Quantum soundness of testing tensor codes

TL;DR

This paper proves that the axis-parallel line vs. point test for tensor codes remains sound even when quantum entangled provers are involved, supporting the robustness of quantum-proof testing in error-correcting codes.

Contribution

It establishes the quantum soundness of the axis-parallel line vs. point test for tensor codes, extending the understanding of quantum-proof property testing.

Findings

The test is sound against quantum entangled provers.

Implication for the quantum-soundness of the low individual degree test.

Generalization to the infinite-dimensional commuting-operator model.

Abstract

A locally testable code is an error-correcting code that admits very efficient probabilistic tests of membership. Tensor codes provide a simple family of combinatorial constructions of locally testable codes that generalize the family of Reed-Muller codes. The natural test for tensor codes, the axis-parallel line vs. point test, plays an essential role in constructions of probabilistically checkable proofs. We analyze the axis-parallel line vs. point test as a two-prover game and show that the test is sound against quantum provers sharing entanglement. Our result implies the quantum-soundness of the low individual degree test, which is an essential component of the MIP* = RE theorem. Our proof also generalizes to the infinite-dimensional commuting-operator model of quantum provers.