Application of sdp to product rules and quantum query complexity

Abstract

In recent years, semidefinite programming has played a vital role in shaping complexity theory and quantum computing. There have been numerous applications ranging from estimating quantum values, over approximating combinatorial quantities, to proving various bounds. This work extends the use of semidefinite programs (SDPs) to proving product rules and to characterizing quantum query complexity. In the first application, we provide a general framework to establishing product rules for quantities that can be expressed (or approximated) using SDPs. We use duality theory to give product rules, which bound the value of the “product” of two problems in terms of their value. Some previous results have implicitly used the properties of SDPs to give such product rules. Here we give sufficient and necessary conditions under which these approaches work, thereby enabling us to capture these previous results under our unified framework. We also include a discussion about alternate definitions of what a “product” means and how they fit into our approach. The second application provides an SDP characterization of quantum query complexity, which is one of the ways in which complexity of a function can be measured. It is known that quantum query complexity can be lower bounded by the so-called “adversary method” which is expressible as a semidefinite program. Recently, Ben Reichardt showed that the adversary method leads to a tight lower bound for boolean functions by converting the solution of this SDP (of adversary method) into an algorithm. We show that a related SDP, called “witness size” in this thesis, provides a tight bound on the quantum query complexity of non boolean functions (total as well as partial). This witness size SDP is also used to give composition results for quantum query complexity. We also show that the witness size is bounded by a constant multiple of the adversary bound. Finally, we briefly explore whether other convex programming paradigms can be useful in complexity theory. One of them is copositive programming. We show that one of the recent result about parallel repetition of unique games, by Barak et.al., can be interpreted as an application of copositive programming.