DNP in Quantum Computing Eisuke Abe Spintronics Research Center, Keio University

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1 DNP in Quantum Computing Eisuke Abe Spintronics Research Center, Keio University Future of Hyper-Polarized Nuclear Osaka

2 DNP in quantum computing Molecule Pseudo-pure state Algorithmic cooling Phosphorus donor in silicon ENDOR Single nuclear spin under control

3 DiVincenzo s criteria. A scalable physical system with well characterized qubits Spin-½ 2. The ability to initialize the state of the qubits to a simple fiducial state, such as Long relevant decoherence times, much longer than the gate operation time 4. A universal set of quantum gates 5. A qubit-specific measurement capability ( RWTH Aachen U.) Fortschr. Phys. 48, 77 (2000) DiVincenzo

DiVincenzo s criteria. A scalable physical system with well characterized qubits Spin-½ 2. The ability to initialize the state of the qubits to a simple fiducial state, such as 000... 00% DNP 3.

4 DiVincenzo s criteria. A scalable physical system with well characterized qubits Spin-½ 2. The ability to initialize the state of the qubits to a simple fiducial state, such as % DNP 3. Long relevant decoherence times, much longer than the gate operation time 4. A universal set of quantum gates Arbitrary unitary operations 5. A qubit-specific measurement capability T 2 ( RWTH Aachen U.) Fortschr. Phys. 48, 77 (2000) DiVincenzo

5 A molecule as a quantum computer Hamiltonian H mol = ν i I z i i + J ii I z i I z j i<j Bromotrifluoroethylene Qubit B ν A,B,C ~ =.7 T 0 J AB = 22. Hz ν A ν B = 3.2 khz Qubit A J BC = 53.8 Hz 9 F Br C C 9 F 9 F J AC = 75.0 Hz ν C ν A = 9.5 khz Qubit C

6 A molecule as a quantum computer Hamiltonian H mol = ν i I z i i + J ii I z i I z j i<j Bromotrifluoroethylene J AB = 22. Hz ν A ν B = 3.2 khz Qubit B ν A,B,C ~ =.7 T J BC = 53.8 Hz 9 F Br C C 9 F 9 F ABC = 000 Qubit A J AC = 75.0 Hz ν C ν A = 9.5 khz Qubit C

7 DiVincenzo s criteria for a molecule. Qubits: Nuclear spins in a molecule (ABC) n polymer* 2. Initialization: Measure and flip 3. Coherence: Good 4. Quantum gates: RF control and spin-spin interaction (always on) 5. Measurement: *Science 26, 569 (993) Lloyd

8 Molecules as a quantum computer Hamiltonian H mol = ν i I z i i + J ii I z i I z j i<j Ensemble of molecules can be measured, but then how to initialize them? Thermal equilibrium ensemble (> 0 8 identical molecules)

9 Spin system in thermal equilibrium Hamiltonian Simplified notation of ρ n H mol = ν i I z i Density matrix ρ eq = Z exp ε = i H mol k B T 500 MHz K 0 5 ρ n = σ z i i + J ii I z i I z j i<j n ε 2n + 2 n ρ n ν 0 I z i Silent majority (no signal, no time evolution) ρ = σ z ρ 2 = σ z + σ z i d ρ 2 = d ρ 3 = ρ 3 = σ z + σ z + σ z d ρ =

Initialization Thermal equilibrium ρ eq = n ε + 2n 2 n ρ n ρ pure = 000 000 00% DNP d ρ pure = 0

10 Initialization Thermal equilibrium ρ eq = n ε + 2n 2 n ρ n ρ pure = % DNP d ρ pure = 0 0 ρ pps = α n 2 n + αρ pure Pseudo-pure state Temporal averaging Spatial averaging Logical labeling

11 Science 275, 350 (997) Gershenfeld & Chuang ( MIT) ( IQC, U. Waterloo) PNAS 94, 634 (997) Cory et al.

Phys. Rev. Lett. 80, 3408 (998) Chuang et al.

