Limits of quantum one-way communication by matrix Hypercontractive Inequality

An important discovery in quantum information processing is that quantum one-way communication protocols can be exponentially more efficient than classical protocols. Those extraordinary quantum advantages were demonstrated through the Hidden Matching Problem and its variants, where the underlying basic task is to determine the parity of some k = 2 bits of an n-bit string. We prove that for larger values of k, the quantum complexities of those problems increase exponentially from O(log n) to Ω(n1−2/k), which is almost tight and renders any super-polynomial quantum-classical gaps impossible. Our results also rule out a “quantum argument”, in the sense of Kerenidis and de Wolf (Journal of Computer and System Sciences, 69(3)395–420, 2004), for proving any super-polynomial lower bound on Locally Decodable Codes for more than 2 queries. Our proofs are new applications of the matrix Hypercontractive Inequality developed by Ben-Aroya, Regev, and de Wolf (FOCS 2008). 1 Background and summary of the main results The lower bound problems and motivations. A central question in quantum information processing is to identify its power and limitations in comparison with classical models. Because of the apparent difficulty of the question, researchers have been focusing on simple yet useful models. One-way communication is one such model: Alice and Bob wish to compute a function f (x, y), for which the input x is known to Alice only and y known to Bob only. Alice sends a single message to Bob, who is required to output their best guess for f (x, y). The one-way communication complexity of f is the smallest integer k such that the function can be computed using a length k message