An introduction to commutative and noncommutative Gro¨bner bases
Selected papers of the second international colloquium on Words, languages and combinatorics
SIAM Review
Global Optimization with Polynomials and the Problem of Moments
SIAM Journal on Optimization
The Quantum Moment Problem and Bounds on Entangled Multi-prover Games
CCC '08 Proceedings of the 2008 IEEE 23rd Annual Conference on Computational Complexity
Algorithmic aspects of sums of Hermitian squares of noncommutative polynomials
Computational Optimization and Applications
Product-state approximations to quantum ground states
Proceedings of the forty-fifth annual ACM symposium on Theory of computing
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We consider optimization problems with polynomial inequality constraints in noncommuting variables. These noncommuting 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.