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NMR: Shor algorithm - Experimental realization Patrik Caspar, Fadri Grnenfelder Patrik Caspar, Fadri Grnenfelder 20.05.2016 1 Outline Motivation Recapitulation: Shors algorithm Examples: N = 15, a = 11, 7 Quantum Part NMR techniques


  1. NMR: Shor algorithm - Experimental realization Patrik Caspar, Fadri Grünenfelder Patrik Caspar, Fadri Grünenfelder 20.05.2016 1

  2. Outline Motivation Recapitulation: Shor’s algorithm Examples: N = 15, a = 11, 7 Quantum Part NMR techniques Experimental setup Molecule Pulses Decoherence Readout Other experiments Patrik Caspar, Fadri Grünenfelder 20.05.2016 2

  3. Motivation • Shor’s algorithm in general: Patrik Caspar, Fadri Grünenfelder 20.05.2016 3

  4. Motivation • Shor’s algorithm in general: – Goal: Efficient prime factorization of L bit number N Patrik Caspar, Fadri Grünenfelder 20.05.2016 3

  5. Motivation • Shor’s algorithm in general: – Goal: Efficient prime factorization of L bit number N – Speedup compared to classical algorithm: Patrik Caspar, Fadri Grünenfelder 20.05.2016 3

  6. Motivation • Shor’s algorithm in general: – Goal: Efficient prime factorization of L bit number N – Speedup compared to classical algorithm: – Tool for breaking public key cryptosystems Patrik Caspar, Fadri Grünenfelder 20.05.2016 3

  7. Motivation • Shor’s algorithm in general: – Goal: Efficient prime factorization of L bit number N – Speedup compared to classical algorithm: – Tool for breaking public key cryptosystems • NMR implementation: Patrik Caspar, Fadri Grünenfelder 20.05.2016 3

  8. Motivation • Shor’s algorithm in general: – Goal: Efficient prime factorization of L bit number N – Speedup compared to classical algorithm: – Tool for breaking public key cryptosystems • NMR implementation: – Demonstration of experimental techniques for quantum computation with NMR Patrik Caspar, Fadri Grünenfelder 20.05.2016 3

  9. Motivation • Shor’s algorithm in general: – Goal: Efficient prime factorization of L bit number N – Speedup compared to classical algorithm: – Tool for breaking public key cryptosystems • NMR implementation: – Demonstration of experimental techniques for quantum computation with NMR – Implementation of Shor’s algorithm for N = 15 Patrik Caspar, Fadri Grünenfelder 20.05.2016 3

  10. Recapitulation: Shor’s algorithm Patrik Caspar, Fadri Grünenfelder 20.05.2016 4

  11. Recapitulation: Shor’s algorithm Patrik Caspar, Fadri Grünenfelder 20.05.2016 4

  12. Recapitulation: Shor’s algorithm Patrik Caspar, Fadri Grünenfelder 20.05.2016 4

  13. Recapitulation: Shor’s algorithm Patrik Caspar, Fadri Grünenfelder 20.05.2016 4

  14. Recapitulation: Shor’s algorithm Patrik Caspar, Fadri Grünenfelder 20.05.2016 4

  15. Recapitulation: Shor’s algorithm Patrik Caspar, Fadri Grünenfelder 20.05.2016 4

  16. Examples: N = 15, a = 11, 7 Patrik Caspar, Fadri Grünenfelder 20.05.2016 5

  17. Shor’s Algorithm - Quantum Part | ψ 1 � = | 0 � n | 1 � m L. M. K. Vandersypen et al. , Nature 414 ,883 (2001) Patrik Caspar, Fadri Grünenfelder 20.05.2016 6

  18. Shor’s Algorithm - Quantum Part | ψ 1 � = | 0 � n | 1 � m 2 n − 1 1 1 2 n / 2 ( | 0 � + | 1 � ) ⊗ n | 1 � m = � | ψ 2 � = | k � n | 1 � m 2 n / 2 k = 0 L. M. K. Vandersypen et al. , Nature 414 ,883 (2001) Patrik Caspar, Fadri Grünenfelder 20.05.2016 6

