1. Introduction to Quantum Computers Goren Gordon The Gordon Residence July 2006

2. Outline Introduction to quantum physics –Superposition Software –Deutsch-Jozsa algorithm –Grover search algorithm –No Shor for you, come back one year!!! Hardware Why arent there any QC around? The weird stuff –Cluster state quantum computers

3. Classical (regular) computer 0 or 1 Introduction to Quantum Physics Classical bit (binary): 00 1 f(00)

4. Classical (regular) computer 0 or 1 Introduction to Quantum Physics Classical bit (binary): 01 2 f(01)

5. Classical (regular) computer 0 or 1 Introduction to Quantum Physics Classical bit (binary): 10 3 f(10)

6. Classical (regular) computer 0 or 1 Introduction to Quantum Physics Classical bit (binary): 11 4 f(11) One computation per input number 2 N computation for N bits Computational complexity: how many computations as a function of number of bits

7. Classical (regular) computer Introduction to Quantum Physics 01f(01) (Classical) Parallel computing 10f(10) 11f(11) 00f(00) Faster Same number of computations Same computational complexity Slow simple computation VERY large parallelism

8. The Dream… Introduction to Quantum Physics 00 and 01 and 10 and 11 1 f(00) and f(01) and f(10) and f(11) One computation for ALL possible numbers How can this happen?

9. Quantum Superposition 0 and 1 A quantum bit (qubit): |0> + |1> You can process all the numbers at the same time !!! Introduction to Quantum Physics In the quantum world you can have: |00> +|01>+|10>+|11> 1 |f(00)>+|f(01)>+|f(10)>+|f(11)>

10. Quantum Superposition What does it mean to have a superposition? A qubit: a|0> + b|1> |0> |1> If I open the boxes, (measurement) Probability a 2 to be in box |0> Probability b 2 to be in box |1> Closed boxes. Contain one particle. Introduction to Quantum Physics a 2 +b 2 =1 Ring a bell? cos 2 +sin 2 =1 Qubit: cos |0> +sin |1> An axiom of QM: Borns Rule a and b are numbers

11. Quantum Measurement Introduction to Quantum PhysicsExample: Polarization of light |0> |1> |0> |1> |0>+|1> |0> 0 |1> 90 |0> + |1> 45 superposition Rotation by 45 |0> OR |1> 0 90 45 OR 135 Measure: 50% |0>, 50% |1> Classical |0> + |1> 45 |1> 90 Measure: 100% |1> Quantum Rotation by 45 |0>+|1> (+|0>+|1>)/2 + (-|0>+|1>)/2 = |1> Cancel out Qubit: cos |0> +sin |1> +

12. Quantum Measurement Distinguishability: two states are distinguishable if I can tell with 100% which state I have. Introduction to Quantum Physics |0>, |1> are distinguishable |0>, |0>+|1> are not distinguishable: if I measure and get |0>, there is 50% that the state was |0>+|1>. Example: Polarization of light |0> |1> |0>+|1>, |0>-|1> are distinguishable |0> |1> |0>+|1> |0>-|1> The sign matters

13. Quantum Logic Gate Introduction to Quantum Physics |0> |1> |0>+|1> |0>-|1> Single qubit logic gate: Rotation Hadamard |0> |0> +|1> |1> |0> - |1> Not|0> |1> |1> |0> Example: Polarization of light |0> |1>

14. Computation with Quantum Superposition A qubit: a|0> + b|1> Introduction to Quantum Physics qubit Logic gate output |0> NOT |1> NOT |0> a|0>+b|1> NOT a|1>+b|0> Two operations of the Gate in one step!!!

15. Computation with Quantum Superposition Many qubits: | >=a 0 |000> + a 1 |001>+…+a 8 |111> Introduction to Quantum Physics | > Complex Computation | >=a 0 |f(000)> + a 1 |f(001)>+…+a 8 |f(111)> Many operations of the function in one step!!! Classical computation: x f(x)

16. Computation with Quantum Superposition Many qubits: | >=a 0 |000> + a 1 |001>+…+a 8 |111> Introduction to Quantum Physics | >=a 0 |f(000)> + a 1 |f(001)>+…+a 8 |f(111)> Result of computation: Problem: We need to measure the result!!! Each measurement gives only one result of the calculation!!! With probability a 0 2 we will get f(000), With probability a 1 2 we will get f(001), … With probability a 8 2 we will get f(111) Does not save time, or number of calculations!!!

