The World of Quantum Information

1. School of Chemistry, Tel Aviv University The World of Quantum Information Marianna Safronova Department of Physics and Astronomy November 16, 2008

2. Outline • Quantum Information: fundamental principles (and how it is different from the classical one). • Bits & Qubits • Quantum weirdness: entanglement, superposition & measurement • Logic gates & Quantum circuits • Cryptography & quantum information • A brief introduction to quantum computing • Real world: what do we need to build a quantum computer/quantum network? • Current status & future roadmap

3. Why quantum information? Information is physical! Any processing of information is always performed by physical means Bits of information obey laws of classical physics.

4. Why quantum information? Information is physical! Any processing of information is always performed by physical means Bits of information obey laws of classical physics.

5. Why Quantum Computers? Computer technology is making devices smaller and smaller… …reaching a point where classical physics is no longer a suitable model for the laws of physics.

6. Fundamental building blocks of classical computers: BITS Fundamental building blocks of quantum computers: Quantum bits or QUBITS Basis states: and Bits & Qubits STATE: Definitely 0 or 1 Superposition:

7. Fundamental building blocks of classical computers: BITS Fundamental building blocks of quantum computers: Quantum bits or QUBITS Basis states: and Bits & Qubits STATE: Definitely 0 or 1

8. Qubit: any suitable two-level quantum system

9. Bits & Qubits:primary differences Superposition

10. Bits & Qubits:primary differences Measurement • Classical bit: we can find out if it is in state 0 or 1 and the measurement will not change the state of the bit. • Qubit: Quantum calculation: number of parallel processes due to superposition FR QO

11. Bits & Qubits:primary differences • Superposition • Measurement • Classical bit: we can find out if it is in state 0 or 1 and the measurement will not change the state of the bit. • Qubit: we cannot just measure a and b and thus determine its state! We get either or with corresponding probabilities |a|2and |b|2. • The measurement changes the state of the qubit!

12. Hilbert space is a big place! - Carlton Caves Multiple qubits

13. Hilbert space is a big place! - Carlton Caves Multiple qubits • Two bits with states 0 and 1 form four definite states 00, 01, 10, and 11. • Two qubits: can be in superposition of four computational basis set states. 2 qubits 4 amplitudes 3 qubits 8 amplitudes 10 qubits 1024 amplitudes 20 qubits 1 048 576 amplitudes 30 qubits 1 073 741 824 amplitudes 500 qubitsMore amplitudes than our estimate of number of atoms in the Universe!!!

14. Results of the measurement First qubit 0 1 Second qubit 0 1 Entanglement Entangled states

15. Quantum cryptography

16. Classical cryptography Scytale - the first known mechanical device to implement permutation of characters for cryptographic purposes

17. Classical cryptography Private key cryptography How to securely transmit a private key?

18. Key distribution A central problem in cryptography: the key distribution problem. • Mathematics solution: public key cryptography. • Physics solution: quantum cryptography. Public-key cryptography relies on the computational difficulty of certain hard mathematical problems (computational security) Quantum cryptography relies on the laws of quantum mechanics (information-theoretical security).

19. Quantum key distribution A quantum communication channel: physical system capable delivering quantum systems more or less intact from one place to another. • Quantum mechanics: quantum bits cannot be copied or monitored. • Any attempt to do so will result in altering it that can not be corrected. • Problems • Authentication • Noisy channels

20. Quantum logic gates

21. Classical NOTgate Quantum NOT gate (X gate) A NOTA Logic gates X Matrix form representation The only non-trivial single bit gate

22. Z H More single qubit gates Any unitary matrix U will produce a quantum gate! Hadamard gate:

23. Single qubit gates,two-qubit gates, three-qubit gates … • How many gates do we need to make? • Do we need three-qubit and four-qubit gates? • Where do we find such physical interactions? • Coming up with one suitable controlled interaction for physical system is already a problem!

24. A A ANDB B A A NANDB B Universality:classical computation Only one classical gate (NAND) is needed to compute any function on bits! A NOTA

25. Universality:quantum computation How many quantum gates do we need to build anyquantum gate? Any n-qubit gate can be made from 2-qubit gates. (Since any unitary n x n matrix can be decomposed to product of two-level matrices.) Only one two-qubit gate is needed! Example: CNOT gate

26. Quantum parallelism I can look at both sides of my coin at a single glance

27. x x Uf y f(x) Quantum parallelism Superposition Single circuit just evaluatedf(x) for both x=0 and 1 simultaneously!

28. Quantum parallelism:a major problem • So we can evaluate functions for all values of x at the same time using just one circuit! • Need only n+1 qubits to evaluate 2n values of x. • But we still get only one answer when we measure the result: it collapses to x,f(x)!!! FR QO

29. Quantum parallelism:a major problem • So we can evaluate functions for all values of x at the same time using just one circuit! • Need only n+1 qubits to evaluate 2n values of x. • But we still get only one answer when we measure the result: it collapses to x,f(x)!!! • We can’t read out the result for specific value of x! • With one qubit we get either f(0) or f(1) and we don’t know which one in advance. This is why we need quantum algorithms!

