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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

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NMR: Shor algorithm - Experimental realization

Patrik Caspar, Fadri Grünenfelder

Patrik Caspar, Fadri Grünenfelder 20.05.2016 1

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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

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Motivation

  • Shor’s algorithm in general:

Patrik Caspar, Fadri Grünenfelder 20.05.2016 3

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Motivation

  • Shor’s algorithm in general:

– Goal: Efficient prime factorization of L bit number N

Patrik Caspar, Fadri Grünenfelder 20.05.2016 3

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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

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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

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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

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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

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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

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Recapitulation: Shor’s algorithm

Patrik Caspar, Fadri Grünenfelder 20.05.2016 4

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Recapitulation: Shor’s algorithm

Patrik Caspar, Fadri Grünenfelder 20.05.2016 4

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Recapitulation: Shor’s algorithm

Patrik Caspar, Fadri Grünenfelder 20.05.2016 4

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Recapitulation: Shor’s algorithm

Patrik Caspar, Fadri Grünenfelder 20.05.2016 4

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Recapitulation: Shor’s algorithm

Patrik Caspar, Fadri Grünenfelder 20.05.2016 4

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Recapitulation: Shor’s algorithm

Patrik Caspar, Fadri Grünenfelder 20.05.2016 4

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Examples: N = 15, a = 11, 7

Patrik Caspar, Fadri Grünenfelder 20.05.2016 5

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Shor’s Algorithm - Quantum Part

|ψ1 = |0n |1m

  • L. M. K. Vandersypen et al., Nature 414,883 (2001)

Patrik Caspar, Fadri Grünenfelder 20.05.2016 6

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Shor’s Algorithm - Quantum Part

|ψ1 = |0n |1m |ψ2 =

1 2n/2 (|0 + |1)⊗n |1m = 1 2n/2

2n−1

  • k=0

|kn |1m

  • L. M. K. Vandersypen et al., Nature 414,883 (2001)

Patrik Caspar, Fadri Grünenfelder 20.05.2016 6

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Shor’s Algorithm - Quantum Part

|ψ1 = |0n |1m |ψ2 =

1 2n/2 (|0 + |1)⊗n |1m = 1 2n/2

2n−1

  • k=0

|kn |1m |ψ3 =

1 2n/2

2n−1

  • k=0

|kn |ak mod Nm

  • L. M. K. Vandersypen et al., Nature 414,883 (2001)

Patrik Caspar, Fadri Grünenfelder 20.05.2016 6

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Shor’s Algorithm - Quantum Part

|ψ3 =

1 2n/2

2n−1

  • k=0

|kn |ak mod Nm

Basis change:

|usm :=

1

√x

x−1

  • k=0

exp −2πisk x

  • |ak mod Nm

|ak mod Nm =

1

√x

x−1

  • s=0

exp 2πisk x

  • |usm
  • L. M. K. Vandersypen et al., Nature 414,883 (2001)

Patrik Caspar, Fadri Grünenfelder 20.05.2016 7

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Shor’s Algorithm - Quantum Part

Therefore: |ψ3 = 1

√x

x−1

  • s=0

1 2n/2

2n−1

  • k=0

|kn exp

  • 2n+1πisk

2nx

  • |usm
  • L. M. K. Vandersypen et al., Nature 414, 883 (2001)

Patrik Caspar, Fadri Grünenfelder 20.05.2016 8

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Shor’s Algorithm - Quantum Part

Therefore: |ψ3 = 1

√x

x−1

  • s=0

1 2n/2

2n−1

  • k=0

|kn exp

  • 2n+1πisk

2nx

  • |usm

|ψ4 =

1

√x

x−1

  • s=0

|2ns/xn |usm

Measurement outcome: 2ns/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

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NMR techniques

Manipulation: H = −

N

  • i=1

ωi

0Ii z −

  • i<j

2πJijIi

zIj z

−

N

  • i=1

γiB1[cos(ωrft + φ)Ii

x − sin(ωrft + φ)Ii y]

  • L. M. K. Vandersypen and I. L. Chuang, Reviews of modern Physics 76,1037

(2004)

Patrik Caspar, Fadri Grünenfelder 20.05.2016 9

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NMR techniques

2 Qubit effective pure state:

αi = ωi/kBT ≈ 10−5 ρ ∝ exp(ωiIi

z/kBT)

ρ = 1

4 + ρ∆ = 1 4 + 1 4      

α1 + α2 α1 − α2 −α1 + α2 −α1 − α2

      Sum over permutations of the diagonal elements:

  • A. Wallraff, Lecture Notes QSIT (2016)

Patrik Caspar, Fadri Grünenfelder 20.05.2016 10

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NMR techniques

Readout: We can measure: µx + iµy = γ Tr[ρ∆(Ix + iIy)]

