Factoring 15 with NMR spectroscopy. Josefine Enkner, Felix Helmrich

Transcription

1 Factoring 15 with NMR spectroscopy Josefine Enkner, Felix Helmrich Josefine Enkner, Felix Helmrich April 23,

2 Introduction: What awaits you in this talk Recap Shor s Algorithm NMR Magnetic Nuclear Resonace Example: Factoring 15 using NMR Placeholder for organisational unit name / logo (edit in slide master via View > Slide Master ) Josefine Enkner, Felix Helmrich April 23,

3 Shor s Algorithm Solves N=p*q where N odd N is not a power of a prime The algorithm essetially consists of two parts: Reduction to a period-finding problem (classical (1) and (2)) A quantum algorithm to solve the orderfinding problem using QFT (3) Procedure: 1. Pick a random number 2. Comput gcd(b,n) 3. If than a is factor 4. Otherwise: find the period r with QFT of 5. If r is odd, go back to step 1 6. If go back to step 1 7. and are both prime factors of N Placeholder for organisational unit name / logo (edit in slide master via View > Slide Master ) Josefine Enkner, Felix Helmrich April 23,

4 Shor s Algorithm In essence we have to pick a number a with the period r such that so same as same as N devides LUCKY! So same as same as N does not divide EVEN LUCKIER! Placeholder for organisational unit name / logo (edit in slide master via View > Slide Master ) Josefine Enkner, Felix Helmrich April 23,

5 Shor s Algorithm (1) Apply a Hadamard transform H to the first n qubits, so the first register reaches (2) Multiply the second register by to get (3) Perform the inverse QFT on the first register, giving (4) Measure the qubits in the first register. The measurement outcome is c2^r for some c, and r can be quickly deduced from c2^n on a classical computer via continued fractions Placeholder for organisational unit name / logo (edit in slide master via View > Slide Master ) Josefine Enkner, Felix Helmrich ) April 23,

6 NMR spectroscopy Nuclear spins I in strong magnetic fields B0 Interaction with radio-frequency fields B1 (t) (100 to 1000 MHz) Discovery in 1946 independently by Felix Bloch and Edward Mills Purcell Richard R. Ernst, Nobel Prize in Chemistry 1991 Since then: evolution to an invaluable analytic technique in chemistry, biology and medicine (MRI) Felix Bloch, Nobel Prize in Physics 1952 Coherent quantum control at room temperature in a lab Kurt Wüthrich, Nobel Prize in Chemistry 2002 Josefine Enkner, Felix Helmrich April 23,

7 Experiment Nuclear spins I in strong homogeneous magnetic field B0 Manipulation with radio-frequency magnetic fields B 1 (t) Read-out: induced voltage in coil Ensemble of quantum computers Josefine Enkner, Felix Helmrich April 23,

8 Static field and chemical shift Zeeman effect H 0 = k γ k I k B0 = k γ kb 0 I k z = k ωk 0 Ik z γ k : gyromagnetic ratio, ω k 0 = γ kb 0 : Larmor frequency Time evolution ψ(t) = e ih0t/ ψ 0 precession around z axis of Bloch sphere Chemical shift Electrons respond to the magnetic field (diamagnetic/paramagnetic) ω k 0 = γ k(1 σ k )B 0 spectral distinction of spins Isotropic in liquid-state NMR Josefine Enkner, Felix Helmrich April 23,

9 Radio-frequency field Control rf field B 1 (t) H control = k γ k B 1 [cos(ω rf t + φ)i k x sin(ω rf t + φ)i k y ] H rot = k ω k 1{cos[(ω rf ω k 0 )t + φ]ik x (Phase φ determines the axis of the rf pulse.) sin[(ω rf ω k 0 )t + φ]ik y }, ω k 1 = γ kb 1 Rotating frame [ ] ψ rot = e iωk 0 t ψ Simplified dynamics nutation about the control field k Josefine Enkner, Felix Helmrich April 23,

