Improved Quantum data analysis

We provide more sample-efficient versions of some basic routines in quantum data analysis, along with simpler proofs. Particularly, we give a quantum ”Threshold Search” algorithm that requires only O((log2 m)/є2) samples of a d-dimensional state ρ. That is, given observables 0 ≤ A1, A2, …, Am ≤ 1 such that (ρ Ai) ≥ 1/2 for at least one i, the algorithm finds j with (ρ Aj) ≥ 1/2−є. As a consequence, we obtain a Shadow Tomography algorithm requiring only O((log2 m)(logd)/є4) samples, which simultaneously achieves the best known dependence on each parameter m, d, є. This yields the same sample complexity for quantum Hypothesis Selection among m states; we also give an alternative Hypothesis Selection method using O((log3 m)/є2) samples.

[1]  Ryan O'Donnell,et al.  The Quantum Union Bound made easy , 2021, SOSA.

[2]  Thomas Steinke,et al.  Private Hypothesis Selection , 2019, IEEE Transactions on Information Theory.

[3]  Ryan O'Donnell,et al.  Quantum Spectrum Testing , 2015, Communications in Mathematical Physics.

[4]  Nengkun Yu.  Sample optimal Quantum identity testing via Pauli Measurements , 2020, 2009.11518.

[5]  Nengkun Yu.  Sample efficient tomography via Pauli Measurements , 2020, 2009.04610.

[6]  2020 IEEE International Symposium on Information Theory (ISIT) , 2020 .

[7]  Sebastien Bubeck,et al.  Entanglement is Necessary for Optimal Quantum Property Testing , 2020, 2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS).

[8]  Patrick Rebentrost,et al.  Robust quantum minimum finding with an application to hypothesis selection , 2020, ArXiv.

[9]  Zhiwei Steven Wu,et al.  Locally Private Hypothesis Selection , 2020, COLT.

[10]  R. Kueng,et al.  Predicting many properties of a quantum system from very few measurements , 2020, Nature Physics.

[11]  Chris Martens,et al.  Theory , 1934, Secrets in Global Governance.

[12]  Guy N. Rothblum,et al.  Gentle measurement of quantum states and differential privacy , 2019, Electron. Colloquium Comput. Complex..

[13]  Nengkun Yu.  Quantum Closeness Testing: A Streaming Algorithm and Applications , 2019, 1904.03218.

[14]  Aaron B. Wagner,et al.  Estimating Quantum Entropy , 2017, IEEE Journal on Selected Areas in Information Theory.

[15]  Ryan O'Donnell,et al.  Quantum state certification , 2017, STOC.

[16]  M. Mézard,et al.  Journal of Statistical Mechanics: Theory and Experiment , 2011 .

[17]  John Watrous,et al.  The Theory of Quantum Information , 2018 .

[18]  Ashwin Nayak,et al.  Online learning of quantum states , 2018, NeurIPS.

[19]  Benjamin Doerr,et al.  An Elementary Analysis of the Probability That a Binomial Random Variable Exceeds Its Expectation , 2017, Statistics & Probability Letters.

[20]  Scott Aaronson,et al.  Shadow tomography of quantum states , 2017, Electron. Colloquium Comput. Complex..

[21]  Ashley Montanaro,et al.  Sequential measurements, disturbance and property testing , 2016, SODA.

[22]  Xiaodi Wu,et al.  Sample-Optimal Tomography of Quantum States , 2015, IEEE Transactions on Information Theory.

[23]  Scott Aaronson,et al.  The Complexity of Quantum States and Transformations: From Quantum Money to Black Holes , 2016, Electron. Colloquium Comput. Complex..

[24]  Raef Bassily,et al.  Algorithmic stability for adaptive data analysis , 2015, STOC.

[25]  Ryan O'Donnell,et al.  Efficient quantum tomography , 2015, STOC.

[26]  Ronald de Wolf,et al.  A Survey of Quantum Property Testing , 2013, Theory Comput..

[27]  Toniann Pitassi,et al.  Preserving Statistical Validity in Adaptive Data Analysis , 2014, STOC.

[28]  Jingliang Gao.  Quantum union bounds for sequential projective measurements , 2014, 1410.5688.

[29]  Aaron Roth,et al.  The Algorithmic Foundations of Differential Privacy , 2014, Found. Trends Theor. Comput. Sci..

[30]  Arleta Szkola,et al.  An asymptotic error bound for testing multiple quantum hypotheses , 2011, 1112.1529.

[31]  Daniel Stefankovic,et al.  Density Estimation in Linear Time , 2007, COLT.

[32]  A. Berlinet,et al.  Divergence criteria for improved selection rules , 2008 .

[33]  G. Mitchison,et al.  The Spectra of Quantum States and the Kronecker Coefficients of the Symmetric Group , 2004, quant-ph/0409016.

[34]  Alison L Gibbs,et al.  On Choosing and Bounding Probability Metrics , 2002, math/0209021.

[35]  Masahito Hayashi,et al.  Quantum universal variable-length source coding , 2002, Physical Review A.

[36]  Luc Devroye,et al.  Combinatorial methods in density estimation , 2001, Springer series in statistics.

[37]  G. Lugosi,et al.  Nonasymptotic universal smoothing factors, kernel complexity and yatracos classes , 1997 .

[38]  G. Lugosi,et al.  A universally acceptable smoothing factor for kernel density estimates , 1996 .

[39]  B. Silverman,et al.  International Statistical Review , 1996 .

[40]  A. Berlinet.  A note on variance reduction , 1995 .

[41]  C. Fuchs,et al.  Mathematical techniques for quantum communication theory , 1995, quant-ph/9604001.

[42]  Y. Yatracos.  Rates of Convergence of Minimum Distance Estimators and Kolmogorov's Entropy , 1985 .

[43]  C. Helstrom.  Quantum detection and estimation theory , 1969 .

[44]  Physical Review , 1965, Nature.