Patrik Caspar, Fadri Grünenfelder 20.05.2016 11

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Experimental setup

B0 = 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

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Quantum computer molecule

  • L. M. K. Vandersypen et al., Nature 414, 883 (2001)

Patrik Caspar, Fadri Grünenfelder 20.05.2016 13

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Quantum computer molecule

  • L. M. K. Vandersypen et al., Nature 414, 883 (2001)

Patrik Caspar, Fadri Grünenfelder 20.05.2016 13

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Refocusing

In rotating frame of qubit A

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Refocusing

In rotating frame of qubit A

Patrik Caspar, Fadri Grünenfelder 20.05.2016 14

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Pulse sequence

For a = 7: ∼300 pulses (0.22 - 2 ms), total ∼720 ms

π 2 X-/Y-rotations,

π −X-rotations (refocusing),

Z-rotations

  • L. M. K. Vandersypen et al. (2001)

Patrik Caspar, Fadri Grünenfelder 20.05.2016 15

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Pulse sequence

For a = 7: ∼300 pulses (0.22 - 2 ms), total ∼720 ms

π 2 X-/Y-rotations,

π −X-rotations (refocusing),

Z-rotations

  • L. M. K. Vandersypen et al. (2001)

Patrik Caspar, Fadri Grünenfelder 20.05.2016 15

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Decoherence

Operator sum representation: ρ →

  • k

EkρE†

k ,

  • k

E†

k Ek = I

  • L. M. K. Vandersypen et al., Nature 414, 883 (2001)

Patrik Caspar, Fadri Grünenfelder 20.05.2016 16

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Decoherence

Operator sum representation: ρ →

  • k

EkρE†

k ,

  • k

E†

k Ek = I

  • Generalized amplitude damping (T1):

p = 1

2 + ω 4kBT , γ = 1 − e−t/T1

E0 = √p

  • 1

√1 − γ

  • ,

E1 = √p

  • √γ
  • E2 =
  • 1 − p

√1 − γ 1

  • ,

E3 =

  • 1 − p
  • √γ
  • L. M. K. Vandersypen et al., Nature 414, 883 (2001)

Patrik Caspar, Fadri Grünenfelder 20.05.2016 16

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Decoherence

Operator sum representation: ρ →

  • k

EkρE†

k ,

  • k

E†

k Ek = I

  • Generalized amplitude damping (T1):

p = 1

2 + ω 4kBT , γ = 1 − e−t/T1

E0 = √p

  • 1

√1 − γ

  • ,

E1 = √p

  • √γ
  • E2 =
  • 1 − p

√1 − γ 1

  • ,

E3 =

  • 1 − p
  • √γ
  • Phase damping (T2):

λ ∼ 1

2(1 + e−t/T2)

E0 = √ λ

  • 1

1

  • ,

E1 = √ 1 − λ

  • 1

−1

  • L. M. K. Vandersypen et al., Nature 414, 883 (2001)

Patrik Caspar, Fadri Grünenfelder 20.05.2016 16

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Readout

Thermal equilibrium state Effective pure ground state by adding multiple experiments

  • L. M. K. Vandersypen et al. (2001)

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Readout

for a = 11

  • L. M. K. Vandersypen et al. (2001)

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Readout

for a = 7

  • L. M. K. Vandersypen et al. (2001)

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Further experiments

  • 2009: Photonic chip (4 qubits)
  • 2012: Josephson phase qubit quantum processor (4 qubits)

Patrik Caspar, Fadri Grünenfelder 20.05.2016 20

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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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References

  • 1. Vandersypen, L. M. K. et al. Experimental realization of Shor’s

quantum factoring algorithm using nuclear magnetic resonance. Nature 414, 883 (2001).

  • 2. Gershenfeld, N. A. and Chuang, I. L. Bulk Spin-Resonance Quantum
  • Computation. Science 275, 350 (1997).
  • 3. Vandersypen L. M. K. and Chuang, I. L. NMR techniques for quantum

control and computation. Review of Modern Physics 76, 1037 (2004).

  • 4. Lucero, E. et al. Computing prime factors with a Josephson phase

qubit quantum processor. Nature Physics 8, 719 (2012).

  • 5. Politi, A., Matthews, J. C. F. and O’Brien, J. L. Shor’s Quantum

Factoring Algorithm on a Photonic Chip. Science 325, 1221 (2009).

  • 6. Chuang, I. L., Gershenfeld, N., Kubinec, M. G. and Leung, D. W. Bulk

Quantum Computation with Nuclear Magnetic Resonance: Theory and Experiment. Proceedings of the Royal Society A 454, pp. 447-467 (1998).

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