Bulk Spin Resonance Quantum Information Processing

1. Bulk Spin Resonance Quantum Information Processing Yael Maguire Physics and Media Group (Prof. Neil Gershenfeld) MIT Media Lab ACAT 2000 Fermi National Accelerator Laboratory, IL 17-Oct-2000

2. Why should we care? • By ~ 2030: transistor = 1 atom, 1 bit = 1 electron, Fab cost = GNP of the planet • Scaling: time (1 ns/ft), space (DNA computers  mass of the planet). • Remaining resource: Hilbert Space.

3. Bits • Classical bit • Analog “bit” • Quantum qubit

4. More Bits • 2 Classical Bits • 2 Quantum Bits • N Classical Bits • N binary values • N Quantum Bits • 2N complex numbers • superposition of states • Hilbert space

5. A B Entanglement • correlated decay • project A • hidden variables? • action at a distance? • information travelling back in time? • alternate universes (many worlds)? • interconnect in Hilbert space – O(2-N) to O(1) AB

6. The Promise • Examples: • Shor’s algorithm (1000 bit number): • O((logN)2+) vs. O(exp(1.923+(logN)1/3(loglogN)2/3) • O(1 yr) @ 1Hz vs. O(107 yrs) @ 1 GFLOP • Grover’s algorithm (8 TB): • O( ) vs. O(N) • 27 min. vs. 1 month @ same clock speed.

7. What do you need to build a quantum computer? • Pure States • Coherence • Universal Family • Readout • Projection Operators • Circuits

8. Previous/Current Attempts • spin chains • quantum dots • isolated magnetic spins • trapped ions • Optical photons • cavity QED • Coherence! • Breakthroughs: • Bulk thermal NMR quantum computers • quantum coherent information bulk thermal ensembles • Quantum Error Correction • Correct for errors without observing. • Add extra qubits syndrome

9. What do you need to build a quantum computer using NMR? Gershenfeld, Chuang, Science (1997) Cory, Havel, Fahmy, PNAS (1997) • Pure States • effective pure states in deviation density matrix • Coherence • nuclear spin isolation, 1-10s • Universal Family • arbitrary rotations (RF pulses) and C-NOT (spin-spin interactions) • Readout • Observable magnetization • Projection Operators • Change algorithms • Circuits • Multiple pulses are gates

10. Quantum Mechanics • wave function • observables • pure state • mixed state • Hamiltonian (energy) • evolution • equilibrium

11. Bulk Density Matrix B0 B1 • ~1023 spindegrees of freedom • rapid tumbling averages inter-molecular interactions • ~N effective degrees of freedom • decoherence averages off-diagonal coherences N spins I (1/2)

12. NMR: “reduced” density matrix Deviation Density Matrix in NMR • high temperature approximation • identity can be ignored • ensemble Fmolecule Fdeviation

13. Spin Hamiltonian • magnetic moment • angular momentum • spin precession • Zeeman splitting • 2 spin interaction Hamiltonian A-B

14. Magnetic Field and Rotation Operators • apply a z field: • evolve in field: • two spins, scalar coupling: • evolution = 3 commuting operators Arbitrary single qubit operations

15. The Controlled-NOT Gate • ENDOR (1957) • electron-nuclear double resonance • INEPT (1979) • insensitive nuclei enhanced by polarization transfer

16. The Controlled-NOT Gate Input thermal density matrix CNOT output

17. Ground State Preparation • We want: where • How? Use degrees of freedom to create an environment for computational spins. • 1. Logical Labeling (Gershenfeld, Chuang) • ancilla spins - submanifolds act as pure states - exponential signal • 2. Spatial Labeling (Cory, Havel, Fahmy) • field gradients dephase density matrix terms - exponential space • 3. Temporal Labeling (Knill, Chuang, Laflamme) • use randomization and averaging over set of experiments - exponential time

18. Algorithms - Grover’s Algorithm • find xn | f(xn) = 1, f(xm)=0 • Initialize L bit registers • Prepare superposition of states • Apply operator that rotatesphase by p if f(x) = 1 • Invert about average • Repeat O(N1/2) times • Measure state

19. NMR Implementation • Pure state preparation • Superposition of all statesH = RyA(90) RyB(90) - RxA(180) RxB(180) • Conditional sign flip (test for both bits up)C = RzAB(270) - RzA(90) - RzB(90) • Invert-about-meanM = H - RzAB(90) - RzA(90) - RzB(90) - H

20. Experimental Implementation of Fast Quantum Searching, I.L. Chuang, N. Gershenfeld, M. Kubinec, Physical Review Letters (80), 3408 (1998).