; published 4 May) Nature 393, 344 (998) Jones et al.

12 Phys. Rev. Lett. 80, 3408 (998) Chuang et al. (Received 2 Nov. 97; published 3 Apr.) Nature 393, 43 (998) Chuang et al. (Received 2 Jan.; accepted 8 Mar.; published 4 May) Nature 393, 344 (998) Jones et al. (Received 6 Mar.; accepted 23 Apr.; published 28 May) J. Chem. Phys. 09, 648 (998) Jones et al. (Received 6 Jan.; accepted 22 Apr.; published Aug.)

13 ( QuTech, TU Delft) ( IBM) Nature 44, 883 (200) Vandersypen et al.

14 DiVincenzo s criteria for NMR QC. Qubits: Nuclear spins in molecules (ABC) n polymers* 2. Initialization: DNP, pseudo-pure state, algorithmic cooling** 3. Coherence: Good 4. Quantum gates: RF control and spin-spin interaction (always on) 5. Measurement: Ensemble-averaged signals *Science 26, 569 (993) Lloyd **Proc. 3st Annu. ACM Symp. Theory Compt. p.322 (999) Schulman & Vazirani

15 Logical labeling Case study: n = 3 d ρ 3 = ρ ABC = 000 +

16 Logical labeling Case study: n = 3 d ρ 3 = 3 3 Unitary operator for LL 3 V = A B C A B Flip A if BC = 0 or 0 A B C B C

17 Logical labeling Case study: n = 3 d ρ 3 = 3 3 ρ 3 Unitary operator for LL 3 V = ABC = 000 +

Logical labeling Case study: n = 3 d ρ 3 = 3 3 d Vρ 3 V = 3 3 0 0 0 Sub-ensemble of BC labeled by A = 0 is pseudo-pure Vρ 3 V Unitary operator for

18 Logical labeling Case study: n = 3 d ρ 3 = 3 3 d Vρ 3 V = Sub-ensemble of BC labeled by A = 0 is pseudo-pure Vρ 3 V Unitary operator for LL 3 V = ABC = 000 +

19 Algorithmic cooling N-qubit array with polarization ϵ 0 Redistribute entropy N cold N = H + ε 0 2 Cold Hot Shannon entropy H p = p log 2 p p log 2 ( p) Proc. 3st Annu. ACM Symp. Theory Compt. p.322 (999) Schulman & Vazirani

20 Algorithmic cooling N cold N = H + ε 0 2 Proc. 3st Annu. ACM Symp. Theory Compt. p.322 (999) Schulman & Vazirani

21 Heat bath algorithmic cooling Iterative procedure to compress and pump out entropy from Qubits to Heat Bath Qubit array Refresh Bath Cold Hot Colder than Bath PNAS 99, 3388 (2002) Boykin at al.

22 HBAC: Partner pairing algorithm R: Thermalize R by contacting with Bath with polarization ϵ b C : Swap A and R ( 00 00, 0 0 ) C 2 : Swap B and R ( 00 00, 0 0 ) C 3 : Swap 0 and 00 Phys. Rev. Lett. 94, 2050 (2005) Schulman at al. A B R Bath R C R C 2 R C 3

23 HBAC: st round d ρ I = 8 R d ρ I R = 8 + ε b ε b + ε b ε b + ε b ε b + ε b ε b C d C ρ I R C = 8 + ε b + ε b + ε b + ε b ε b ε b ε b ε b p(a): 0 ϵ b p(b): 0 0 p(r): 0 0 A B R Bath R C R C 2 R C 3

24 HBAC: st round R d (C ρ I R C ) R = ε b 2 ε b 2 + ε b 2 ε b 2 ε b 2 ε b 2 ε b 2 ε b C 2 d C 2 (C ρ I R C ) R C 2 = ε b 2 + ε b 2 ε b 2 ε b 2 ε b 2 ε b 2 ε b 2 ε b p(a): ϵ b ϵ b p(b): 0 ϵ b p(r): 0 0 A B R Bath R C R C 2 R C 3