10 J coupling Scalar spin-spin coupling in the liquid state H coupling = 2π j<k J jki j zi k z Fermi contact interaction A = 2 3 µ 0 µ e µ n Ψ(0) 2 J in the range of 10 Hz to 100 Hz Splitting of spectral lines Coupling through one or more bonds Josefine Enkner, Felix Helmrich April 23,

11 Refocusing "turning off" unwanted J couplings Two spins (rotating frame) Four spins: only J 12 remains active Only qubit 1 shown, solid arrow qubit 2 in 0, dashed arrow qubit 2 in 1 + spin in original position, spin inverted Josefine Enkner, Felix Helmrich April 23,

12 C-NOT gate Matrix representation CNOT 12 = CNOT 21 is similar with swapped control and target qubits. Angle for delay τ: πjτ/2 Josefine Enkner, Felix Helmrich April 23,

13 Temporal averaging Mixed initial state with ω 0 k B T ρ 0 = e H/(kBT ) tr1 Create pseudo-pure state by permuting the populations and average different measurements. a b 0 0 ρ 0 = 0 0 c d trρ 0 = a + b + c + d = 1 Permutation matrix P = CNOT 1 CNOT 2 a ρ 1 = Pρ 0 P 0 c d b a ρ 2 = P 0 d 0 0 ρ 0 P 0 0 b c Josefine Enkner, Felix Helmrich April 23,

14 Temporal averaging Propagate each ρ i by unitary U (quantum circuit) and measure the observable A (traceless). C i = A ρi = tr(uρ i U A) Averaged measurement C = 1 3 i C i By linearity C = tr( Uρ i U A) i 3a b + c + d 0 0 = tru 0 0 c + d + b 0 U A d + b + c = tr[u(4a 1) U A] + tr[u(1 a)1u A] = (4a 1)tr[U U A]. Same as for a pure initial state! Improved schemes only need (2 n 1)/n (instead of 2 n 1) experiments. Josefine Enkner, Felix Helmrich April 23,

15 Example: Factoring 15 Example N=15 and a=7 Remember: pick an a and find r (even) such that And and a For N=15, a may be 2, 4, 7, 8, 11, 13, or 14 a=11 easy case a=7 difficult case find the period r=4 compute giving 5 and 3 We need 7 qubits to implement this algorithm: n=3 qubits for the first register m=4 qubits for the second register Placeholder for organisational unit name / logo (edit in slide master via View > Slide Master ) Josefine Enkner, Felix Helmrich April 23,

16 The molecule Why such a complicated molecule? No magnetically equivalent spins asymmetry J coupling between all spins Sufficiently large coherence times. ω k 0 /2π (Hz) in 11.7 T field relative to 470 MHz (19 F), 127 MHz ( 13 C) reference. T 1 and T 2 times in s. J coupling constants in Hz. Josefine Enkner, Felix Helmrich April 23,

17 Implementation: Pulse pattern Placeholder for organisational unit name / logo (edit in slide master via View > Slide Master ) Josefine Enkner, Felix Helmrich April 23,

18 State estimation: NMR spectroscopy Easy case: Qubit 1 and 2 are in and 3 is in an equal mixture of and First register is thus in a mixture of and or and giving a periodicity 4 for the amplitude Period is 2 from And we find Placeholder for organisational unit name / logo (edit in slide master via View > Slide Master ) Josefine Enkner, Felix Helmrich April 23,

19 State estimation: NMR spectroscopy Difficult case: Qubit 1 is in and 2, 3 are in a mixture of and First register is thus in a mixture of and, or and giving a periodicity of 2 The period then is 2 And we find Placeholder for organisational unit name / logo (edit in slide master via View > Slide Master ) Josefine Enkner, Felix Helmrich April 23,

20 Conclusion First experimental demonstration of Shor s quantum factoring algorithm. Scalability remains difficult. Josefine Enkner, Felix Helmrich April 23,