Lower Bounds on Matrix Rigidity Via a Quantum Argument

The rigidity of a matrix measures how many of its entries need to be changed in order to reduce its rank to some value. Good lower bounds on the rigidity of an explicit matrix would imply good lower bounds for arithmetic circuits as well as for communication complexity. Here we reprove the best known bounds on the rigidity of Hadamard matrices, due to Kashin and Razborov, using tools from quantum computing. Our proofs are somewhat simpler than earlier ones (at least for those familiar with quantum) and give slightly better constants. More importantly, they give a new approach to attack this longstanding open problem.

[1]  A. Razborov,et al.  Improved lower bounds on the rigidity of Hadamard matrices , 1998 .

[2]  Joel Friedman,et al.  A note on matrix rigidity , 1993, Comb..

[3]  Scott Aaronson,et al.  Quantum computing, postselection, and probabilistic polynomial-time , 2004, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[4]  Dorit Aharonov,et al.  A lattice problem in quantum NP , 2003, 44th Annual IEEE Symposium on Foundations of Computer Science, 2003. Proceedings..

[5]  Daniel A. Spielman,et al.  A Remark on Matrix Rigidity , 1997, Inf. Process. Lett..

[6]  Iordanis Kerenidis,et al.  Quantum multiparty communication complexity and circuit lower bounds , 2005, Mathematical Structures in Computer Science.

[7]  Noga Alon,et al.  The Probabilistic Method , 2015, Fundamentals of Ramsey Theory.

[8]  Kyriakos Kalorkoti.  ALGEBRAIC COMPLEXITY THEORY (Grundlehren der Mathematischen Wissenschaften 315) , 1999 .

[9]  Leslie G. Valiant,et al.  Graph-Theoretic Arguments in Low-Level Complexity , 1977, MFCS.

[10]  Scott Aaronson,et al.  Lower bounds for local search by quantum arguments , 2003, STOC '04.

[11]  Gatis Midrijanis,et al.  Three lines proof of the lower bound for the matrix rigidity , 2005, ArXiv.

[12]  Noga Alon,et al.  The Probabilistic Method, Second Edition , 2004 .

[13]  Александр Александрович Разборов,et al.  Новые нижние оценки устойчивости матриц Адамара@@@Improved lower bounds on the rigidity of Hadamard matrices , 1998 .

[14]  Satyanarayana V. Lokam.  On the rigidity of Vandermonde matrices , 2000, Theor. Comput. Sci..

[15]  Satyanarayana V. Lokam.  Spectral methods for matrix rigidity with applications to size-depth tradeoffs and communication complexity , 1995, Proceedings of IEEE 36th Annual Foundations of Computer Science.

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

[17]  Michael Clausen,et al.  Algebraic complexity theory , 1997, Grundlehren der mathematischen Wissenschaften.

[18]  Dorit Aharonov,et al.  Lattice Problems in NP cap coNP , 2004, FOCS.

[19]  Ronald de Wolf,et al.  Improved Lower Bounds for Locally Decodable Codes and Private Information Retrieval , 2004, ICALP.

[20]  Stasys Jukna,et al.  Extremal Combinatorics , 2001, Texts in Theoretical Computer Science. An EATCS Series.

[21]  Ashwin Nayak,et al.  Optimal lower bounds for quantum automata and random access codes , 1999, 40th Annual Symposium on Foundations of Computer Science (Cat. No.99CB37039).

[22]  Troy Lee,et al.  THE QUANTUM ADVERSARY METHOD AND CLASSICAL FORMULA SIZE LOWER BOUNDS , 2005, 20th Annual IEEE Conference on Computational Complexity (CCC'05).

[23]  Satyanarayana V. Lokam.  Quadratic Lower Bounds on Matrix Rigidity , 2006, TAMC.

[24]  Satyanarayana V. Lokam.  Spectral Methods for Matrix Rigidity with Applications to Size-Depth Trade-offs and Communication Complexity , 2001, J. Comput. Syst. Sci..

[25]  Thierry Paul,et al.  Quantum computation and quantum information , 2007, Mathematical Structures in Computer Science.

[26]  Dorit Aharonov,et al.  Lattice problems in NP ∩ coNP , 2005, JACM.