  19. Shor’s Algorithm - Quantum Part | ψ 1 � = | 0 � n | 1 � m 2 n − 1 1 1 2 n / 2 ( | 0 � + | 1 � ) ⊗ n | 1 � m = � | ψ 2 � = | k � n | 1 � m 2 n / 2 k = 0 2 n − 1 1 | k � n | a k mod N � m � | ψ 3 � = 2 n / 2 k = 0 L. M. K. Vandersypen et al. , Nature 414 ,883 (2001) Patrik Caspar, Fadri Grünenfelder 20.05.2016 6

  20. Shor’s Algorithm - Quantum Part 2 n − 1 1 | k � n | a k mod N � m � | ψ 3 � = 2 n / 2 k = 0 Basis change: x − 1 � − 2 π isk 1 � | a k mod N � m � √ x | u s � m := exp x k = 0 x − 1 1 � 2 π isk � | a k mod N � m = � √ x | u s � m exp x s = 0 L. M. K. Vandersypen et al. , Nature 414 ,883 (2001) Patrik Caspar, Fadri Grünenfelder 20.05.2016 7

  21. Shor’s Algorithm - Quantum Part 2 n − 1 x − 1 � � 2 n + 1 π isk 1 1 � � Therefore: | ψ 3 � = √ x | k � n exp | u s � m 2 n / 2 2 n x s = 0 k = 0 L. M. K. Vandersypen et al. , Nature 414 , 883 (2001) Patrik Caspar, Fadri Grünenfelder 20.05.2016 8

  22. Shor’s Algorithm - Quantum Part 2 n − 1 x − 1 � � 2 n + 1 π isk 1 1 � � Therefore: | ψ 3 � = √ x | k � n exp | u s � m 2 n / 2 2 n x s = 0 k = 0 x − 1 1 � | 2 n s / x � n | u s � m √ x | ψ 4 � = s = 0 Measurement outcome: 2 n s / x for some s in 0, ..., x − 1 L. M. K. Vandersypen et al. , Nature 414 , 883 (2001) Patrik Caspar, Fadri Grünenfelder 20.05.2016 8

  23. NMR techniques Manipulation: N z I j � � ω i 0 I i � 2 π J ij I i H = − z − z i = 1 i < j N � � γ i B 1 [ cos ( ω rf t + φ ) I i x − sin ( ω rf t + φ ) I i − y ] i = 1 L. M. K. Vandersypen and I. L. Chuang, Reviews of modern Physics 76 ,1037 (2004) Patrik Caspar, Fadri Grünenfelder 20.05.2016 9

  24. NMR techniques 2 Qubit effective pure state: α i = � ω i / k B T ≈ 10 − 5 ρ ∝ exp ( ω i I i z / k B T )   α 1 + α 2 0 0 0 4 + 1  α 1 − α 2  ρ = 1 4 + ρ ∆ = 1 0 0 0     4 − α 1 + α 2 0 0 0     − α 1 − α 2 0 0 0 Sum over permutations of the diagonal elements: Patrik Caspar, Fadri Grünenfelder 20.05.2016 10 A. Wallraff, Lecture Notes QSIT (2016)

  25. NMR techniques Readout: We can measure: � µ x + i µ y � = � γ Tr [ ρ ∆ ( I x + iI y )] Patrik Caspar, Fadri Grünenfelder 20.05.2016 11

  26. Experimental setup B 0 = 11.7 T I. L. Chuang et al., Proceedings of the Royal Society A 454 , pp. 447-467 (1998). Patrik Caspar, Fadri Grünenfelder 20.05.2016 12

