qubits in the Bell basis and determine the classical information sent by Alice

32. Superdense coding can be viewed as a statement about the interchangeability of physical resources. Superdense coding

33. Teleportation AliceBob

34. Teleportation AliceBob 01

35. Teleportation AliceBob

36. Teleportation AliceBob 01

37. How to measure in the EPR basis? Output states Measurement with 100% probability Input EPR states

38. Teleportation can be viewed as a statement about the interchangeability of physical resources. Compare with superdense coding:

39. Recent achievements in teleportation Dutch research group led by Prof. Ronald Hanson, Inst. of Nanoscience, Delft Univ. of Technology, managed to teleport information encoded in diamond spin qubits between two points spaced 3 meters apart from each other with 100% accuracy (DOI: 10.1126/science.1253512, May 29,2014) Hanson: “If you consider that we are nothing more than a collection of atoms joined together in a certain way, it seems theoretically possible to teleport ourselves from one place to another. Practically, this would be very unlikely, but not impossible. I would not exclude it simply because there is no fundamental natural law that prevents it. But if it ever will be possible, it will take place in the distant future”.

40. Recent achievements in teleportation (cont.) S.Filippov (Moscow Institute of Science and Technology) together with M.Ziman (Institute of Physics, Bratislava) found a way to keep quantum entanglement during the attenuation and amplification at a long-distance transmission via a noisy channel Phys. Rev. A 90, 010301(R) (2014) –July 14, 2014

41. What fundamental problems are addressed by quantum information science?

42. What high-level principles are implied by quantum mechanics?

43. Superfluidity, like the fractional quantum Hall effect, is an emergent phenomenon – a low-energy collective effect of huge numbers of particles that cannot be deduced from the microscopic equations of motion in a rigorous way and that disappears completely when the system is taken apart (Anderson, 1972)” “I give my class of extremely bright graduate students, who have mastered quantum mechanics but are otherwise unsuspecting and innocent, a take-home exam in which they are asked to deduce superfluidity from first principles. There is no doubt a special place in hell being reserved for me at this very moment for this mean trick, for the task is impossible. Robert B. Laughlin, 1998 Nobel Lecture

44. Quantum processes teleportation communication cryptography Shor’s algorithm quantum error-correction quantum phase transitions Quantum information science as an approach to the study of complex quantum systems theory of entanglement

45. Problem statement Problem: Description of entanglement space Relevant mathematical objects: Group polynomial invariants Computer algebra based computational tool: Gröbner bases and solvers of polynomial inequalities

46. Density matrix and orbit space A complete information on N -dimensional quantum system is accumulated in its density N × N matrix ρ ρ = ρ † – (self-adjointness) ρ ≥ 0 – (positive semi-definiteness) Tr (ρ) = 1 – (unit trace) Let Ρ be a set of all possible density matrices and N = N 1 × N 2 (bipartite system). Then the entanglement space is the factor set

47. Example: 4×4 density matrix Gerdt, Palii, Khvedelidze’11

48. Factoring Problem Given a positive integer N, find a nontrivial factor of N if N is a composite number. Best-known classical algorithm (for n > 100): general number field sieve (GNFS) Best-known quantum algorithm: Shor’s factoring algorithm (Shor 1994) P.Shor

49. Factoring in theory P NP NP- complete NPI Integer factoring

50. Factorization in practice http://www.crypto-world.com/FactorRecords.html numberdigitsdate completedsieving timealgorithm C1161161990275 MIPS years mpqs RSA-120120June, 1993830 MIPS years mpqs RSA-129129April, 19945000 MIPS years mpqs RSA-130130April, 19961000 MIPS years gnfs RSA-140140February, 19992000 MIPS years gnfs RSA-155155August, 19998000 MIPS years gnfs C158158January, 2002 3.4 Pentium 1GHz CPU years gnfs RSA-160160March, 2003 2.7 Pentium 1GHz CPU years gnfs RSA-576174December, 2003 13.2 Pentium 1GHz CPU years gnfs C176176May, 2005 48.6 Pentium 1GHz CPU years gnfs RSA-200200May, 2005 121 Pentium 1GHz CPU years gnfs RSA-768232Dec, 2009 3,300 Opteron 1GHz CPU years gnfs

51. Factoring of a 300-digit integer. The best classical algorithm would take 5 x 10 24 steps, or about 150,000 years at terahertz speed. A quantum computer would take only 5 x 10 10 steps, or less then a second at terahertz speed. M.Nielsen. Scientific American 67, 2002