Solving Commutative Relaxations of Word Problems

We present an algebraic characterization of the standard commutative relaxation of the word problem in terms of a polynomial equality. We then consider a variant of the commutative word problem, referred to as the “Zero-to-All reachability” problem. We show that this problem is equivalent to a finite number of commutative word problems, and we use this insight to derive necessary conditions for Zero-to-All reachability. We conclude with a set of illustrative examples

Algorithm 950: Ncpol2sdpa---Sparse Semidefinite Programming Relaxations for Polynomial Optimization Problems of Noncommuting Variables

A hierarchy of semidefinite programming (SDP) relaxations approximates the global optimum of polynomial optimization problems of noncommuting variables. Generating the relaxation, however, is a computationally demanding task, and only problems of commuting variables have efficient generators. We develop an implementation for problems of noncommuting problems that creates the relaxation to be solved by SDPA -- a high-performance solver that runs in a distributed environment. We further exploit the inherent sparsity of optimization problems in quantum physics to reduce the complexity of the resulting relaxations. Constrained problems with a relaxation of order two may contain up to a hundred variables. The implementation is available in Python. The tool helps solve problems such as finding the ground state energy or testing quantum correlations.Comment: 17 pages, 3 figures, 1 table, 2 algorithms, the algorithm is available at http://peterwittek.github.io/ncpol2sdpa

A paradox in bosonic energy computations via semidefinite programming relaxations

We show that the recent hierarchy of semidefinite programming relaxations based on non-commutative polynomial optimization and reduced density matrix variational methods exhibits an interesting paradox when applied to the bosonic case: even though it can be rigorously proven that the hierarchy collapses after the first step, numerical implementations of higher order steps generate a sequence of improving lower bounds that converges to the optimal solution. We analyze this effect and compare it with similar behavior observed in implementations of semidefinite programming relaxations for commutative polynomial minimization. We conclude that the method converges due to the rounding errors occurring during the execution of the numerical program, and show that convergence is lost as soon as computer precision is incremented. We support this conclusion by proving that for any element p of a Weyl algebra which is non-negative in the Schrodinger representation there exists another element p' arbitrarily close to p that admits a sum of squares decomposition.Comment: 22 pages, 4 figure

Convergent relaxations of polynomial optimization problems with non-commuting variables

We consider optimization problems with polynomial inequality constraints in non-commuting variables. These non-commuting variables are viewed as bounded operators on a Hilbert space whose dimension is not fixed and the associated polynomial inequalities as semidefinite positivity constraints. Such problems arise naturally in quantum theory and quantum information science. To solve them, we introduce a hierarchy of semidefinite programming relaxations which generates a monotone sequence of lower bounds that converges to the optimal solution. We also introduce a criterion to detect whether the global optimum is reached at a given relaxation step and show how to extract a global optimizer from the solution of the corresponding semidefinite programming problem.Comment: 35 pages. v2: Improved notation and revised proof of Theorem

The matricial relaxation of a linear matrix inequality

Given linear matrix inequalities (LMIs) L_1 and L_2, it is natural to ask: (Q1) when does one dominate the other, that is, does L_1(X) PsD imply L_2(X) PsD? (Q2) when do they have the same solution set? Such questions can be NP-hard. This paper describes a natural relaxation of an LMI, based on substituting matrices for the variables x_j. With this relaxation, the domination questions (Q1) and (Q2) have elegant answers, indeed reduce to constructible semidefinite programs. Assume there is an X such that L_1(X) and L_2(X) are both PD, and suppose the positivity domain of L_1 is bounded. For our "matrix variable" relaxation a positive answer to (Q1) is equivalent to the existence of matrices V_j such that L_2(x)=V_1^* L_1(x) V_1 + ... + V_k^* L_1(x) V_k. As for (Q2) we show that, up to redundancy, L_1 and L_2 are unitarily equivalent. Such algebraic certificates are typically called Positivstellensaetze and the above are examples of such for linear polynomials. The paper goes on to derive a cleaner and more powerful Putinar-type Positivstellensatz for polynomials positive on a bounded set of the form {X | L(X) PsD}. An observation at the core of the paper is that the relaxed LMI domination problem is equivalent to a classical problem. Namely, the problem of determining if a linear map from a subspace of matrices to a matrix algebra is "completely positive".Comment: v1: 34 pages, v2: 41 pages; supplementary material is available in the source file, or see http://srag.fmf.uni-lj.si