A quantum algorithm to estimate the Gowers U2 norm and linearity testing of Boolean functions

We propose a quantum algorithm to estimate the Gowers $U_2$ norm of a Boolean function, and extend it into a second algorithm to distinguish between linear Boolean functions and Boolean functions that are $\epsilon$-far from the set of linear Boolean functions, which seems to perform better than the classical BLR algorithm. Finally, we outline an algorithm to estimate Gowers $U_3$ norms of Boolean functions.

[1]  Pantelimon Stanica,et al.  Cryptographic Boolean Functions and Applications , 2009 .

[2]  Manuela Herman,et al.  Quantum Computing: A Gentle Introduction , 2011 .

[3]  Victor Y. Chen.  The Gowers norm in the testing of Boolean functions , 2009 .

[4]  W. T. Gowers,et al.  A new proof of Szemerédi's theorem , 2001 .

[5]  Manuel Blum,et al.  Self-testing/correcting with applications to numerical problems , 1990, STOC '90.

[6]  T. Sanders,et al.  Analysis of Boolean Functions , 2012, ArXiv.

[7]  Subhamoy Maitra,et al.  Deutsch-Jozsa Algorithm Revisited in the Domain of Cryptographically Significant Boolean Functions , 2004 .

[8]  Ben Green,et al.  AN INVERSE THEOREM FOR THE GOWERS U4-NORM , 2005, Glasgow Mathematical Journal.

[9]  Terence Tao,et al.  An inverse theorem for the Gowers U^{s+1}[N]-norm , 2010 .

[10]  I. Chuang,et al.  Quantum Computation and Quantum Information: Bibliography , 2010 .

[11]  Daowen Qiu,et al.  Quantum algorithms on Walsh transform and Hamming distance for Boolean functions , 2018, Quantum Inf. Process..

[12]  Claude Carlet,et al.  Recursive Lower Bounds on the Nonlinearity Profile of Boolean Functions and Their Applications , 2008, IEEE Transactions on Information Theory.

[13]  Ben Green,et al.  AN INVERSE THEOREM FOR THE GOWERS $U^3(G)$ NORM , 2008, Proceedings of the Edinburgh Mathematical Society.

[14]  Leonid A. Levin,et al.  A hard-core predicate for all one-way functions , 1989, STOC '89.

[15]  W. Hoeffding.  Probability Inequalities for sums of Bounded Random Variables , 1963 .

[16]  Erika Andersson,et al.  Quantum tests for the linearity and permutation invariance of Boolean functions , 2011, 1106.4831.

[17]  Noam Nisan,et al.  Constant depth circuits, Fourier transform, and learnability , 1993, JACM.

[18]  Leslie G. Valiant,et al.  A theory of the learnable , 1984, STOC '84.

[19]  Subhamoy Maitra,et al.  Efficient Quantum Algorithms Related to Autocorrelation Spectrum , 2018, INDOCRYPT.

[20]  D. Deutsch,et al.  Rapid solution of problems by quantum computation , 1992, Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences.