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Recovering Low-Rank Matrices From Few Coefficients in Any Basis
- D. Gross
- Computer ScienceIEEE Transactions on Information Theory
- 12 October 2009
It is shown that an unknown matrix of rank can be efficiently reconstructed from only randomly sampled expansion coefficients with respect to any given matrix basis, which quantifies the “degree of incoherence” between the unknown matrix and the basis.
Quantum state tomography via compressed sensing.
- D. Gross, Yi-Kai Liu, S. Flammia, Stephen Becker, J. Eisert
- PhysicsPhysical Review Letters
- 18 September 2009
These methods are specialized for quantum states that are fairly pure, and they offer a significant performance improvement on large quantum systems, and are able to reconstruct an unknown density matrix of dimension d and rank r using O(rdlog²d) measurement settings, compared to standard methods that require d² settings.
Hudson's theorem for finite-dimensional quantum systems
- D. Gross
- 1 February 2006
We show that, on a Hilbert space of odd dimension, the only pure states to possess a non-negative Wigner function are stabilizer states. The Clifford group is identified as the set of unitary…
Corrigendum: Negative quasi-probability as a resource for quantum computation
The full text of this article is available in the PDF provided.
Index Theory of One Dimensional Quantum Walks and Cellular Automata
If a one-dimensional quantum lattice system is subject to one step of a reversible discrete-time dynamics, it is intuitive that as much “quantum information” as moves into any given block of cells…
Quantum tomography via compressed sensing: error bounds, sample complexity and efficient estimators
This paper shows how to certify the accuracy of a low-rank estimate using direct fidelity estimation, and describes a method for compressed quantum process tomography that works for processes with small Kraus rank and requires only Pauli eigenstate preparations and Pauli measurements.
Most quantum States are too entangled to be useful as computational resources.
It is shown that quantum states can be too entangled to be useful for the purpose of computation, in that high values of the geometric measure of entanglement preclude states from offering a universal quantum computational speedup.
Efficient quantum state tomography.
Two tomography schemes that scale much more favourably than direct tomography with system size are presented, one of them requires unitary operations on a constant number of subsystems, whereas the other requires only local measurements together with more elaborate post-processing.
Evenly distributed unitaries: On the structure of unitary designs
We clarify the mathematical structure underlying unitary t-designs. These are sets of unitary matrices, evenly distributed in the sense that the average of any tth order polynomial over the design…
Novel schemes for measurement-based quantum computation.
It is shown that there exist resource states which are locally arbitrarily close to a pure state, and comment on the possibility of tailoring computational models to specific physical systems.