17. Deutsch-Jozsa algorithm Quantum Algorithms Function: f(x) Either constant: f(x) = +1 always f(x) = -1 always Or balanced:f(x) = +1 half of the time -1 half of the time x=0-8 (3 bits) f(000) = +1 f(001) = -1 f(010) = +1 f(011) = +1 f(100) = -1 f(101) = -1 f(110) = +1 f(111) = -1 Balanced f(000) = +1 f(001) = +1 f(010) = +1 f(011) = +1 f(100) = +1 f(101) = +1 f(110) = +1 f(111) = +1 Constant

18. Deutsch-Jozsa algorithm Quantum Algorithms Function: f(x) Either constant, or balanced. x=0-8 (3 bits) How many calculations of f(x) do I need to do to know if it is constant or balanced? On a classical computer we need: At worst 5 calculations (more than half) f(000) = +1 f(001) = +1 f(010) = +1 f(011) = +1 f(100) = -1 Balanced Computational complexity: 2 N-1 +1

19. Deutsch-Jozsa algorithm Quantum Algorithms Function: f|x>=±|x> Either constant, or balanced. | >=|000>+|001>+|010>+|011>+|100>+|101>+|110>+|111> On a quantum computer: Only one calculation: Constant: f| >=±(|000>+|001>+|010>+|011>+|100>+|101>+|110>+|111>) Balanced: f| >=|000>-|001>+|010>+|011>-|100>-|101>+|110>-|111>

20. Deutsch-Jozsa algorithm Quantum Algorithms Constant: f c | >=±(|000>+|001>+|010>+|011>+|100>+|101>+|110>+|111>) Balanced: f b | >=|000>-|001>+|010>+|011>-|100>-|101>+|110>-|111> |0>+|1>, |0>-|1> are distinguishable |0> |1> |0>+|1> |0>-|1> The sign matters Reminder:

21. Deutsch-Jozsa algorithm Quantum Algorithms Constant: f c | >=±(|000>+|001>+|010>+|011>+|100>+|101>+|110>+|111>) Balanced: f b | >=|000>-|001>+|010>+|011>-|100>-|101>+|110>-|111> These two states are distinguishable One can make a measurement to distinguish between the two states. Only one calculation of the function is needed to know if it is constant or balanced!!!

22. Grovers Search Algorithm Quantum Algorithms Find a specific number out of N numbers. Example: Searching in a database 1 3 2 5 6 8 7 4 Is it the right number? NO YES!!!

23. Grovers Search Algorithm Quantum Algorithms Find a specific number out of N numbers. Example: Searching in a database Given: f(x) = -1 for a specific (unknown) x +1 for all other x Goal: Find x Classical computer: Worst case: Go over all x until you find. Quantum computer: Use superposition to shorten the search

24. Grovers Search Algorithm Quantum Algorithms Stage 1: Prepare | > = |000>+|001>+…+|111> (superposition of all states) Stage 2: Do: 2.a. Apply f| >calculate f 2.b. Apply 2| >< |-Ido another simple calculation Stage 3: measure Example: | >=a|0>+b|1> 2.b. (2a-1)|0> + (2b-1)|1>

25. Grovers Search Algorithm Quantum Algorithms Example: 2 qubits, f(|01>)=-1 Stage 1: ½|00>+½|01>+½|10>+½|11> Stage 2.a. ½|00>-½|01>+½|10>+½|11> Stage 2.b.|01> Stage 3. Measure |01> (2x½-1)=0, (2x(-½)-1)=-1 We applied f(x) only once !!! Normalization: (½) 2 +(½) 2 +(½) 2 +(½) 2 =1 f(|01>)=-1

26. Grovers Search Algorithm Quantum Algorithms Stage 1: Prepare | > = |000>+|001>+…+|111> (superposition of all states) Stage 2: Do N times: 2.a. Apply f| > 2.b. Apply 2| >< |-I Stage 3: measure |x> | > |x > Final result: Instead of N times in the classical computer You need N times in the quantum computer Desired Unknown state Orthogonal to desired state Very small error

27. One word on Shor The algorithm that started everything Proves that a quantum computers can break the RSA code in polynomial times Uses Fourier Transforms (and other mathematical stuff) Too complicated to show it here Quantum Algorithms

