Low rank matrix recovery from Clifford orbits

We prove that low-rank matrices can be recovered efficiently from a small number of measurements that are sampled from orbits of a certain matrix group. As a special case, our theory makes statements about the phase retrieval problem. Here, the task is to recover a vector given only the amplitudes of its inner product with a small number of vectors from an orbit. Variants of the group in question have appeared under different names in many areas of mathematics. In coding theory and quantum information, it is the complex Clifford group; in time-frequency analysis the oscillator group; and in mathematical physics the metaplectic group. It affords one particularly small and highly structured orbit that includes and generalizes the discrete Fourier basis: While the Fourier vectors have coefficients of constant modulus and phases that depend linearly on their index, the vectors in said orbit have phases with a quadratic dependence. In quantum information, the orbit is used extensively and is known as the set of stabilizer states. We argue that due to their rich geometric structure and their near-optimal recovery properties, stabilizer states form an ideal model for structured measurements for phase retrieval. Our results hold for $m\geq C \kappa_r r d \log(d)$ measurements, where the oversampling factor k varies between $\kappa_r=1$ and $\kappa_r = r^2$ depending on the orbit. The reconstruction is stable towards both additive noise and deviations from the assumption of low rank. If the matrices of interest are in addition positive semidefinite, reconstruction may be performed by a simple constrained least squares regression. Our proof methods could be adapted to cover orbits of other groups.

[1]  Rick P. Millane,et al.  Phase retrieval in crystallography and optics , 1990 .

[2]  R. Kueng Low rank matrix recovery from few orthonormal basis measurements , 2015, 2015 International Conference on Sampling Theory and Applications (SampTA).

[3]  Massimo Fornasier,et al.  Low-rank Matrix Recovery via Iteratively Reweighted Least Squares Minimization , 2010, SIAM J. Optim..

[4]  Thomas W. Parks,et al.  The Weyl correspondence and time-frequency analysis , 1994, IEEE Trans. Signal Process..

[5]  D. M. Appleby Symmetric informationally complete–positive operator valued measures and the extended Clifford group , 2005 .

[6]  Stephanie Koch,et al.  Harmonic Analysis In Phase Space , 2016 .

[7]  T. Tao An uncertainty principle for cyclic groups of prime order , 2003, math/0308286.

[8]  Isaac L. Chuang,et al.  Quantum Computation and Quantum Information (10th Anniversary edition) , 2011 .

[9]  R. Balan,et al.  Painless Reconstruction from Magnitudes of Frame Coefficients , 2009 .

[10]  Andreas Klappenecker,et al.  Mutually unbiased bases are complex projective 2-designs , 2005, Proceedings. International Symposium on Information Theory, 2005. ISIT 2005..

[11]  C. Caves,et al.  Minimal Informationally Complete Measurements for Pure States , 2004, quant-ph/0404137.

[12]  Zak Webb,et al.  The Clifford group forms a unitary 3-design , 2015, Quantum Inf. Comput..

[13]  L. Demanet,et al.  Stable Optimizationless Recovery from Phaseless Linear Measurements , 2012, Journal of Fourier Analysis and Applications.

[14]  Palina Salanevich,et al.  Polarization based phase retrieval for time-frequency structured measurements , 2015, 2015 International Conference on Sampling Theory and Applications (SampTA).

[15]  John I. Haas,et al.  Achieving the orthoplex bound and constructing weighted complex projective 2-designs with Singer sets , 2015, 1509.05333.

[16]  Xiaodong Li,et al.  Solving Quadratic Equations via PhaseLift When There Are About as Many Equations as Unknowns , 2012, Found. Comput. Math..

[17]  Emmanuel J. Candès,et al.  Exact Matrix Completion via Convex Optimization , 2008, Found. Comput. Math..

[18]  Markus Grassl,et al.  The Clifford group fails gracefully to be a unitary 4-design , 2016, 1609.08172.

[19]  Daniel Gottesman,et al.  Stabilizer Codes and Quantum Error Correction , 1997, quant-ph/9705052.

