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

Abstract

A locally decodable code encodes n-bit strings x in m-bit codewords C(x), in such a way that one can recover any bit xi 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 need exponential length: m=2Ω(n). 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 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 PIR scheme with O(n3/10) qubits of communication, improving upon the O(n1/3) bits of communication of the best known classical 2-server PIR.