28. Software: Conclusions The quantum computer uses superposition The quantum algorithms are only useful for global, or collective results (Deustch- Jozsa) There are many (many) new quantum algorithms, which are exponentially faster than classical computers There isnt any quantum computer, yet Quantum Algorithms

29. Building a Quantum Computer Problems 1.Distinguishable qubitssingle particles 1.Preparationreproducibility 2.Readoutdeterministic 2.Single qubit gatescontrol, short 3.Two-qubits gatesinteraction 4.(noise issues)always Quantum Hardware

30. Building a Quantum Computer 1.Opticsqubits = polarization 2.Atoms qubits = electron energy levels 3.Moleculesqubits = nuclear spins 4.QDotsqubits = electron charge Quantum Hardware

31. Optics Quantum Hardware Distinguishable qubits: Polarization of single photons Preparation: single photon sources Usually, two photon sources © Stanford University Laser Strong filter Single photons Laser Special material Entangled photon pair D Single photons Polarization of light |0> |1> http://www.qcaustralia.org/crp_sl.htm

32. Optics Quantum Hardware Distinguishable qubits: Polarization of single photons Readout: single photon detectors © LC Technologies There are two ways a detector can fail: 1. It counts too few photons (loss); 2. it counts too many photons (dark counts). Polarization of light |0> |1> http://www.qcaustralia.org/crp_sl.htm

33. Optics Quantum Hardware Single photon gates: polarization rotation Two-photon gates: THE PROBLEM Photons do not interact. Solutions: 1.Non-liner materials – low efficiency 2.Non-deterministic schemes – probabilistic, requires auxiliary resources Non-linear |1> |0> XOR gate Polarization of light |0> |1> http://www.qcaustralia.org/crp_sl.htm

34. Optics Quantum Hardware Polarization of light |0> |1> © Stanford University © LC Technologies The whole setup Single photon source Single photons Computation Single photon detection http://www.qcaustralia.org/crp_sl.htm

35. Optics Pros: There are demonstrations of quantum computations with optics Low noise – good!!! Cons: Requires too many resources Not scalable (yet) Quantum Hardware Polarization of light |0> |1> http://www.qcaustralia.org/crp_sl.htm

36. Neutral Atoms Quantum Hardware THE PROBLEM: Working with single atoms Optical Tweezer Scattering force: Gradient Force: http://www.iap.uni-bonn.de/ag_meschede/english/index_eng.html

37. Neutral Atoms Quantum Hardware THE PROBLEM: Working with single atoms Source of atom beam Single atom Magneto Optical Trap (MOT) lasers magnets http://www.iap.uni-bonn.de/ag_meschede/english/index_eng.html

38. Neutral Atoms Quantum Hardware Optical conveyer belt Single atoms Moving and controlling single atoms http://www.iap.uni-bonn.de/ag_meschede/english/index_eng.html

39. Neutral Atoms Quantum Hardware Moving and controlling single atoms http://www.iap.uni-bonn.de/ag_meschede/english/index_eng.html

40. Neutral Atoms Quantum Hardware Moving and controlling single atoms http://www.iap.uni-bonn.de/ag_meschede/english/index_eng.html

41. Neutral Atoms Quantum Hardware Distinguishable qubits: electrons energy levels |0> |1> Energy levels electron |0> |1> |0>+|1> http://www.iap.uni-bonn.de/ag_meschede/english/index_eng.html

42. Neutral Atoms Quantum Hardware Distinguishable qubits: electrons energy levels |0> |1> Energy levels Resonant LASER: A laser with a specific frequency that Matches the energy levels laser Creates transition between the two levels |0> /4 pulse |0>+|1> |0> /2 pulse |1> http://www.iap.uni-bonn.de/ag_meschede/english/index_eng.html

43. Neutral Atoms Quantum Hardware Distinguishable qubits: electrons energy levels |0> |1> Energy levels Preparation: All electrons decay to |0> Readout: Usually fluorescence |0> |1> laser |0> |1> laser light Nothing happens Non-resonant Resonant http://www.iap.uni-bonn.de/ag_meschede/english/index_eng.html

44. Neutral Atoms Quantum Hardware |0> |1> Energy levels laser Single qubit gates: laser pulses |0> |1> Two qubit gate: Instead of LASER: Interaction between two atoms And EM field in the cavity |00> |10>|01> |11> laser SWAP gate: |01> |10> http://www.iap.uni-bonn.de/ag_meschede/english/index_eng.html

