CS: Compressive Sampling for Large-Scale Open Quantum Systems.

109 pages for an approach to quantum sensing using random projections in Practical recipes for the model order reduction, dynamical simulation, and compressive sampling of large-scale open quantum systems by John Sidles, Joseph Garbini, L. E. Harrell, Alfred Hero, Jon Jacky, Joe Malcomb, A. G. Norman and A. M. Williamson promise us. The abstract reads:

This article presents practical numerical recipes for simulating high-temperature and nonequilibrium quantum spin systems that are continuously measured and controlled. The notion of a \spin system" is broadly conceived, in order to encompass macroscopic test masses as the limiting case of large-j spins. The simulation technique has three stages: first the deliberate introduction of noise into the simulation, then the conversion of that noise into an informatically equivalent continuous measurement and control process, and finally, projection of the trajectory onto a Kahlerian state-space manifold having reduced dimensionality and possessing a Kahler potential of multilinear (i.e., product-sum) functional form. These state-spaces can be regarded as ruled algebraic varieties upon which a projective quantum model order reduction (QMOR) is performed. The Riemannian sectional curvature of ruled Kahlerian varieties is analyzed, and proved to be non-positive upon all sections that contain a rule. It is further shown that the class of ruled Kahlerian state-spaces includes the Slater determinant wave-functions of quantum chemistry as a special case, and that these Slater determinant manifolds have a Fubini-Study metric that is Kahler-Einstein; hence they are solitons under Ricci flow. It is suggested that these negative sectional curvature properties geometrically account for the fidelity, efficiency, and robustness of projective trajectory simulation on ruled Kahlerian state-spaces. Some implications of trajectory compression for geometric quantum mechanics are discussed. The resulting simulation formalism is used to construct a positive P-representation for the thermal density matrix and to derive a quantum limit for force noise and measurement noise in monitoring both macroscopic and microscopic test-masses; this quantum noise limit is shown to be consistent with well established quantum noise limits for linear amplifiers and for monitoring linear dynamical systems. Single-spin detection by magnetic resonance force microscopy (MRFM) is then simulated, and the data statistics are shown to be those of a random telegraph signal with additive white noise, to all orders, in excellent agreement with experimental results. Then a larger-scale spin-dust model is simulated, having no spatial symmetry and no spatial ordering; the high-fidelity projection of numerically computed quantum trajectories onto low-dimensionality Kahler state-space manifolds is demonstrated. Finally, the high-fidelity reconstruction of quantum trajectories from sparse random projections is demonstrated, the onset of Donoho-Stodden breakdown at the Candes Tao sparsity limit is observed, and methods for quantum state optimization by Dantzig selection are given.

Paragraph 4.6 page 79 is the one most focused on CS:

.....We will conclude our survey of spin-dust simulations with some concrete calculations that are motivated by recent advances in the theory of compressive sampling (CS) and sparse reconstruction. It will become apparent that synoptic simulations of quantum trajectories mesh very naturally with CS methods and ideas. To the best of our knowledge, this is the first description of CS methods applied to quantum state-spaces. Our analysis will mainly draw upon the ideas and methods of Donoho [61] and of Candes and Tao [37], and our discussion will assume a basic familiarity with these and similar CS articles [30, 31, 32, 34, 36], especially a recent series of articles and commentaries on the Dantzig selector [17, 29, 38, 64, 79, 137, 167]. Our analysis can alternatively be viewed as an extension to the quantum domain of the approach of Baraniuk, Hegde, and Wakin [11, 47, 194] to manifold learning [188] from sparse random projections.
Our objectives in this section are:

• establish that synoptically simulated wave functions 0 are compressible objects in the
• sense of Candes and Tao [37],
• Establish that high fidelity quantum state reconstruction from sparse random projections
• is algorithmically tractable,
• Describe how nonlinear GK projection can be described as an embedding within a larger linear state-space of a convex optimization problem, and thereby
• Specify algorithms for optimization over quantum states in terms of the Dantzig selector (a linear convex optimization algorithm) of Candes and Tao [37].

At the time of writing, the general field of compressive sensing, sampling and simulation is evolving rapidly"Nowadays, novel exciting results seem to come out at a furious pace, and this testifies to the vitality and intensity of the field "[8] and our overall goal is to provide mathematical recipes by which researchers in quantum sensing, sampling and simulation can participate in this enterprise.
4.6.1. Establishing that quantum states are compressible objects .....