Algorithmic Cooling in Liquid-state NMR

Yuval Elias / Gilles Brassard Group

NTRQ Meeting November 2015

Château Bromont Algorithmic Cooling in Liquid-state NMR

Yuval Elias / Gilles Brassard Group

NTRQ Meeting November 2015

Château Bromont Bonne-Fête Anniversaire Gilles!

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http://menoummm.centerblog.net/6123670-Bonne-Fete-Anniversaire- https://s3.amazonaws.com/uploads.startups.fm/wp-content/uploads/ 2013/08/erwin-schrodinger.jpg Overview Novel Cooling from Spin cooling: enhancing polarization in NMR Polarization compression: generalized polarization transfer Open systems: -bath cooling, algorithmic cooling Heat-bath Cooling Experiments Model system: 13C-labeled trichloroethylene (TCE) 13C-labeled glutamate and glycine (Glu, Gly) Algorithmic Cooling Experiments AC in solid-state NMR AC in liquid-state NMR - building blocks AC in liquid-state NMR – process 1 (cooling C1) AC in liquid-state NMR – processes 2,3 (cooling C2, C3) Future prospects – -based Brain

Outline

The advent of NMR quantum computing in the late 1990s led to renewed interest in manipulations for spin cooling. Spin Cooling – Enhancing Polarization in NMR

Polarization bias : excess polarization of spins aligned with field

1 2 0 EEEE    T 0   hB 0 1  tanh2kT 2 kT 2 kT 0 2 

Effective spin : for any bias (also non-thermal)

11.7T (500 MHz), 25°C   ~ 10-5  Spin-cooling: transient increase in polarization, beyond thermal

Abragam and Goldman, Nuclear : Order and Disorder (1982) Polarization Compression

Shannon’s bound on total entropy Shannon, Bell Syst. Tech. J. (1948) H( X ) p log 1  i 2 pi Coin flip: H(fair) = 1; H(¼,¾) = 0.811; H(heads) = 0 Fair biased coin: flip twice, use first toss if different J. von Neumann, Appl. Math. Ser. (1951).

NMR: rf pulses for optimal polarization transfer Sørensen, Prog. Nucl. Mag. Res. Spec. (1989)

“Molecular scale heat engine”: loss-less data compression Schulman & Vazirani, Proc. 31st ACM Symp. Theory Comput. (1999) Polarization Compression – Shannon’s Bound

Shannon’s bound on single spin entropy:

11 1  2 H(1) p logpp  p log  1   ln4 Shannon’s bound for n spin-system:

2 2 H() n nH (1)  n 1 11max  (1)1 n  n  ln 4  ln 4  max

 Over a billion spins required to produce one pure spin! abc P(abc) 3 3-Bit Polarization Compression 000 pp3   001 p2

a 010 p2 p 3B-Comp 011 1 a = 1.50 100 p2 101 p1 b c  = = 0.5 110 p1 a =b=c= 0 b c 0 3 111 pp0   132 3  321 0 0 P a03  p  p  p   22 a 

abc   3-bit compression with non-uniform bias:  a  2 Elias, Fernandez, Mor & Weinstein, Isr. J. Chem. (2006) Spin Cooling in Open Systems selective-reset(cr): PT to selected computation spin c followed by reset of reset spin r

1 Environment Reset spin ( H), eq  40

PT PT WAIT

13 Computation spins ( C), eq = 0

 Assumption: computation spins relax infinitely faster than reset spins Heat-bath Cooling of Spins Heat-bath cooling: one or more selective-reset steps  cool spin-system entirely or partially

Reset spin (1H),   4 H eq 0

SR(HC1) SR(HC2)

C2 C1 13 Computation spins ( C), eq = 0

 Improved cooling of both 13C by avoiding final reset

Fernandez, Lloyd, Mor & Roychowdhury, Int. J. Quant. Inf. (2004) Algorithmic Cooling of Spins Heat-bath algorithmic cooling: bypass entropy bound combine polarization compression and heat-bath cooling!