[20]  Martin Ehler,et al.  Phase retrieval using random cubatures and fusion frames of positive semidefinite matrices , 2015, 1505.05003.

[21]  R. Kueng,et al.  Spherical designs as a tool for derandomization: The case of PhaseLift , 2015, 2015 International Conference on Sampling Theory and Applications (SampTA).

[22]  Joel A. Tropp,et al.  Convex recovery of a structured signal from independent random linear measurements , 2014, ArXiv.

[23]  Gabriele Nebe,et al.  Self-dual codes and invariant theory , 2009, Algebraic Aspects of Digital Communications.

[24]  Richard Kueng,et al.  Qubit stabilizer states are complex projective 3-designs , 2015, ArXiv.

[25]  B. Moor,et al.  Stabilizer states and Clifford operations for systems of arbitrary dimensions and modular arithmetic , 2004, quant-ph/0408190.

[26]  O. Bunk,et al.  Phase retrieval and differential phase-contrast imaging with low-brilliance X-ray sources , 2006 .

[27]  Yonina C. Eldar,et al.  Phase Retrieval via Matrix Completion , 2011, SIAM Rev..

[28]  Christina Gloeckner Foundations Of Time Frequency Analysis , 2016 .

[29]  Emmanuel J. Candès,et al.  Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information , 2004, IEEE Transactions on Information Theory.

[30]  J R Fienup,et al.  Phase retrieval algorithms: a comparison. , 1982, Applied optics.

[31]  Holger Boche,et al.  Phaseless Signal Recovery in Infinite Dimensional Spaces Using Structured Modulations , 2013, ArXiv.

[32]  D. Gross Hudson's theorem for finite-dimensional quantum systems , 2006, quant-ph/0602001.

[33]  Stephen Becker,et al.  Quantum state tomography via compressed sensing. , 2009, Physical review letters.

[34]  Pablo A. Parrilo,et al.  Guaranteed Minimum-Rank Solutions of Linear Matrix Equations via Nuclear Norm Minimization , 2007, SIAM Rev..

[35]  B. Bodmann,et al.  Algorithms and error bounds for noisy phase retrieval with low-redundancy frames , 2014, 1412.6678.

[36]  J. Neumann Die Eindeutigkeit der Schrödingerschen Operatoren , 1931 .

[37]  Holger Rauhut,et al.  Stable low-rank matrix recovery via null space properties , 2015, ArXiv.

[38]  R. Hudson When is the wigner quasi-probability density non-negative? , 1974 .

[39]  David Gross,et al.  Recovering Low-Rank Matrices From Few Coefficients in Any Basis , 2009, IEEE Transactions on Information Theory.

[40]  Emmanuel J. Candès,et al.  Near-Optimal Signal Recovery From Random Projections: Universal Encoding Strategies? , 2004, IEEE Transactions on Information Theory.

[41]  D. Gross,et al.  Focus on quantum tomography , 2013 .

[42]  Jonas Helsen,et al.  Representations of the multi-qubit Clifford group , 2016, Journal of Mathematical Physics.

[43]  Dustin G. Mixon,et al.  Phase Retrieval with Polarization , 2012, SIAM J. Imaging Sci..

[44]  Maarten Van den Nest,et al.  The LU-LC conjecture, diagonal local operations and quadratic forms over GF(2) , 2007, Quantum Inf. Comput..

[45]  Holger Rauhut,et al.  A Mathematical Introduction to Compressive Sensing , 2013, Applied and Numerical Harmonic Analysis.

[46]  Joseph M. Renes,et al.  Symmetric informationally complete quantum measurements , 2003, quant-ph/0310075.

[47]  Weiyu Xu,et al.  Null space conditions and thresholds for rank minimization , 2011, Math. Program..

[48]  Andris Ambainis,et al.  Quantum t-designs: t-wise Independence in the Quantum World , 2007, Twenty-Second Annual IEEE Conference on Computational Complexity (CCC'07).

[49]  Felix Krahmer,et al.  Improved Recovery Guarantees for Phase Retrieval from Coded Diffraction Patterns , 2014, arXiv.org.

