A Lattice Problem in Quantum NP
Abstract
We consider coGapSVP_\sqrt{n}, a gap version of the shortest vector in a lattice problem. This problem is known to be in AM\cap coNP but is not known to be in NP or in MA. We prove that it lies inside QMA, the quantum analogue of NP. This is the first nontrivial upper bound on the quantum complexity of a lattice problem. The proof relies on two novel ideas. First, we give a new characterization of QMA, called QMA+. Working with the QMA+ formulation allows us to circumvent a problem which arises commonly in the context of QMA: the prover might use entanglement between different copies of the same state in order to cheat. The second idea involves using estimations of autocorrelation functions for verification. We make the important observation that autocorrelation functions are positive definite functions and using properties of such functions we severely restrict the prover's possibility to cheat. We hope that these ideas will lead to further developments in the field.
 Publication:

arXiv eprints
 Pub Date:
 July 2003
 arXiv:
 arXiv:quantph/0307220
 Bibcode:
 2003quant.ph..7220A
 Keywords:

 Quantum Physics