Experimental Methods for Quantum Control in Nuclear Spin Systems

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Dorthy McKenzie
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Control in Nuclear Spin Systems Together with: Jonathan Baugh, Hyang Joon Cho, Paola

1 Tim Havel and David Cory Dept. of Nuclear Engineering Massachusetts Institute of Technology Experimental Methods for Quantum Control in Nuclear Spin Systems Together with: Jonathan Baugh, Hyang Joon Cho, Paola Cappellaro,, Nicolas Boulant,, Joseph Emerson, Jonathan Hodges, Raymond Laflamme, Tirthahalli Mahesh, Chandrasekhar Ramanathan, Suddhasattwa Sinha,, and Yaakov Weinstein

2 NMR QIP: It really works! 2

3 In Principle or In Reality? Evolution: U (1 + ε )U = 1 + ε (U ( )( )( U ) Measurement: tr ((1 + ε ) M ) = ε M ε ε/ E ε/3 E ε/3 0 0 E +ε +ε 1 3

4 Where Are We Now? Fully programmable systems on 3-6 qubits Molecules are 13 C-labeled Alanine & Crotonic Acid Efficient, modular imple- mentations of logic gates Achieved via strongly modulated RF pulses Designed by simulation & numerical optimization to correct incoherent errors J 12 C1 J 13 C1 C2 C3 J 12 J 23 C2 J 23 J 34 C3 C4 4

Control of Incoherent Errors Errors in quantum control may be classified as: Coherent (inaccurate) Incoherent (imprecise) Decoherent (micro- scopically random) Incoherent are correctable by

5 Control of Incoherent Errors Errors in quantum control may be classified as: Coherent (inaccurate) Incoherent (imprecise) Decoherent (micro- scopically random) Incoherent are correctable by refocusing, as shown by composite pulses But strong modulation can get the entire unitary right x z ρ out = dα p V α U α ρ in U α distribution p α can be measured & incorpor- ated into pulse design Approximate Modulation Sequence Simulate Unitary (or set thereof) Target Unitary Matrix Comparison Function New Simplex Point RF Coil Gradient Coil Voxals of Spatially Distributed Sample y Simplex Algorithm U α Physical Constraints on Sequence 5

6 Achievements of NMR QIP The first successful demonstrations of: Quantum logic gates on superpositions Application of entangling unitary operations Simple quantum algorithms (QFT, Grover, &c) Quantum error correcting codes (on 3 qubits) Decoherence-free subspaces and systems Quantum simulation as proposed by Feynman Semi-classical / quantum simulations of decoherence Quantum process tomography (on 3 qubits) 6

7 Quantum Process Tomography { S ρ eq in } {ρ in} {m in} S op op { ro { U ro } U } M obs {ρ } {m } op m xx tr( U ro ( ρ xx ) σ ) Due to errors, particularly incoherent, measured superoperator may NOT be completely positive U xx ( ρ) dqu xx (q) (q)@ ρu xx Calculating a superoperator from experimental data: S in in in out out out ρ 1 ρ ρ N ρ1 ρ ρ N = superoperator matrix vs.. induced columns are stacked columns are stacked basis is unknown input density matrices output density matrices 7

8 Natural Relaxation H H Our first attempt at QPT sought to determine the natural relaxation superoperator of a 2-spin system, 2,3-dibromothiophene This means to finding all of its relax- ation rates, i.e. or Lindblad operators Done by first determining the super- propagator at 4 time points, & fitting the superoperator to these data Fit was ill-conditioned until complete positivity constraint was imposed 2,3-Dibromothiophene 8

9 3-qubit QFT with our SM-pulses The plots below show the theoretical, simulated and experimental superoperator on the 3-qubit QFT as implemented by strongly modulating pulses in alanine, all versus the product operator (σ( α σζ ) basis Correl. Coeff = 0.99 Correl. Coeff =

10 Identifying Errors via the Model Kraus Operator Amplitudes (compensated control sequences) simulated from model (see text) from experimental data The lack of complete positivity,, though significant, could be fixed without large changes in the superoperator s s eigenvalues,, implying that the incoherent errors were not large (as we had hoped!) The clustering of the eigenvalues at [ 0, i, 1, i ] was much improved by finding the product of qubit rotations that maximized the correlation with the simulated, implying that the main coherent errors were the cumulative result of many small single qubit errors 10

11 Where Are We Going with NMR? Solid-state NMR offers several practical benefits: Longer decoherence times and faster gates Adjustable chemical shifts to address more qubits The ability to perform Dynamic Nuclear Polarization Liquid -State NMR QIP e e Dilute Crystal of Spin-Labelled Organic Molecules e e Engineered Systems? 11

Restricting Dipole Couplings in Crystal Lattices to Nearest Neighbors In dilute spin crystals, this would improve isolation of spins in different molecules, and greatly reduce the complexity of the

12 Restricting Dipole Couplings in Crystal Lattices to Nearest Neighbors In dilute spin crystals, this would improve isolation of spins in different molecules, and greatly reduce the complexity of the intramolecular Hamiltonian In cubic lattices of spins (e.g. CaF 2 ), it would enable simulation of massive 3D quantum Ising models The figure shows this effect in the case of Gypsum, a crystal containing strongly-coupled pairs of protons with many weak interpair couplings 12

13 2 1.5 How Do We Do It? The dipolar Hamiltonian of a a pair of spins has eigen- values [ 1,0,1,2] D Thus on-resonance RF cosine modulated at 3D/2 tracks the spins natural evolution An RF power of D/2 is enough to average any weaker couplings to zero fraction of cycle time for weaker coupling fraction of cycle time for weaker coupling 13

14 Conclusions and Acknowledgements We hope that as our experience with coherent control of nuclear spins in the solid state improves, a a route to a truly scalable architecture, based on spatial addressing, will open up though we are not ready to promise scalability today Those interested in further details should drop by either my poster or Jonathan Baugh s This work supported largely by &. 14

15 Advertisement A recent Kluwer-Springer journal awaits your papers! Free sample issues avail- able at poster sessions (one per customer, please)! A quadruple issue on exp- erimental aspects of QIP is now in press, with Our Patron Saint, Henry Everitt, as the guest editor (also to be published as an independent book)! 15

arxiv:quant-ph/ v1 30 Jun 2004

arxiv:quant-ph/ v1 30 Jun 2004 Quantum Process Tomography of the Quantum Fourier Transform Yaakov S. Weinstein, 1, Timothy F. Havel, 1 Joseph Emerson, 1, Nicolas Boulant, 1 Marcos Saraceno, 2 Seth Lloyd, 3 and David G. Cory 1 1 Massachusetts