Dissertation/ Thesis
Προβολή σε EDS

Some Aspects of Noncommutativity in Polynomial Optimization

Λεπτομέρειες βιβλιογραφικής εγγραφής
Τίτλος: Some Aspects of Noncommutativity in Polynomial Optimization
Συγγραφείς: Mousavi Haji, Seyyed Hamoon
Έτος έκδοσης: 2023
Συλλογή: Columbia University: Academic Commons
Θεματικοί όροι: Computer science, Quantum computers, Mathematical optimization--Computer programs, Polynomials, Noncommutative algebras, Quantum theory--Mathematics
Περιγραφή: Most combinatorial optimization problems from theoretical computer science have a natural framing as optimization of polynomials in commuting variables. Noncommutativity is one of the defining features of quantum mechanics. So it is not surprising that noncommutative polynomial optimization plays an equally important role in quantum computer science. Our main goal here is to understand the relative hardness of commutative versus noncommutative polynomial optimization. At a first glance it might seem that noncommutative polynomial optimization must be more complex. However this is not always true and this question of relative hardness is substantially more subtle than might appear at the outset. First in this thesis we show that the general noncommutative polynomial optimization is complete for the class $\Pi_2$; this class is in the second level of the arithmetical hierarchy and strictly contains both the set of recursively enumerable languages and its complement. On the other hand, commutative polynomial optimization is decidable and belongs to $\PSPACE$. We then provide evidence that for polynomials arising from a large class of constraint satisfaction problems the situation is reversed: the noncommutative polynomial optimization is an easier computational problem compared to its commutative analogue. A second question we are interested in is about whether we could extract good commutative solutions from noncommutative solutions? This brings us to the second theme of this thesis which is about understanding the algebraic structure of the solutions of noncommutative polynomial optimization. We show that this structural insight then could shed light on the optimal commutative solutions and thereby paves the path in understanding the relationships between the commutative and noncommutative solutions. Here we first use the sum-of-squares framework to understand the algebraic relationships that are present between operators in any optimal noncommutative solution of a class of polynomial optimization problems ...
Τύπος εγγράφου: thesis
Γλώσσα: English
DOI: 10.7916/x62q-ha25
Διαθεσιμότητα: https://doi.org/10.7916/x62q-ha25
Αριθμός Καταχώρησης: edsbas.A2AE47C0
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  Data: Some Aspects of Noncommutativity in Polynomial Optimization
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  Data: <searchLink fieldCode="DE" term="%22Computer+science%22">Computer science</searchLink><br /><searchLink fieldCode="DE" term="%22Quantum+computers%22">Quantum computers</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematical+optimization--Computer+programs%22">Mathematical optimization--Computer programs</searchLink><br /><searchLink fieldCode="DE" term="%22Polynomials%22">Polynomials</searchLink><br /><searchLink fieldCode="DE" term="%22Noncommutative+algebras%22">Noncommutative algebras</searchLink><br /><searchLink fieldCode="DE" term="%22Quantum+theory--Mathematics%22">Quantum theory--Mathematics</searchLink>
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  Data: Most combinatorial optimization problems from theoretical computer science have a natural framing as optimization of polynomials in commuting variables. Noncommutativity is one of the defining features of quantum mechanics. So it is not surprising that noncommutative polynomial optimization plays an equally important role in quantum computer science. Our main goal here is to understand the relative hardness of commutative versus noncommutative polynomial optimization. At a first glance it might seem that noncommutative polynomial optimization must be more complex. However this is not always true and this question of relative hardness is substantially more subtle than might appear at the outset. First in this thesis we show that the general noncommutative polynomial optimization is complete for the class $\Pi_2$; this class is in the second level of the arithmetical hierarchy and strictly contains both the set of recursively enumerable languages and its complement. On the other hand, commutative polynomial optimization is decidable and belongs to $\PSPACE$. We then provide evidence that for polynomials arising from a large class of constraint satisfaction problems the situation is reversed: the noncommutative polynomial optimization is an easier computational problem compared to its commutative analogue. A second question we are interested in is about whether we could extract good commutative solutions from noncommutative solutions? This brings us to the second theme of this thesis which is about understanding the algebraic structure of the solutions of noncommutative polynomial optimization. We show that this structural insight then could shed light on the optimal commutative solutions and thereby paves the path in understanding the relationships between the commutative and noncommutative solutions. Here we first use the sum-of-squares framework to understand the algebraic relationships that are present between operators in any optimal noncommutative solution of a class of polynomial optimization problems ...
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      – TitleFull: Some Aspects of Noncommutativity in Polynomial Optimization
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