30. Quantum algorithms Unique features of quantum computation • Superposition: n qubits can represent 2n integers. • Problem: if we read the outcome we lose the superposition and we can’t know with certainty which one of the values we will obtain. • Entanglement: measurements of states of different qubits may be highly correlated.

31. Quantum algorithms Strategy: Use superposition to calculate 2n values of function simultaneously and do not read out the result until a useful result is expected with reasonably high probability. Use entanglement

32. Current advantages of quantum computation • Shor's quantum Fourier transform provides exponential speedup over known classical algorithms. • Applications: solving discrete logarithm and factoring problems which enables a quantum computer to break public key cryptosystems such as RSA. • Quantum searching (Grover's algorithm) allows quadratic speedup over classical computers. • Simulations of quantum systems.

33. How to factor 15? • Pick a number less then 15: 7 • Calculate 7n mod 15: • Calculate R=4

34. Shor’s algorithm for N=15 • Choose n such as 2n<15: n=4 • Choose y: y=7 • Initialize two four-qubit register • Create a superposition of states of the first register • Compute the function f(k)=7k mod 15 on the second register. • Operate on the first register by a Fourier transform • Measure the state of the first register: u=0, 4, 8, 12 are only non-zero results. • Two cases give period R=4, therefore the procedure succeeds with probability 1/2 after one run.

35. Back to the real world: What do we need to build a quantum computer? • Qubitswhich retaintheir properties. Scalable array of qubits. • Initialization: ability to prepare one certain state repeatedly on demand. Need continuous supply of . • Universal set of quantum gates. A system in which qubits can be made to evolve as desired. • Long relevant decoherence times. • Ability to efficiently read out the result.

36. Real world strategy “…If X is very hard it can be substituted with more of Y. Of course, in many cases both X and Y are beyond the present experimental state of the art …” David P. DiVincenzo The physical implementation of quantum computation.

37. P1/2 D5/2 „quantum bit“ S1/2 Quantum Computer(Innsbruck) Courtesy of the R. Blatt’s Innsbruck group

38. Experimental proposals • Liquid state NMR • Trapped ions • Cavity QED • Trapped atoms • Solid state schemes • And other ones …

39. The DiVincenzo criteria Quantum Information Processing with Neutral Atoms

40. Quantum computing with neutral atoms

41. 1. A scalable physical system with well characterized qubits: memory • Internal atomic state qubits: • ground hyperfine states of neutral trapped atoms • well characterized • Very long lived! MF=-2,-1,0,1,2 F=2 5s1/2 87Rb: Nuclear spin I=3/2 6.8 GHz F=1 MF=-1,0,1

42. 1. A scalable physical system with well characterized qubits: memory (b) Motional qubits : quantized levels in the trapping potential also well characterized

43. 1. A scalable physical system with well characterized qubits: memory • Internal atomic state qubits • (b) Motional qubits Advantages: very long decoherence times! Internal states are well understood: atomic spectroscopy & atomic clocks.

44. 1. A scalable physical system with well characterized qubits Optical lattices: loading of one atom per site may be achieved using Mott insulator transition. Scalability: the properties of optical lattice system do not change in the principal way when the size of the system is increased. Designer lattices may be created (for example with every third site loaded). Advantages: inherent scalability and parallelism. Potential problems: individual addressing.

45. 2: Initialization Internal state preparation: putting atoms in the ground hyperfine state Very well understood (optical pumping technique is in use since 1950) Very reliable (>0.9999 population may be achieved) Motional states may be cooled to motional ground states (>95%) Loading with one atom per site: Mott insulator transition and other schemes. Zero’s may be supplied during the computation (providing individual or array addressing).

46. 3: A universal set of quantum gates • Single-qubit rotations: well understood and had been carried out • in atomic spectroscopy since 1940’s. • 2. Two-qubit gates: none currently implemented • (conditional logic was demonstrated) • Proposed interactions for two-qubit gates: • Electric-dipole interactions between atoms • Ground-state elastic collisions • Magnetic dipole interactions • Only one gate proposal does not involve moving atoms (Rydberg gate). • Advantages: possible parallel operations • Disadvantages: decoherence issues during gate operations

47. Rydberg gate scheme Gate operations are mediated by excitation of Rydberg states Jaksch et al., Phys. Rev. Lett. 85, 2208 (2000) Why Rydberg gate?

48. Rydberg gate scheme Gate operations are mediated by excitation of Rydberg states Jaksch et al., Phys. Rev. Lett. 85, 2208 (2000) Do not need to move atoms! FAST!

49. Local blockade of Rydberg excitations Excitations to Rydberg states are suppressed due to a dipole-dipole interaction or van der Waals interaction http://www.physics.uconn.edu/~rcote/

50. D 1 2 Rydberg gate scheme FAST! Rb 40p Apply a series of laser pulses to realize the following logic gate: 5s Jaksch et al., Phys. Rev. Lett. 85, 2208 (2000)