[50]  A. Walther The Question of Phase Retrieval in Optics , 1963 .

[51]  R. Kueng,et al.  Distinguishing quantum states using Clifford orbits , 2016, 1609.08595.

[52]  F. Brandão,et al.  Local random quantum circuits are approximate polynomial-designs: numerical results , 2012, 1208.0692.

[53]  W. H. Benner,et al.  Femtosecond diffractive imaging with a soft-X-ray free-electron laser , 2006, physics/0610044.

[54]  Weiyu Xu,et al.  Necessary and sufficient conditions for success of the nuclear norm heuristic for rank minimization , 2008, 2008 47th IEEE Conference on Decision and Control.

[55]  T. Heinosaari,et al.  Quantum Tomography under Prior Information , 2011, 1109.5478.

[56]  J. H. Seldin,et al.  Hubble Space Telescope characterized by using phase-retrieval algorithms. , 1993, Applied optics.

[57]  M. Fazel,et al.  Iterative reweighted least squares for matrix rank minimization , 2010, 2010 48th Annual Allerton Conference on Communication, Control, and Computing (Allerton).

[58]  J. Seidel,et al.  SPHERICAL CODES AND DESIGNS , 1991 .

[59]  Holger Rauhut,et al.  Low rank matrix recovery from rank one measurements , 2014, ArXiv.

[60]  Yi-Kai Liu,et al.  Universal low-rank matrix recovery from Pauli measurements , 2011, NIPS.

[61]  Steven T. Flammia,et al.  Quantum tomography via compressed sensing: error bounds, sample complexity and efficient estimators , 2012, 1205.2300.

[62]  Eiichi Bannai,et al.  A survey on spherical designs and algebraic combinatorics on spheres , 2009, Eur. J. Comb..

[63]  Jan Peřina,et al.  Quantum optics and fundamentals of physics , 1994 .

[64]  Felix Krahmer,et al.  Phase Retrieval Without Small-Ball Probability Assumptions , 2016, IEEE Transactions on Information Theory.

[65]  J. Niel de Beaudrap,et al.  A linearized stabilizer formalism for systems of finite dimension , 2011, Quantum Inf. Comput..

[66]  N. Mermin Quantum theory: Concepts and methods , 1997 .

[67]  S. Frick,et al.  Compressed Sensing , 2014, Computer Vision, A Reference Guide.

[68]  Andrea Montanari,et al.  Matrix completion from a few entries , 2009, 2009 IEEE International Symposium on Information Theory.

[69]  Scott Aaronson,et al.  Improved Simulation of Stabilizer Circuits , 2004, ArXiv.

[70]  Felix Krahmer,et al.  A Partial Derandomization of PhaseLift Using Spherical Designs , 2013, Journal of Fourier Analysis and Applications.

[71]  Emmanuel J. Candès,et al.  PhaseLift: Exact and Stable Signal Recovery from Magnitude Measurements via Convex Programming , 2011, ArXiv.

[72]  Huangjun Zhu Multiqubit Clifford groups are unitary 3-designs , 2015, 1510.02619.

[73]  W. Duke On codes and Siegel modular forms , 1993 .

[74]  Xiaodong Li,et al.  Phase Retrieval from Coded Diffraction Patterns , 2013, 1310.3240.

[75]  E. Candès,et al.  Stable signal recovery from incomplete and inaccurate measurements , 2005, math/0503066.

[76]  Shahar Mendelson,et al.  Learning without Concentration , 2014, COLT.

[77]  Emmanuel J. Candès,et al.  The Power of Convex Relaxation: Near-Optimal Matrix Completion , 2009, IEEE Transactions on Information Theory.

[78]  Michael Kech,et al.  Explicit Frames for Deterministic Phase Retrieval via PhaseLift , 2015, Applied and Computational Harmonic Analysis.

[79]  G. Pfander Gabor Frames in Finite Dimensions , 2013 .

[80]  V. Koltchinskii,et al.  Bounding the smallest singular value of a random matrix without concentration , 2013, 1312.3580.

[81]  G. Mackey The theory of unitary group representations , 1976 .