Reset spin (1H),   4 H eq 0

HBC Comp Scalable NMRQC

C2 C1 13 Computation spins ( C), eq = 0  = 60  beyond H(3)!

 PAC – practicable algorithmic cooling

Boykin, Mor, Roychowdhury, Vatan & Vrijen, PNAS (2002) Fernandez, Lloyd, Mor & Roychowdhury, Int. J. Quant. Inf. (2004) Mor, Roychowdhury, Fernandez, Lloyd & Weinstein, US Patent #6,873,154 (2005) Algorithmic Cooling – Second Cooling Level

Reset spin (1H),   4 H eq 0 H

3B-Comp

C4 C3 C2 C1 13 Computation spins ( C), 1 = 60 2 = 90 (160)  PAC with n spins  (3/2)n [2n]

Fernandez, Lloyd, Mor & Roychowdhury, Int. J. Quant. Inf. (2004) Spin-cooling For n spins, one reset spin and 2Q = n-1 computation spins, ideally: leftmost - reset spin; rightmost - coldest PAC : {1 1 3/2 3/2 9/4 9/4 27/8 ... (3/2)Q}  53 resets Optimal algorithm (PPA): {1 1 2 4 8 16 32 ... 2n-2}

“Fibonacci” algorithm: {1 1 2 3 5 8 13 ... Fn}

“Tribonacci” algorithm: {1 1 2 4 7 13 24 ... Tn} Semi-optimal 4PAC: {1 1 1.94 1.94 3.75 3.75 7.27 }  911 resets Semi -optimal 4Fib: {1 1 1.88 2.70 4.28 6.54 10.2 }

Fernandez, Lloyd, Mor & Roychowdhury, Int. J. Quant. Inf. (2004) Schulman, Mor & Weinstein, Phys. Rev. Lett. (2005) Elias, Fernandez, Mor & Weinstein, Isr. J. Chem. (2006) Elias, Mor & Weinstein, Phys. Rev. A, (2011) Relaxation-Time Ratios Reset spins should repolarize much faster than computation spins relax Preserve enhanced polarization during reset

(c,r) = T1 (c) / T1 (r) >> d * Nr (#reset-steps) Preferably - (c,r) > 10*Nr (typically d ~ 1-5)

Elias, Mor & Weinstein, Phys. Rev. A, (2011) Brassard, Elias, Mor & Weinstein, EPJ+ (2014) Heat-bath Cooling Model System 3.5s Cl H 13C2-TCE C1 C2 20s C1 904 T1(C1) ~ 40s 13C 13C C2 103 0 C1 C2 H 9 201 Cl Cl Non-selective (“hard”) pulses

chloroform-d / acetone–d6  Paramagnetic reagent Cr(acac)3 (13C,1H)  10 T2 >> 1/J  < 1 kHz  hard pulses

Fernandez, Mor & Weinsten, Int. J. Quant. Inf. (2005) Brassard, Elias, Fernandez, Gilboa, Jones, Mor, Weinstein & Xiao, arXiv:quant-ph/0511156 (2005) Heat-bath Cooling – POTENT pulse sequence

1) 1st Selective Reset(C1H) 2) 2nd Selective Reset(C2H)

POTENT pulse sequence: Cl H POlarization Transfer via ENvironment Thermalization 13 13 C C 1 2

1H Cl Cl

* 13C H C2 H C2

13C C2 C1 d2 d3

T1(H) < d2,d3 < T1(C2) < T1(C1) Brassard, Elias, Fernandez, Gilboa, Jones, Mor, Weinstein & Xiao, arXiv:quant-ph/0511156 (2005), EPJP (2014)

Annotated POTENT pulse sequence

Brassard, Elias, Fernandez, Gilboa, Jones, Mor, Weinstein & Xiao, EPJP (2014) Heat-bath Cooling of TCE – Entropy Reduction C2

C1 1H Cl H

13C 13C After POTENT:

C1 C2 1H Cl Cl

Equilibrium:

