Exponential lower bound for 2-query locally decodable codes via a quantum argument

A locally decodable code encodes <i>n</i>-bit strings <i>x</i> in <i>m</i>-bit codewords <i>C(x)</i>, in such a way that one can recover any bit <i>x<sub>i</sub></i> from a corrupted codeword by querying only a few bits of that word. We use a <i>quantum</i> argument to prove that LDCs with 2 classical queries need exponential length: <i>m=2<sup>Ω(n)</sup></i>. Previously this was known only for linear codes (Goldreich et al. 02). Our proof shows that a 2-query LDC can be decoded with only 1 quantum query, and then proves an exponential lower bound for such 1-query locally quantum-decodable codes. We also show that <i>q</i> quantum queries allow more succinct LDCs than the best known LDCs with <i>q</i> classical queries. Finally, we give new classical lower bounds and quantum upper bounds for the setting of private information retrieval. In particular, we exhibit a quantum 2 server PIR scheme with <i>O(n<sup>3/10</sup>)</i> qubits of communication, improving upon the <i>O(n<sup>1/3</sup>)</i> bits of communication of the best known classical 2-server PIR.

[1]  Peter Bro Miltersen,et al.  Are bitvectors optimal? , 2000, STOC '00.

[2]  Daniel A. Spielman,et al.  Nearly-linear size holographic proofs , 1994, STOC '94.

[3]  Richard Beigel,et al.  The polynomial method in circuit complexity , 1993, [1993] Proceedings of the Eigth Annual Structure in Complexity Theory Conference.

[4]  Andris Ambainis,et al.  On Lower Bounds for the Communication Complexity of Private Information Retrieval ∗ , 2000 .

[5]  Ronitt Rubinfeld,et al.  Self-testing/correcting for polynomials and for approximate functions , 1991, STOC '91.

[6]  Madhu Sudan,et al.  Highly Resilient Correctors for Polynomials , 1992, Inf. Process. Lett..

[7]  Yuval Ishai,et al.  Breaking the O(n/sup 1/(2k-1)/) barrier for information-theoretic Private Information Retrieval , 2002, The 43rd Annual IEEE Symposium on Foundations of Computer Science, 2002. Proceedings..

[8]  Jaikumar Radhakrishnan,et al.  The Quantum Complexity of Set Membership , 2002, Algorithmica.

[9]  Joan Feigenbaum,et al.  Random-Self-Reducibility of Complete Sets , 1993, SIAM J. Comput..

[10]  Richard J. Lipton,et al.  New Directions In Testing , 1989, Distributed Computing And Cryptography.

[11]  Jaikumar Radhakrishnan,et al.  Better lower bounds for locally decodable codes , 2002, Proceedings 17th IEEE Annual Conference on Computational Complexity.

[12]  Noam Nisan,et al.  BPP has subexponential time simulations unless EXPTIME has publishable proofs , 1991, [1991] Proceedings of the Sixth Annual Structure in Complexity Theory Conference.

[13]  Ronald de Wolf,et al.  Quantum lower bounds by polynomials , 2001, JACM.

[14]  Srinivasan Venkatesh,et al.  Lower Bounds in the Quantum Cell Probe Model , 2001, ICALP.

[15]  Kenji Obata,et al.  Optimal Lower Bounds for 2-Query Locally Decodable Linear Codes , 2002, RANDOM.

[16]  D. Spielman,et al.  Expander codes , 1996 .

[17]  Renato Renner,et al.  New Bounds in Secret-Key Agreement: The Gap between Formation and Secrecy Extraction , 2003, EUROCRYPT.

[18]  Nicolas Gisin,et al.  Linking Classical and Quantum Key Agreement: Is There a Classical Analog to Bound Entanglement? , 2002, Algorithmica.

[19]  Hoi-Kwong Lo,et al.  Insecurity of Quantum Secure Computations , 1996, ArXiv.

[20]  Luca Trevisan,et al.  Pseudorandom generators without the XOR Lemma (extended abstract) , 1999, STOC '99.

[21]  Wim van Dam,et al.  Quantum Oracle Interrogation: Getting All Information for Almost Half the Price , 1999 .

[22]  Eli Ben-Sasson,et al.  Robust PCPs of Proximity, Shorter PCPs, and Applications to Coding , 2004, SIAM J. Comput..

[23]  Lance Fortnow,et al.  BPP has subexponential time simulations unlessEXPTIME has publishable proofs , 2005, computational complexity.

[24]  Luca Trevisan,et al.  Lower bounds for linear locally decodable codes and private information retrieval , 2002, Proceedings 17th IEEE Annual Conference on Computational Complexity.

[25]  Umesh V. Vazirani,et al.  Quantum complexity theory , 1993, STOC.

[26]  Avi Wigderson,et al.  Extractors: optimal up to constant factors , 2003, STOC '03.

[27]  Leonid A. Levin,et al.  Checking computations in polylogarithmic time , 1991, STOC '91.

[28]  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).

[29]  Elizabeth D Mann.  Private access to distributed information , 1998 .

[30]  Lance Fortnow,et al.  Nearly tight bounds for private information retrieval systems , 2002 .

[31]  Andrew Chi-Chih Yao,et al.  Quantum Circuit Complexity , 1993, FOCS.

[32]  Jonathan Katz,et al.  On the efficiency of local decoding procedures for error-correcting codes , 2000, STOC '00.

[33]  Joan Feigenbaum,et al.  Hiding Instances in Multioracle Queries , 1990, STACS.

[34]  Jaikumar Radhakrishnan,et al.  Privacy and interaction in quantum communication complexity and a theorem about the relative entropy of quantum states , 2002, The 43rd Annual IEEE Symposium on Foundations of Computer Science, 2002. Proceedings..

[35]  Luca Trevisan,et al.  Pseudorandom generators without the XOR lemma , 1999, Proceedings. Fourteenth Annual IEEE Conference on Computational Complexity (Formerly: Structure in Complexity Theory Conference) (Cat.No.99CB36317).

[36]  Eyal Kushilevitz,et al.  Private information retrieval , 1998, JACM.

[37]  Hartmut Klauck,et al.  Interaction in quantum communication and the complexity of set disjointness , 2001, STOC '01.

[38]  M. Sipser,et al.  Limit on the Speed of Quantum Computation in Determining Parity , 1998, quant-ph/9802045.

[39]  Yuval Ishai,et al.  Information-Theoretic Private Information Retrieval: A Unified Construction , 2001, ICALP.