45. Neutral Atoms Quantum Hardware |0> |1> Energy levels The whole setup laser Single atom source computation readout http://www.iap.uni-bonn.de/ag_meschede/english/index_eng.html

46. Neutral Atoms Quantum Hardware |0> |1> Energy levels Pros: Single atom manipulation Scalability to many atoms Cons: Two-atom gate not accomplished yet A lot of noise http://www.iap.uni-bonn.de/ag_meschede/english/index_eng.html

47. Molecules Quantum Hardware Liquid state NMR (nuclear magnetic resonance) : Room temperature liquid of molecules Each one of the atoms of the molecule can be a qubit Distinguishable qubits: nuclear spins http://qso.lanl.gov/qc/

48. Several words on Spins Quantum Hardware Spin: self magnetic field Can have two values: Up / Down Energy level One of the spins have less energy In a magnetic field http://qso.lanl.gov/qc/

49. Molecules Quantum Hardware Spin: self magnetic field RF field Strong magnetic field RF (radio frequency) field Spins can become qubits |0> |1> http://qso.lanl.gov/qc/

50. Molecules Quantum Hardware Liquid state NMR (nuclear magnetic resonance) : A room temperature liquid of molecules Each one of the atoms of the molecule can be a qubit Distinguishable qubits: nuclear spins |0> |1> http://qso.lanl.gov/qc/

51. Molecules Quantum Hardware Preparation: THE PROBLEM At room temperature, the nuclear spins are a mess We want We have Some solutions: cool the liquid (new solid state NMR) |0> |1> http://qso.lanl.gov/qc/

52. Molecules Quantum Hardware Readout: Spectroscopy |0> |1> |0> |1> frequency intensity 91.8 http://qso.lanl.gov/qc/

53. Molecules Quantum Hardware Readout: spectroscopy |0> |1> http://qso.lanl.gov/qc/

54. Molecules Quantum Hardware Single and two-qubit gates: RF fields RF field |0> |1> http://qso.lanl.gov/qc/

55. Molecules Quantum Hardware |0> |1> The whole setup RF field PreparationComputationReadout http://qso.lanl.gov/qc/

56. Molecules Quantum Hardware |0> |1> Pros: Easy gates and readout Easy access to single qubits Cons: Not scalable: never more than 12 qubits Preparation problematic http://qso.lanl.gov/qc/

57. Quantum Dots Quantum Hardware Fabricated nanostructure trapping single electrons Single electron Distinguishable qubits: electron charge http://www.qcaustralia.org/crp_asd.htm

58. Quantum Dots Quantum Hardware Preparation: Putting the electrons in the right place Readout: reading the voltage of the circuit Single and two-qubits gates: applying the right voltages http://www.qcaustralia.org/crp_asd.htm

59. Quantum Dots Quantum Hardware Pros: Scalable Easy manufacture Cons: Hard to create two-qubit gate 3 qubits computation not yet demonstrated http://www.qcaustralia.org/crp_asd.htm

60. Why arent there any QC around? Noise \ loss \ decoherence The quantum information is lost due to interaction with environment: –Fluctuation in magnetic fields –Collision with hot particles Systems: –Photons polarization fluctuates with time –Electrons decay to lower levels \ lose phase –Nuclear spins fluctuates with time –Electrons spins fluctuates

61. Why arent there any QC around? The cohernece time = how long until 1% of the information is lost Quantum computation possible only when Coherence time >> Computation time In all systems, this is a problem

62. Why arent there any QC around? Scalability –if N qubits requires X resources, do 2N qubits require 2X? All systems are not yet scalable The status today: 1.NMR: 12 qubits 2.Ions: 8 qubits 3.Photons, Qdots, atoms: 1-3 qubits

63. The Weird Stuff Cluster state quantum computer All qubits are entangled Measure one qubit Change another according to result of measurement Continue until final result is left measureConditional gate result Final result of Computation http://arxiv.org/abs/quant-ph/0504097

64. Summary Quantum computers can do magic We are only in the beginning Software more advanced than hardware Competition between different setups Real quantum computers in 10-100 years Thank you!!!

65. Topics for next lecture 1.Entanglement and non-locality 2.Shor & Co. algorithms 3.Specific implementation of QC 4.Quantum Games