Brassard, Elias, Fernandez, Gilboa, Jones, Mor, Weinstein & Xiao, EPJP (2014) Heat-bath Cooling – Amino Acids (Gly, Glu)

1) Selective Reset(C1H) 13C -Glu/Gly Gd-DTPA (Magnevist) 2) PT(C2H) 2 D2O, K3PO4, pH~8 Truncated POTENT

 Glutamate (Glu) – major excitatory 2.7s (1.3s) neurotransmitter, Alzheimer’s disease O H 4s (2s)

13 13 + 30s (13s) C CNH3 1 2

O- R = H, CH CH COOD 2 2  Glycine - inhibitory neurotransmitter (low conc.) T1(H) ~ T1(C2) < d2 << T1(C1) T2 >> 1/J Short spin-selective pulses (1 ms BURP)

Elias, Gilboa, Mor & Weinstein, Chem. Phys. Lett. (2011) Cooling Amino Acids Beyond the Entropy Bound

O H Glycine 297K Glutamatic acid

13 13 + C C NH3 1 2 After cooling O - R

1H 1 rec Acq 13Caliphatic HCC Equilibrium HCC  1 rec d3 Acq 13Ccarbonyl d3 ~7T1(H) = 10s (Glu) / 20s (Gly) Both spin-systems cooled (C1 by 1.900.01)

Elias, Gilboa, Mor & Weinstein, Chem. Phys. Lett. (2011) Heat-bath Cooling of Both Backbone

O H 13C2-glutamate 310K +50 µM Magnevist 13 13 + C C NH3 1 2 O - R   Truncated POTENT 1 3

1 rec 1 1 3 Acq HCC d2 d5 d3 HCC E-BURP1 E-BURP1 E-BURP1 1 rec   3 Acq d14 2 d14 d5

I-BURP1 I-BURP1

d2 = 2-3T1(H), d3 ~ T2(H) = 0.5s-1s Both carbons cooled about 2.5-fold (to ~120K)

Elias, Gilboa, Mor & Weinstein, Chem. Phys. Lett. (2011) Algorithmic Cooling in Solid State NMR H H 13C3-malonic acid Hm2 Hm1 13C  Spin diffusion – fast repolarization Cm (msec), T1(13C) > 100 s  HOO13C 13COOH C1 C2  Numerically optimized pulses 1) 1st Selective Reset (C1Hm1) 2) 2nd Selective Reset (C2Hm1) 3) Selective PT(CmHm1) Single crystal, ~3% labeled 4) 3-Bit Compression (C1,Cm,C2) Heat-bath – protons in crystal C1 cooled by factor of 4.0 ± 0.1 C2 cooled beyond heat-bath (4.59, …, 5.58) Baugh, Moussa, Ryan, Nayak & Laflamme, Nature (2005) Ryan, Moussa, Baugh & Laflamme, Phys. Rev. A (2008) Algorithmic Cooling in Liquid-state NMR – Building Block 1

Numerically-optimized pulses (GRAPE) SIMPSON open source simulation package

PE (polarization exchange) : C2 cooled by 3.76 ± 0.02

Glaser, Reiss, Kehlet & Schulte-Herbrüggen, J. Magn. Reson. (2005) Tosner, Vosegaard, Kehlet, Khaneja, Glaserd & Nielsen, J. Magn. Reson. (2009) Atia, Elias, Mor & Weinstein, Int. J. Quant. Inf. Algorithmic Cooling in Liquid-state NMR – Building Block 2

Numerically-optimized pulses (GRAPE) SIMPSON open source simulation package

COMP : C1 cooled by 2.76 ± 0.02

Glaser, Reiss, Kehlet & Schulte-Herbrüggen, J. Magn. Reson. (2005) Tosner, Vosegaard, Kehlet, Khaneja, Glaserd & Nielsen, J. Magn. Reson. (2009) Atia, Elias, Mor & Weinstein, Int. J. Quant. Inf. AC in Liquid-state NMR – Process 1 Overview

Cl H

13 13 C C 1 2 Cl Cl

Atia, Elias, Mor & Weinstein, arXiv:1411.4641v2 [quant-ph], submitted to Phys. Rev. A AC in Liquid-state NMR – Process 1

