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

Abstract

A locally decodable code (LDC) encodes n-bit strings x in m-bit codewords C( x) in such a way that one can recover any bit x i from a corrupted codeword by querying only a few bits of that word. We use a quantum argument to prove that LDCs with 2 classical queries require exponential length: m=2 Ω(n) . Previously, this was known only for linear codes (Goldreich et al., in: Proceedings of 17th IEEE Conference on Computation Complexity, 2002, pp. 175–183). The proof proceeds by showing that a 2-query LDC can be decoded with a single quantum query, when defined in an appropriate sense. It goes on to establish an exponential lower bound on any ‘1-query locally quantum-decodable code’. We extend our lower bounds to non-binary alphabets and also somewhat improve the polynomial lower bounds by Katz and Trevisan for LDCs with more than 2 queries. Furthermore, we show that q quantum queries allow more succinct LDCs than the best known LDCs with q 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 private information retrieval (PIR) scheme with O( n 3/10) qubits of communication, beating the O( n 1/3) bits of communication of the best known classical 2-server PIR.