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The Problem of Translation

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Noncommuting Polynomials

SDP Relaxations

The Problem of Translation

Sparsity

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Limitations

Conclusions

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The Problem of Translation

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Algorithmic Conversion of Polynomial Optimization Problems of Noncommuting Variables to Semidefinite Programming Relaxations

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A hierarchy of semidefinite programming (SDP) relaxations approximates the global optimum of polynomial optimization problems of noncommuting variables. Typical applications include determining genuine multipartite entanglement, finding mutually unbiased bases, and solving ground-state energy problems. Generating the relaxations, however, is a computationally demanding task, and only problems of commuting variables have efficient generators. A closer look at the applications reveals sparse structures: either the monomial base can be reduced or the moment matrix has an inherent structure. Exploiting sparsity leads to scalable generation of relaxations, and the resulting SDP problem is also easier to solve. The current algorithm is able to generate relaxations of optimization problems of a hundred noncommuting variables and a quadratic number of constraints.

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Transcript of "Algorithmic Conversion of Polynomial Optimization Problems of Noncommuting Variables to Semidefinite Programming Relaxations"

  1. 1. Noncommuting Polynomials SDP Relaxations The Problem of Translation Sparsity Scalability Limitations Conclusions Algorithmic Conversion of Polynomial Optimization Problems of Noncommuting Variables to Semidefinite Programming Relaxations Sparsity and Scalability Peter Wittek ˚ University of Boras December 10, 2013 Peter Wittek Noncommuting Polynomial Optimization Problems to SDPs
  2. 2. Noncommuting Polynomials SDP Relaxations The Problem of Translation Sparsity Scalability Limitations Conclusions Outline 1 Polynomial Optimization Problems of Noncommuting Variables 2 SDP Relaxations 3 The Problem of Translation 4 Sources of Sparsity 5 Scalability 6 Limitations 7 Conclusions Peter Wittek Noncommuting Polynomial Optimization Problems to SDPs
  3. 3. Noncommuting Polynomials SDP Relaxations The Problem of Translation Sparsity Scalability Limitations Conclusions Problem Statement We are interested in finding the global minimum p∗ of a polynomial p(x): PK → p∗ := min p(x), x∈K (1) where K is a not necessarily convex set defined by polynomial inequalities gi (x) ≥ 0, i = 1, . . . , r . Peter Wittek Noncommuting Polynomial Optimization Problems to SDPs
  4. 4. Noncommuting Polynomials SDP Relaxations The Problem of Translation Sparsity Scalability Limitations Conclusions Bounding Quantum Correlations max φ, E,φ cij Ei Ej φ ij subject to ||φ|| = 1 Ei Ej Ei = δij Ei ∀i, j = 1 i [Ei , Ej ] = 0 ∀i, j. ´ Navascues, M.; Pironio, S. & Ac´n, A. Bounding the set of ı quantum correlations. Physical Review Letters, 2007, 98, 1040. Peter Wittek Noncommuting Polynomial Optimization Problems to SDPs
  5. 5. Noncommuting Polynomials SDP Relaxations The Problem of Translation Sparsity Scalability Limitations Conclusions cr† cr , (2) Ground-state Energy min φ, Hφ † † cr† cs + cs cr − γ(cr† cs + cs cr ) − 2λ Hfree = r <rs> where < rs > goes through nearest neighbour pairs in the lattice. The fermionic operators are subject to the following constraints: † {cr , cs } = δrs Ir {cr , cs } = 0. Corboz, P.; Evenbly, G.; Verstraete, F. & Vidal, G. Simulation of interacting fermions with entanglement renormalization. Physics Review A, 2010, 81, 010303. Peter Wittek Noncommuting Polynomial Optimization Problems to SDPs
  6. 6. Noncommuting Polynomials SDP Relaxations The Problem of Translation Sparsity Scalability Limitations Conclusions Detection of Multipartite Entanglement Partitions: s = {AB|C, AC|B, BC|A}. P(abc|xyz) is biseparable iff it equals to s s s s s tr[Ma|x ⊗ Mb|y ⊗ Mc|z ρ ], Where measurement operators for an isolated party AB|C AB|C commute: [Mc|z , Mc |z ]. Bancal, J.-D.; Gisin, N.; Liang, Y.-C. & Pironio, S. Device-Independent Witnesses of Genuine Multipartite Entanglement. Physics Review Letters, 2011, 106, 250404. Peter Wittek Noncommuting Polynomial Optimization Problems to SDPs
  7. 7. Noncommuting Polynomials SDP Relaxations The Problem of Translation Sparsity Scalability Limitations Conclusions Mutually Unbiased Bases Mutually unbiased bases in Hilbert space Cd are two orthonormal bases {|e1 , . . . , |ed } and {|f1 , . . . , |fd } such that 1 | ej |fk |2 = d , ∀j, k ∈ {1, . . . , d}. Translating it to a noncommutative optimization problem: 1 Πx = (Πx )∗ ; i i 2 Πx Πx = δij Πx ; i j i 3 4 5 d x i=1 Πi = I; 1 Πx Πx Πx = d Πx , for x = x ; i i j i x O Πx , Πx O Πx ] = 0, for any x, i, and any pair [Πi 1 i i 2 i monomials O1 and O2 of {Πx } of length three. i Peter Wittek of Noncommuting Polynomial Optimization Problems to SDPs
