A Hypercontractive Inequality for Matrix-Valued Functions with Applications to Quantum Computing and LDCs

TL;DR

This paper extends the hypercontractive inequality to matrix-valued functions on the Boolean cube and applies it to quantum encoding limits and locally decodable codes, providing new bounds and proofs in quantum computing and coding theory.

Contribution

It introduces a matrix-valued hypercontractive inequality and applies it to derive bounds on quantum encodings and locally decodable codes, bridging Fourier analysis and quantum information.

Findings

Quantum encoding success probability is exponentially small if m<0.7n.

Strong direct product theorems for quantum communication complexity of Disjointness.

Locally decodable codes with 2 queries require exponential length.

Abstract

The Bonami-Beckner hypercontractive inequality is a powerful tool in Fourier analysis of real-valued functions on the Boolean cube. In this paper we present a version of this inequality for matrix-valued functions on the Boolean cube. Its proof is based on a powerful inequality by Ball, Carlen, and Lieb. We also present a number of applications. First, we analyze maps that encode $n$ classical bits into $m$ qubits, in such a way that each set of $k$ bits can be recovered with some probability by an appropriate measurement on the quantum encoding; we show that if $m<0.7 n$, then the success probability is exponentially small in $k$. This result may be viewed as a direct product version of Nayak's quantum random access code bound. It in turn implies strong direct product theorems for the one-way quantum communication complexity of Disjointness and other problems. Second, we prove that error-correcting codes that are locally decodable with 2 queries require length exponential in the length of the encoded string. This gives what is arguably the first ``non-quantum'' proof of a result originally derived by Kerenidis and de Wolf using quantum information theory, and answers a question by Trevisan.