Product Rules in Semidefinite Programming

Troy Lee, Rajat Mittal

Thomas Hofmeister, Martin Hühne

Franz Rendl

Mittal, Rajat

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.

Hamza Fawzi, Pablo A. Parrilo

There has been a lot of interest recently in proving lower bounds on the size of linear programs needed to represent a given polytope P. In a breakthrough paper Fiorini et al. [Proceedings of 44th ACM Symposium on Theory of Computing 2012, pages 95-106] showed that any linear programming formulation of maximum-cut must have exponential size. A natural question to ask is whether one can prove such strong lower bounds for semidefinite programming formulations. In this paper we take a step towards this goal and we prove strong lower bounds for a certain class of SDP formulations, namely SDPs over the Cartesian product of fixed-size positive semidefinite cones. In practice this corresponds to semidefinite programs with a block-diagonal structure and where blocks have constant size d. We show that any such extended formulation of the cut polytope must have exponential size (when d is fixed). The result of Fiorini et al. for LP formulations is obtained as a special case when d=1. For blocks of size d=2 the result rules out any small formulations using second-order cone programming. Our study of SDP lifts over Cartesian product of fixed-size positive semidefinite cones is motivated mainly from practical considerations where it is well known that such SDPs can be solved more efficiently than general SDPs. The proof of our lower bound relies on new results about the sparsity pattern of certain matrices with small psd rank, combined with an induction argument inspired from the recent paper by Kaibel and Weltge [arXiv:1307.3543] on the LP extension complexity of the correlation polytope.

Giulio Chiribella

We investigate the optimal estimation of a quantum process that can possibly consist of multiple time steps. The estimation is implemented by a quantum network that interacts with the process by sending an input and processing the output at each time step. We formulate the search for the optimal network as a semidefinite program and use duality theory to give an alternative expression for the maximum payoff achieved by estimation. Combining this formulation with a technique devised by Mittal and Szegedy we prove a general product rule for the joint estimation of independent processes, stating that the optimal joint estimation can be achieved by estimating each process independently, whenever the figure of merit is of a product form. We illustrate the result in several examples and exhibit counterexamples showing that the optimal joint network may not be the product of the optimal individual networks if the processes are not independent or if the figure of merit is not of the product form. In particular, we show that entanglement can reduce by a factor K the variance in the estimation of the sum of K independent phase shifts.