21. Quantum Error Correction • 3-bit phase error correcting code - Cory et al, PRL, 81, 2152 (1998) - alanine

22. Quantum Simulation • Feynman/Lloyd- quantum simulations more efficient on a quantum computer • Waugh- average Hamiltonian theory • Dynamics of truncated quantum harmonic oscillator with NMR- Samaroo et al. PRL, 82, 5381.

23. Scaling Issues • Sensitivity vs. System resources • Decoherence per gate • Number of qubits

24. Scaling

25. N  1 0.25 2 0.11 3 0.04 4 1.2x10-2 5 3.4x10-3 6 9.1x10-4 7 2.4x10-4 8 6.0x10-5 9 1.5x10-5 10 3.8x10-6 Scaling is separable if • Is it quantum? Schack, Caves, Braunstein, Linden, Popescu, … • Initial conditions vs quantumevolution • But, Boltzmann limit is not scalable

26. Polarization Enhancement - Optical Pumping • Error correction as well (or phonon)

27. acetone -d6 13C1HCl3 solvent ZLI-1167 215 Hz J J+2D 1706 Hz 25 s T1 (13C) 2 s 19 s T1 (1H) 1.4 s 0.3 s T2 (13C) 0.2 s 7 s T2 (1H) 0.7 s Decoherence per gate • Steady state error correction - 10-4 - 10-6 C. Yannoni, M. Sherwood, L. Vandersypen, D. Miller, M. Kubinec, I. Chuang, Nuclear Magnetic Resonance Quantum Computing Using Liquid Crystal Solvents quant-ph/9907063, July 1999

28. Number of Qubits • Seth Lloyd, Science, 261, 1569 (1993) - SIMD CA • D-A-B-C-A-B-C-A-B-C.... • at worst linear, but may be polylogarithmic • Shulman, Vazirani (quant-ph/980460) - using SIMD CA • can distill qubits where SNR independent of system size

29. Our goals $500,000 • Develop the instrumentation and algorithms needed to manipulate information in natural systems • Table-Top (size & cost) • investigate scaling issues $50,000 $5,000

30. Magnet Design • Halbach arrays using Nd2Fe14B: 1.2T  2.0T • Fermi Lab - iron is a good spatial filter

31. Compilation • Multiplexed Add: • function program = madd(cnumif0, cnumif1, enabindex, selindex, inputbits, outputbits, • BOOLlowisleft) % outputbits MUST be zeros • %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% • % madd.m • % Implements adding a classical number to a quantum number, mod 2^L. • % If N is the thing we want to factor, then selindex says whether N-cnum is less than or • % greater than B: N-cnum>b --> add cnum, else N-cnum<b --> add cnum - N + 2^L • % Enabindex must all be 1, else choose the classical addend to be zero. • % Edward Boyden, e@media.mit.edu • % INPUT • % cnum classical number to be added • % indices column vector of indices on which to operate • % carryindex carry qubit that you're using • %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% • L = length(outputbits); %It's an L-bit adder: contains L-1 MUXFAs and 1 MUXHA • if (L!=length(inputbits)) %MAKE SURE OF THIS! • program = 'Something''s wrong.'; • return; • end; • cbitsif0 = binarize(cnumif0); % BINARIZE! • cbitsif1 = binarize(cnumif1); • cL = length(cbitsif0); • if (cL>L) Can you implement? gcc grover.c -o chloroform

32. Nature is a Computer IBM Dr. Isaac Chuang Dr. Nabil Amer MIT Prof. Neil Gershenfeld Prof. Seth Lloyd U.C. Berkeley Prof. Alex Pines Dr. Mark Kubinec Stanford Prof. James Harris Prof. Yoshi Yamamoto