25 HBAC: st round R C 3 d C 3 (C 2 (C ρ I R C ) R C 2 ) R C 3 = ε b + ε 2 b ( ε b ) + ε 2 b ( ε b ) + ε 2 b ( ε b ) 2 ( + ε b ) ε b 2 ( + ε b ) ε b 2 ( + ε b ) ε b 3 ε b p(a): ϵ b.5ϵ b 0.5ϵ b 3 p(b): ϵ b 0.5ϵ b + 0.5ϵ b 3 p(r): 0 0.5ϵ b + 0.5ϵ b 3 A B R Bath R C R C 2 R C 3

26 HBAC: Multiple rounds R p(a):.5ϵ b 0.5ϵ b 3... p(b): 0.5ϵ b + 0.5ϵ b 3... p(r): 0.5ϵ b + 0.5ϵ b 3 ϵ b A B R Bath R C R C 2 R C 3 m

27 HBAC: Multiple rounds p(a) 2ϵ b (?) A B R Bath R C R C 2 R C 3 m

28 HBAC: Achievable polarization

29 HBAC: Achievable polarization p A max = 2 e 2n 2 ln( +ε b ε b ) + Phys. Rev. Lett. 6, 7050 (206) Rodriguez-Briones & Laflamme

30 Algorithmic cooling N cold N = H + ε 0 2 Proc. 3st Annu. ACM Symp. Theory Compt. p.322 (999) Schulman & Vazirani

31 HBAC: Experiments R A Bath Nature 438, 470 (2005) Baugh et al. Phys. Rev. Lett. 00, 4050 (2008) Ryan et al.

32 Scalable approaches Large DNP on ensemble Hard... HBAC on ensemble Harder...? Measure-and-flip on individual spins Hardest...??

33 Scalable approaches Large DNP on ensemble Hard... HBAC on ensemble Harder...? Measure-and-flip on individual spins Hardest...?? In a few rare systems (Si:P, NV...), we can. The technique used is closely related to DNP.

34 Phosphorus donor in silicon e III (3) IV (4) V (5) B C N Al Si P Ga Ge As 29 Si 3 P 28/30 Si 28 Si : 29 Si (I = ½) : 30 Si = 92.2% : 4.7% : 3.% 3 P (I = ½) = 00%

35 Phosphorus donor in silicon e III (3) IV (4) V (5) B C N Al Si P Ga Ge As 3 P Isotopically purified 28 Si (99.995%) 3 P (I = ½) = 00%

36 Energy levels of Si:P Hamiltonian H 0 = γ e B 0 S z γ P B 0 I z + a 0 S z I z B 0 ~ 350 mt (X-band) e,n = ν n2 γ e = GHz/T γ P = 7.23 MHz/T ν e ν e2 a 0 = 7.53 MHz a 0 /γ e = 4.2 mt ν n ν e = γ e B 0 a 0 /2 ν e2 = γ e B 0 + a 0 /2 Field-sweep ESR spectrum ν n = a 0 /2 + γ P B 0 ν n2 = a 0 /2 γ P B 0

37 ( R.A. Icaacson) Phys. Rev. 03, 500 (956) Feher

38 Phys. Rev. 03, 50 (956) Feher & Gere

39 (π-pulse on e-spin) Phys. Rev. 03, 50 (956) Feher & Gere

40 Electron Nuclear DOuble Resonance (π-pulse on e-spin) (π-pulse on n-spin) Phys. Rev. 03, 50 (956) Feher & Gere

ENDOR & parity non-conservation [ ] In the fall of 956, I gave a colloquium at Columbia University on the nuclear polarization

Lee excitedly tried to persuade me to measure the asymmetry of β-decay in a polarized sample of donor nuclei in silicon. T. D.