Process 1: D1=150s, D2=5s, D3=3s

Atia, Elias, Mor & Weinstein, arXiv:1411.4641v2 [quant-ph], submitted to Phys. Rev. A AC in Liquid-state NMR – Process 1 Buildup

Atia, Elias, Mor & Weinstein, arXiv:1411.4641v2 [quant-ph], submitted to Phys. Rev. A AC in Liquid-state NMR – Process 1 (7 Rounds)

C1 cooled by factor of about 4.6, ICC1 = 21.25 ± 0.18

Atia, Elias, Mor & Weinstein, arXiv:1411.4641v2 [quant-ph], submitted to Phys. Rev. A AC in Liquid-state NMR – Process 2

Process 2: D1=150s, D2=5s, D3=3s, D4=5s

Atia, Elias, Mor & Weinstein, arXiv:1411.4641v2 [quant-ph], submitted to Phys. Rev. A AC in Liquid-state NMR – Process 2 Results (7 Rounds)

C1, C2 cooled by factors of 3.8, 3.4 (ICC1,C2=25.9 ± 0.2)

Atia, Elias, Mor & Weinstein, arXiv:1411.4641v2 [quant-ph], submitted to Phys. Rev. A AC in Liquid-state NMR – Process 3

Process 3: D1=150s, D2=5s, D3=3s, D4=6s, D5=6s

Atia, Elias, Mor & Weinstein, arXiv:1411.4641v2 [quant-ph], submitted to Phys. Rev. A AC in Liquid-state NMR – Process 3 Results (7 Rounds)

{C1, C2, H} = {2.87, 2.64, 3.58} (± 0.02), IC=28.0 ± 0.2

Atia, Elias, Mor & Weinstein, arXiv:1411.4641v2 [quant-ph], submitted to Phys. Rev. A Future Prospects – Carbon-based Brain Spectroscopy

 Suitable isotopomers  Moderate cooling  In vivo – replenish  Monitor slow

Review: Rodrigues & Cerdán, Concepts Magn. Reson. (2005) Future Prospects – Carbon-based Brain Spectroscopy

 Dorith Goldsher MD (Rambam) - 3T MRI

 Dr. Itamar Kahn (Technion) - 9.4T MRI

 Dr. Andrew Webb (Leiden) - 7T MRI

Sailasuta, Robertson, Harris, Gropman, Allen & Ross, J. Mag. Res. (2008) Collaborators

Technion Rambam Health Center Department of Computer Science Dorith Goldsher MD Prof Tal Mor Dr Alexandra Nemirovski Dr Matty Katz Yosi Atia Université de Montréal Département IRO Schulich Faculty of Chemistry Prof Gilles Brassard Prof Haggai Gilboa Dr Yael Balasz Funding Department of Computer Science Mafat, the Israeli Ministry of Defense Dr Yossi Weinstein Wolfson Foundation NTRQ NSERC Thanks for listening! NMR Quantum Computing (NMRQC) Experiments since 1997 leading QC implementation – 13 (2010) Qubits: nuclear spins, usually spin-½ (1H, 13C, 15N, 19F, 31P) Single gates: spin-selective rf pulses Multi-qubit gates: J-coupling, dipolar coupling Algorithm: pulse sequence – universal set of gates Sample: ensemble of many identical molecules Pseudopure state – not scalable! Review: Jones, “Quantum Computing with NMR” preprint arXiv:1011.1382v1 (2010) HCC relay using soft vs hard pulses

1 2

Waltz-16 1H 2d4 2

1 d5 13Caliphatic E-BURP1 2

2 2 1 d5 2d7 2d7 13Ccarbonyl I-BURP1 I-BURP1 E-BURP1 1= x; 2= -y

Elias, Gilboa, Mor & Weinstein, Chem. Phys. Lett. (2011) Brassard, Elias, Fernandez, Gilboa, Jones, Mor, Weinstein & Xiao, EPJP (2014)