NMR Quantum Computation C CS Phys 191 Quantum Information Science and Technology 11 13 2003 Thaddeus Ladd Department of Applied Physics Stanford University tladd stanford edu Solution NMR Quantum Computation 4 Measurement is facilitated by huge ensemble of independent copies B RF 3 Dipolar interactions controllable by RF 1 A small number of stable qubits provided by distinct spin 1 2 nuclei 5 Decoherence times are long since nuclei interact only weakly with environment 2 Perfect initialization is not possible Instead pseudopure states must be used N A Gershenfeld and I Chuang Science 275 350 1997 D G Cory A F Fahmy and T F Havel Proc Natl Acad Sci USA 94 1634 1997 Solution NMR QC Why Bother From the start 1997 researchers knew solution NMR would not surpass the 10 qubit level Why bother going through the trouble of making solution NMR quantum computers then Because classical computers are too slow to factor 15 and decide whether 3 bit functions are constant or balanced These problems will soon to be solved by coffee cup NMR quantum computers in every home Right answer NMR is easy due to the 50 year history and extensive commercial engineering of the technique If we cannot do quantum computing with this plug and play high Q system then our chances with atoms semiconductor electrons etc are slim NMR gives us a head start on solving the problems in quantum control that we will face with other architectures Outline Why bother with solution NMR quantum computing Basic physics of NMR and ingredients for QC Nuclear magnetic moments in static and RF magnetic fields Nuclear nuclear couplings Weak ensemble measurement Experimental Overview Initialization Physical polarization Algorithmic cooling Pseudo pure states Scaling Quantum logic RF implementations of the universal gate set Refocussing Survey of results The future of NMR quantum computing Definitions and Units Subscripts indicate nuclei from different atoms of the molecule All Hamiltonians in units of angular frequency The Hamiltonian Bigger smaller Independent of spin Term Size Role Motion kT 6 THz Bath for thermal relaxation Affects decoherence Zeeman 0 100 MHz Allows polarization Establishes resonance SNR HyperFine A 1 MHz Chemical shielding and J coupling RadioFrequency 1 10 kHz The knob Allows spin rotations Dipolar D 1 kHz Decoherence source Applied Magnetic Fields and the Rotating Frame RF frequency is chosen to be different from resonant frequency 0 B0 by frame z Rotating rotates at Spin precession at 0 New frame rotates at in spin subspace Fastest uninteresting motion rotation about z axis at removed Terms oscillating at 2 are neglected y x Spin precession at x RF oscillating at z RF counterrotating at 2 y Stationary RF Change 1 and to achieve arbitrary rotation angle and axis Single Spin Rotations with the Density Matrix Consider example of a spin in pure state Suppose we apply on resonant RF 0 with phase 0 for time z M y x The last step is the result of some simple matrix algebra it is worth deriving in your spare time Result is independent of representation End result is standard rotation of a vector Thus we write Of particular importance are pulses corresponding to which invert M and 2 pulses corresponding to 2 which are Hadamard gates The Hamiltonian Bigger smaller Independent of spin Term Size Role Motion kT 6 THz Bath for thermal relaxation Affects decoherence Zeeman 0 100 MHz Allows polarization Establishes resonance SNR HyperFine A 1 MHz Chemical shielding and J coupling RadioFrequency 1 10 kHz The knob Allows spin rotations Dipolar D 1 kHz Decoherence source The Chemical Shift Although the hyperfine coupling constants can be large MHz this term ends up small because the sums over spin Sk and orbital Lk angular momentum of electrons vanish in molecular orbital ground states The chemical shift arises as a first order perturbation of the ground state wave function due to the magnetic field Tiny electron currents oppose magnetic fields Magnetic field seen by nucleus is reduced by a few parts per million depending on the molecular orbital state Varies from nucleus to nucleus according to symmetry of molecule Gives quantitative information about molecular structure Allows qubits of same nuclear species to be distinguished Molecular orbital nucleus j j The J Coupling The J coupling arises as a second order perturbation to the ground state wavefunction as a result of the effective field due to the nuclei Magnetic field of one nucleus distorts electron orbital Tiny electron currents change magnetic field at second nucleus The matrix J gives quantitative information about molecular structure Nonlinear term used for quantum logic j j Beff j Iz 0 j j Beff j Iz 0 Components of effective magnetic field due to Jcoupling not in z direction time average to zero due to rapid Larmor rotatation of nuclei in large applied magnetic field The Hamiltonian Bigger smaller Independent of spin Term Size Role Motion kT 6 THz Bath for thermal relaxation Affects decoherence Zeeman 0 100 MHz Allows polarization Establishes resonance SNR HyperFine A 1 MHz Chemical shielding and J coupling RadioFrequency 1 10 kHz The knob Allows spin rotations Dipolar D 1 kHz Decoherence source The Dipolar Coupling The Advantage of Liquids Couples nuclear spins Coupling strength is 1 kHz Includes relative distance r between nuclear spins The advantage of liquids is that due to the thermal motion of the molecules in the liquid the dipolar coupling averages to zero in first order Residual dipolar coupling is leading cause of Decoherence T2 Random magnetic fields due to nuclear dipoles on other molecules apply random z rotations Initially parallel spins go out of phase due to random rotations Nuclear magnetization from different molecules destructively interferes Thermal relaxation T1 Sometimes moving molecules interact for timescale of order 1 0 Larmor period Such a random interaction is like a random resonant RF applied field Tips spins with energy compensated by motional bath The Hamiltonian in the Rotating Frame Bigger We have complete control over single spin rotations Spin couplings and relaxation are always present smaller Select spin with RF frequency Select angle with RF amplitude Select axis with RF phase Free Induction Decay FID Consider a two spin molecule Neglect T1 Suppose spins begin in ground state parallel to field Set 1 0 by choosing RF frequency to match chemical shifted Larmor frequency of first nucleus Apply 2 pulse about y axis z y x z Ry 2 y x