  27. Quantum computer molecule L. M. K. Vandersypen et al., Nature 414 , 883 (2001) Patrik Caspar, Fadri Grünenfelder 20.05.2016 13

  28. Quantum computer molecule L. M. K. Vandersypen et al., Nature 414 , 883 (2001) Patrik Caspar, Fadri Grünenfelder 20.05.2016 13

  29. Refocusing In rotating frame of qubit A Patrik Caspar, Fadri Grünenfelder 20.05.2016 14

  30. Refocusing In rotating frame of qubit A Patrik Caspar, Fadri Grünenfelder 20.05.2016 14

  31. Pulse sequence For a = 7: ∼ 300 pulses (0.22 - 2 ms), total ∼ 720 ms π π − X-rotations (refocusing), 2 X-/Y-rotations, Z-rotations L. M. K. Vandersypen et al. (2001) Patrik Caspar, Fadri Grünenfelder 20.05.2016 15

  32. Pulse sequence For a = 7: ∼ 300 pulses (0.22 - 2 ms), total ∼ 720 ms π π − X-rotations (refocusing), 2 X-/Y-rotations, Z-rotations L. M. K. Vandersypen et al. (2001) Patrik Caspar, Fadri Grünenfelder 20.05.2016 15

  33. Decoherence Operator sum representation: �� � � E k ρ E † E † ρ → k E k = I k , k k L. M. K. Vandersypen et al., Nature 414 , 883 (2001) Patrik Caspar, Fadri Grünenfelder 20.05.2016 16

  34. Decoherence Operator sum representation: �� � � E k ρ E † E † ρ → k E k = I k , k k � ω p = 1 4 k B T , γ = 1 − e − t / T 1 Generalized amplitude damping ( T 1 ) : 2 + √ γ E 0 = √ p � � E 1 = √ p � � 1 0 0 √ 1 − γ , 0 0 0 � √ 1 − γ � � � 0 0 0 � � √ γ E 2 = 1 − p E 3 = 1 − p , 0 1 0 L. M. K. Vandersypen et al., Nature 414 , 883 (2001) Patrik Caspar, Fadri Grünenfelder 20.05.2016 16

  35. Decoherence Operator sum representation: �� � � E k ρ E † E † ρ → k E k = I k , k k � ω p = 1 4 k B T , γ = 1 − e − t / T 1 Generalized amplitude damping ( T 1 ) : 2 + √ γ E 0 = √ p � � E 1 = √ p � � 1 0 0 √ 1 − γ , 0 0 0 � √ 1 − γ � � � 0 0 0 � � √ γ E 2 = 1 − p E 3 = 1 − p , 0 1 0 λ ∼ 1 2 ( 1 + e − t / T 2 ) Phase damping ( T 2 ) : √ √ � � � � 1 0 1 0 E 0 = λ , E 1 = 1 − λ 0 1 0 − 1 L. M. K. Vandersypen et al., Nature 414 , 883 (2001) Patrik Caspar, Fadri Grünenfelder 20.05.2016 16

  36. Readout Thermal equilibrium state Effective pure ground state by adding multiple experiments L. M. K. Vandersypen et al. (2001) Patrik Caspar, Fadri Grünenfelder 20.05.2016 17

  37. Readout for a = 11 L. M. K. Vandersypen et al. (2001) Patrik Caspar, Fadri Grünenfelder 20.05.2016 18

  38. Readout for a = 7 L. M. K. Vandersypen et al. (2001) Patrik Caspar, Fadri Grünenfelder 20.05.2016 19

  39. Further experiments • 2009: Photonic chip (4 qubits) • 2012: Josephson phase qubit quantum processor (4 qubits) Patrik Caspar, Fadri Grünenfelder 20.05.2016 20

  40. Summary • First experimental realization of Shor’s factoring algorithm • Advantages: – long coherence times – high degree of control • Problems: – scaling – constant coupling Patrik Caspar, Fadri Grünenfelder 20.05.2016 21

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