Yang had circulated a preprint of an article in which they suggested that one of the conservation laws of physics, parity, did

41 ENDOR & parity non-conservation [ ] In the fall of 956, I gave a colloquium at Columbia University on the nuclear polarization scheme. After the colloquium, C. S. Wu and T. D. Lee excitedly tried to persuade me to measure the asymmetry of β-decay in a polarized sample of donor nuclei in silicon. T. D. Lee and C. N. Yang had circulated a preprint of an article in which they suggested that one of the conservation laws of physics, parity, did not hold in the case of weak interactions. [...] I listened politely with limited interest and promised them I would get to it as soon as I finished the ENDOR experiments [...] C. S. Wu ( AIP Emilio Segre Visual Archives) T. D. Lee ( Nobel Foundation) C. N. Yang ( Nobel Foundation) Annu. Rev. Biophys. Biomol. Struct. 3, (2002) Feher

42 ENDOR & parity non-conservation [ ] In the fall of 956, I gave a colloquium at Columbia University on the nuclear polarization scheme. After the colloquium, C. S. Wu and T. D. Lee excitedly tried to persuade me to measure the asymmetry of β-decay in a polarized sample of donor nuclei in silicon. T. D. Lee and C. N. Yang had circulated a preprint of an article in which they suggested that one of the conservation laws of physics, parity, did not hold in the case of weak interactions. [...] I listened politely with limited interest and promised them I would get to it as soon as I finished the ENDOR experiments [...] After finishing these at the end of 956, I took an extended skiing vacation in the West. On the way back I stopped off at the University of Pittsburgh where I gave a colloquium [...] At the conclusion, I mentioned that I would like to test Lee & Yang s hypothesis of parity nonconservation. [...] it felt as if the temperature of the room had dropped by 0 degrees. Finally, G. C. Wick said, "But don t you know that parity nonconservation has already been proven by several groups?". Of course, I did not know; I had been skiing for a month. Annu. Rev. Biophys. Biomol. Struct. 3, (2002) Feher

43 Science 270, 255 (995) DiVincenzo

44 ν e π/2 π π π π Echo ν n2 τ τ π π π Transfer to n-spin Transfer to e-spin ( UCL) Nature 455, 085 (2008) Morton et al.

45 Phys. Rev. B 82, 220 (200) Abe et al. Phys. Rev. B 93, 6202 (206) Petersen et al.

46 ( JQI) Nature 393, 33 (998) Kane Cf. Nature 393, 43 (998) Chuang et al. Experimental realization of a quantum algorithm

47 ( UNSW) Nature 497, 687 (200) Morello et al.

48 Nature 497, 687 (200) Morello et al.

49 Nature 489, 54 (202) Pla et al.

50 ν e,2 = γ e B 0 a 0 /2 are dependent on the n-spin state ESR does not change the n-spin state Quantum nondemolition (QND) measurement Nature 496, 334 (203) Pla et al.

51 Nature 496, 334 (203) Pla et al.

52 Single nuclear spin under control Rabi Ramsey DiVincenzo s criteria Qubit array Initialization Coherence Quantum gates Measurement Hahn echo Nature 496, 334 (203) Pla et al.

53 Ensemble to single, single to... Ensemble (copy) of identical spins, global control Scalable quantum computer Single spin, local control Multiple single-spins, individual controls

54 DNP in quantum computing Molecule Preparation of pseudo-pure states costs resources exponentially, deeming NMR QC non-scalable. HBAC is a quantum information theoretic approach to DNP. Phosphorus donor in silicon Pulsed ENDOR, a well-established technique for population transfer, is a quantum gate operation. Single nuclear spins can be read out non-destructively. The real challenge is how to scale up the system.

Experimental Realization of Shor s Quantum Factoring Algorithm

Experimental Realization of Shor s Quantum Factoring Algorithm M. Steffen1,2,3, L.M.K. Vandersypen1,2, G. Breyta1, C.S. Yannoni1, M. Sherwood1, I.L.Chuang1,3 1 IBM Almaden Research Center, San Jose, CA