On the power of quantum proofs
Ran Raz, Amir Shpilka · 2004
We study the power of quantum proofs, or more precisely, the power of Quantum Merlin-Arthur (QMA) protocols, in two well studied models of quantum computation: the black box model and the communication complexity model. Our main results are obtained for the communication complexity model. For this model, we identify a complete promise problem for QMA protocols, the Linear Subspaces Distance problem. The problem is of geometrical nature: Each player gets a linear subspace of R m and considers the sphere of unit vectors in that subspace. Their goal is to output 1 if the distance between the two spheres is very small (say, smaller than 0.1 · √ 2) and 0 if the distance is very large (say, larger than 0.9 · √ 2). We show that: 1. The QMA communication complexity of the problem is O(log m). 2. The (classical) MA communication complexity of the problem is Ω(m ɛ) (for some ɛ> 0). 3. The (standard) quantum communication complexity of the problem is Ω ( √ m). In particular, this gives an exponential separation between QMA communication complexity and MA communication complexity.