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We report on the experimental simulation of spin squeezing using a liquid-state nuclear magnetic resonance (NMR) quantum information processor. This was done by identifying the energy levels within the symmetric subspace of a system of n spin-1/2 nuclei with the energy levels of the simulated spin-(n/2) system. The results obtained for our simulations of spin-1 and spin-3/2 systems are consistent with earlier theoretical studies of spin squeezing, and illustrate interesting relations between the degree of squeezing and the strength of the correlations among the underlying spin-1/2 particles. PACS: 03.65.Bz; 03.67.-a; 03.67.Lx

We introduce a nonsymmetric real matrix which contains all the information that the usual Hermitian density matrix does, and which has exactly the same tensor product structure. The properties of this matrix are analyzed in detail in the case of multi-qubit (e.g., spin = 1/2) systems, where the transformation between the real and Hermitian density matrices is given explicitly as an operator sum, and used to convert the essential equations of the density matrix formalism into the real domain. PACS: 03.65.Ca; 03.67-a; 33.25.+k; 02.10.Xm

We experimentally explore the reduction of decoherence via concatenating quantum error correction (QEC) with decoupling in liquid-state NMR quantum information processing. Decoupling provides an efficient means of suppressing decoherence from noise sources with long correlation times, and then QEC can be used more profitably for the remaining noise sources. PACS: 03.67.Lx, 03.65.Bz

This paper surveys our recent research on quantum information processing by nuclear magnetic resonance (NMR) spectroscopy. We begin with a geometric introduction to the NMR of an ensemble of indistinguishable spins, and then show how this geometric interpretation is contained within an algebra of multispin product operators. This algebra is used throughout the rest of the paper to demonstrate that it provides a facile framework within which to study quantum information processing more generally. The implementation of quantum algorithms by NMR depends upon the availability of special kinds of mixed states, called pseudo-pure states, and we consider a number of different methods for preparing these states, along with analyses of how they scale with the number of spins. The quantum-mechanical nature of processes involving such macroscopic pseudo-pure states also is a matter of debate, and in order to discuss this issue in concrete terms we present the results of NMR experiments which constitute a macroscopic analogue Hardy's paradox. Finally, a detailed product operator description is given of recent NMR experiments which demonstrate a three-bit quantum error correcting code, using field gradients to implement a precisely-known decoherence model.

This paper presents a new distance geometry algorithm for calculating atomic coordinates from estimates of the interatomic distances, which maintains the positions of the atoms in a known rigid substructure. Given an M × 3 matrix of coordinates X for the rigid substructure, this problem consists of finding the N × 3 matrix Y which yields the global minimum of the so-called STRAIN, i.e., $$\mathop {\min }\limits_Y \left\| {\left[ {\begin{array}{*{20}c} {XX^T } \\ {YX^T } \\ \end{array} \begin{array}{*{20}c} {XY^T } \\ {YY^T } \\ \end{array} } \right] - \left[ {\begin{array}{*{20}c} A \\ {B^T } \\ \end{array} \begin{array}{*{20}c} B \\ C \\ \end{array} } \right]} \right\|_F^2 ,$$ where $$A = XX^{\text{{\rm T}}} $$ , and B, C are matrices of inner products calculated from the estimated distances. The vanishing of the gradient of the STRAIN is shown to be equivalent to a system of only six nonlinear equations in six unknowns for the inertial tensor associated with the solution Y. The entire solution space is characterized in terms of the geometry of the intersection curves between the unit sphere and certain variable ellipsoids. Upon deriving tight bilateral bounds on the moments of inertia of any possible solution, we construct a search procedure that reliably locates the global minimum. The effectiveness of this method is demonstrated on realistic simulated and chemical test problems.

A new computational method for simultaneously estimating all the proton-proton coupling constants in a molecule from COSY spectra [Yang, J.-X. and Havel, T.F. (1994) J. Biomol. NMR, 4, 807-826] is applied to experimental data from two polypeptides. The first of these is a cyclic hexapeptide denoted as VDA (-D-Ala1-Phe2-Trp3-Lys(Z)4-Val5-Phe6-), in deuterated DMSO, while the second is a 39-residue protein, called decorsin, in aqueous solution. The effect of different data processing strategies and different initial parameter values on the accuracy of the coupling constants was explored. In the case of VDA, most of the coupling constants did not depend strongly on the initial values chosen for the optimization or on how the data were processed. This, together with our previous experience using simulated data, implies strongly that these values are accurate estimates of the coupling constants. They also differ by an average of only 0.36 Hz from the values of the 14 coupling constants that could be measured independently by established methods. In the case of decorsin, many of the coupling constants exhibited a moderate dependence on their initial values and a strong dependence on how the data were processed. With the most successful data processing strategy, the amide-alpha coupling constants differed by an average of 1.11 Hz from the 21 values that could be measured by established methods, while two thirds of the three-bond coupling constants fell within 1.0 Hz of the ranges obtained by applying the Karplus relation to an independently computed ensemble of distance geometry structures. The averages of the coupling constants over multiple optimizations using random initial values were computed in order to obtain the best possible estimates of the coupling constants. Most clearly incorrect averages can be identified by large standard deviations in the coupling constants or the associated line widths and chemical shifts, and can be explained by strong coupling and/or overlap with the water signal, the diagonal peaks or other cross peaks.

A method is proposed for defining a probability distribution on an ensemble of protein conformations from a 2D NOE spectrum, while at the same time back-calculating the experimental spectrum from the ensemble. This enables one to assess the relative quality and significance of the conformations, and to test the consistency of the ensemble as a whole with the experimental spectrum. The method eliminates the need to integrate the cross-peak intensities and is surprisingly insensitive to random noise in the spectrum. In this communication, these advantages are demonstrated by applying the method to simulated data, for which the correct result is already known.

The mathematics of distance geometry constitutes the basis of a group of algorithms for revealing the structural consequences of diverse forms of information about a macromolecule's conformation. These algorithms are of proven utility in the analysis of experimental conformational data. This paper presents the basic theorems of distance geometry in Euclidean space and gives formal proofs of the correctness and, where possible, of the complexity of these algorithms. The implications of distance geometry for the energy minimization of macromolecules are also discussed.

DISGEO is a new implementation of a distance geometry algorithm which has been specialized for the calculation of macromolecular conformation from distance measurements obtained by two-dimensional nuclear Overhauser enhancement spectroscopy. The improvements include (1) a decomposition of the complete embedding process into two successive, more tractable calculations by the use of “substructures”, (2) a compact data structure for storing incomplete distance information on a structure, (3) a more efficient shortest-path algorithm for computing the triangle inequality limits on all distances from this information, (4) a new algorithm for selecting random metric spaces from within these limits, (5) the use of chirality constraints to obtain good covalent geometry without the use ofad hoc weights or excessive optimization. The utility of the resultant program with nuclear magnetic resonance data is demonstrated by embedding complete spatial structures for the protein basic pancreatic trypsin inhibitor vs all 508 intramolecular, interresidue proton-proton contacts shorter than 4.0 A that were present in its crystal structure. The crystal structure could be reproduced from this data set to within 1.3 A minimum root mean square coordinate difference between the backbone atoms. We conclude that the information potentially available from nuclear magnetic resonance experiments in solution is sufficient to define the spatial structure of small proteins.