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