NMR: Shor algorithm - Experimental realization Patrik Caspar, Fadri - PowerPoint PPT Presentation

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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