Parallelized Solution to Semidefinite Programmings in Quantum Complexity Theory

Abstract: In this paper we present an equilibrium value based framework for solving SDPs via the multiplicative weight update method which is different from the one in Kale's thesis \cite{Kale07}. One of the main advantages of the new framework is that we can guarantee the convertibility from approximate to exact feasibility in a much more general class of SDPs than previous result. Another advantage is the design of the oracle which is necessary for applying the multiplicative weight update method is much simplified in general cases. This leads to an alternative and easier solutions to the SDPs used in the previous results \class{QIP(2)}$\subseteq$\class{PSPACE} \cite{JainUW09} and \class{QMAM}=\class{PSPACE} \cite{JainJUW09}. Furthermore, we provide a generic form of SDPs which can be solved in the similar way. By parallelizing every step in our solution, we are able to solve a class of SDPs in \class{NC}. Although our motivation is from quantum computing, our result will also apply directly to any SDP which satisfies our conditions. In addition to the new framework for solving SDPs, we also provide a novel framework which improves the range of equilibrium value problems that can be solved via the multiplicative weight update method. Before this work we are only able to calculate the equilibrium value where one of the two convex sets needs to be the set of density operators. Our work demonstrates that in the case when one set is the set of density operators with further linear constraints, we are still able to approximate the equilibrium value to high precision via the multiplicative weight update method.