Efficient Learning of Non-Interacting Fermion Distributions

We give an efficient algorithm that recovers the distribution of a non-interacting fermion state over the standard basis, given measurements in additional bases. For a system of n non-interacting fermions and m modes, we show that O ( m 2 n 2 log(1 /δ ) /(cid:15) 2 ) copies of the input state and O ( m 3 n 2 log(1 /δ ) /(cid:15) 2 ) time are sufficient to learn the original distribution to total variation distance (cid:15) with probability at least 1 − δ . Our algorithm empirically estimates one-mode correlations in O ( m ) different measurement bases and uses them to reconstruct a succinct description of the entire distribution efficiently.

[1]  G. Low.  Classical shadows of fermions with particle number symmetry , 2022, 2208.08964.

[2]  Theodore J. Yoder,et al.  Optimal algorithms for learning quantum phase states , 2022, ArXiv.

[3]  B. O’Gorman.  Fermionic tomography and learning , 2022, arXiv.org.

[4]  Joonho Lee,et al.  Matchgate Shadows for Fermionic Quantum Simulation , 2022, 2207.13723.

[5]  Mauro E. S. Morales,et al.  Fermion Sampling: A Robust Quantum Computational Advantage Scheme Using Fermionic Linear Optics and Magic Input States , 2020, PRX Quantum.

[6]  R. O'Donnell,et al.  Improved Quantum data analysis , 2020, STOC.

[7]  A. Miyake,et al.  Fermionic Partial Tomography via Classical Shadows. , 2020, Physical review letters.

[8]  C. Canonne.  A short note on learning discrete distributions , 2020, 2002.11457.

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

[10]  Joel A. Tropp,et al.  Fast state tomography with optimal error bounds , 2018, Journal of Physics A: Mathematical and Theoretical.

[11]  Andrea Rocchetto,et al.  Stabiliser states are efficiently PAC-learnable , 2017, Quantum Inf. Comput..

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

[13]  Ashley Montanaro,et al.  Learning stabilizer states by Bell sampling , 2017, 1707.04012.

[14]  Ankur Moitra,et al.  Learning Determinantal Point Processes with Moments and Cycles , 2017, ICML.

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

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

[17]  Ben Taskar,et al.  An efficient algorithm for the symmetric principal minor assignment problem , 2015 .

[18]  Alex Arkhipov,et al.  BosonSampling is robust against small errors in the network matrix , 2014, 1412.2516.

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

[20]  R. Gillan.  New Editor-in-Chief for Journal of Physics A: Mathematical and Theoretical , 2014 .

[21]  Lance Fortnow,et al.  Proceedings of the forty-third annual ACM symposium on Theory of computing , 2011, STOC 2011.

[22]  Scott Aaronson,et al.  The computational complexity of linear optics , 2010, STOC '11.

[23]  D. Gross,et al.  Efficient quantum state tomography. , 2010, Nature communications.

[24]  I. Ial,et al.  Nature Communications , 2010, Nature Cell Biology.

[25]  M. Ehler.  Applied and Computational Harmonic Analysis , 2008 .

[26]  Scott Aaronson,et al.  The learnability of quantum states , 2006, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[27]  Vlatko Vedral,et al.  Introduction to Quantum Information Science , 2006 .

[28]  Leslie G. Valiant,et al.  Quantum Circuits That Can Be Simulated Classically in Polynomial Time , 2002, SIAM J. Comput..

[29]  Emanuel Knill,et al.  Fermionic Linear Optics and Matchgates , 2001, ArXiv.

[30]  David P. DiVincenzo,et al.  Classical simulation of noninteracting-fermion quantum circuits , 2001, ArXiv.

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

[32]  Charles H. Bennett,et al.  Quantum information and computation , 1995, Nature.

[33]  Axthonv G. Oettinger,et al.  IEEE Transactions on Information Theory , 1998 .

[34]  Reck,et al.  Experimental realization of any discrete unitary operator. , 1994, Physical review letters.

[35]  Audra E. Kosh,et al.  Linear Algebra and its Applications , 1992 .

[36]  Volker Strassen,et al.  Algebraic Complexity Theory , 1991, Handbook of Theoretical Computer Science, Volume A: Algorithms and Complexity.

[37]  Euclid,et al.  Advances in Applied Probability , 1981, Journal of Applied Probability.

[38]  O. Macchi.  The coincidence approach to stochastic point processes , 1975, Advances in Applied Probability.

[39]  P. Schönemann,et al.  A generalized solution of the orthogonal procrustes problem , 1966 .

[40]  Physical Review , 1965, Nature.