  8. 8. Noncommuting Polynomials SDP Relaxations The Problem of Translation Sparsity Scalability Limitations Conclusions Words and Involution Given n noncommuting variables, words are sequences of ∗ ∗ ∗ letters of x = (x1 , x2 , . . . , xn ) and x ∗ = (x1 , x2 , . . . , xn ). ∗ E.g., w = x1 x2 . Involution: similar to a complex conjugation on sequences of letters. A polynomial is a linear combination of words p = w pw w. Hermitian moment matrix. Hermitian variables. Versus commutative case. Peter Wittek Noncommuting Polynomial Optimization Problems to SDPs
  9. 9. Noncommuting Polynomials SDP Relaxations The Problem of Translation Sparsity Scalability Limitations Conclusions The Target SDP We replace the optimization problem (1) by the following SDP: min w y s.t. pw yw M(y) M(gi y) (3) 0, 0, i = 1, . . . , r . A truncated Hankel matrix. ´ Pironio, S.; Navascues, M. & Ac´n, A. Convergent relaxations of ı polynomial optimization problems with noncommuting variables. SIAM Journal on Optimization, SIAM, 2010, 20, 2157–2180. Peter Wittek Noncommuting Polynomial Optimization Problems to SDPs
  10. 10. Noncommuting Polynomials SDP Relaxations The Problem of Translation Sparsity Scalability Limitations Conclusions The Complexity of Translation Generating the moment and localizing matrices is not a trivial task. The number of words – the monomial basis – grows exponentially in the order of relaxation. The number of elements in the moment matrix is the square of that. Peter Wittek Noncommuting Polynomial Optimization Problems to SDPs
  11. 11. Noncommuting Polynomials SDP Relaxations The Problem of Translation Sparsity Scalability Limitations Conclusions Commutative Examples SparsePOP Approximative, up to 1,000 variables in optimal cases. Gloptipoly 3. Philipp Rostalski’s convex algebraic geometry package. Peter Wittek Noncommuting Polynomial Optimization Problems to SDPs
  12. 12. Noncommuting Polynomials SDP Relaxations The Problem of Translation Sparsity Scalability Limitations Conclusions Supporting Libraries Efficient noncommutative symbolic operations Undocumented and dying feature in Yalmip (Matlab). SymPy and its quantum physics extension. SymbolicC++. NCAlgebra in Mathematica. Singular, GAP, etc. Peter Wittek Noncommuting Polynomial Optimization Problems to SDPs
  13. 13. Noncommuting Polynomials SDP Relaxations The Problem of Translation Sparsity Scalability Limitations Conclusions The Kinds of SDPs Equality constraints Solving the equalities is less efficient. Structural Sparsity Sparsity in the generated SDP Peter Wittek Noncommuting Polynomial Optimization Problems to SDPs
  14. 14. Noncommuting Polynomials SDP Relaxations The Problem of Translation Sparsity Scalability Limitations Conclusions Reduce the Monomial Basis We often encounter equality constraints of the following form: a + b = 0, where a and b are monomials. Remove a from the monomial basis, and substitute it with −b when encountered. Peter Wittek Noncommuting Polynomial Optimization Problems to SDPs
  15. 15. Noncommuting Polynomials SDP Relaxations The Problem of Translation Sparsity Scalability Limitations Conclusions Toy Example: Polynomial Optimization Consider the following polynomial optimization problem: min 2x1 x2 x∈R2 such that 2 −x2 + x2 + 0.5 ≥ 0 2 x1 − x1 = 0. Peter Wittek Noncommuting Polynomial Optimization Problems to SDPs
  16. 16. Noncommuting Polynomials SDP Relaxations The Problem of Translation Sparsity Scalability Limitations Conclusions Toy Example: Corresponding SDP min 2y12 y such that          1 y1 y2 y11 y12 y22 −y22 + y2 + 0.5  − y122 + y12 + 0.5y1 −y222 + y22 + 0.5y2  y1 y11 y12 y111 y112 y122 y2 y12 y22 y112 y122 y222 y11 y111 y112 y1111 y1112 y1122 y12 y112 y122 y1112 y1122 y1222 −y122 + y12 + 0.5y1 −y1122 + y112 + 0.5y11 −y1222 + y122 + 0.5y12 y11 − y1  y111 − y11 y112 − y12 y111 − y11 y1111 − y111 y1112 − y112 Peter Wittek y22 y122 y222 y1122 y1222 y2222         0  −y222 + y22 + 0.5y2 −y1222 + y122 + 0.5y12  −y2222 + y222 + 0.5y22 0.  y112 − y12 y1112 − y112  = 0. y1122 − y122 Noncommuting Polynomial Optimization Problems to SDPs
  17. 17. Noncommuting Polynomials SDP Relaxations The Problem of Translation Sparsity Scalability Limitations Conclusions Toy Example: Reduced SDP min 2y12 y such that  1  y1   y2   y12 y22  −y22 + y2 + 0.5  − y122 + y12 + 0.5y1 −y222 + y22 + 0.5y2 y1 y1 y12 y12 y122 y2 y12 y22 y122 y222 y12 y12 y122 y122 y1222 y22 y122 y222 y1222 y2222 −y122 + y12 + 0.5y1 −y122 + y12 + 0.5y1 −y1222 + y122 + 0.5y12 Peter Wittek       0  −y222 + y22 + 0.5y2 −y1222 + y122 + 0.5y12  −y2222 + y222 + 0.5y22 0. Noncommuting Polynomial Optimization Problems to SDPs
  18. 18. Noncommuting Polynomials SDP Relaxations The Problem of Translation Sparsity Scalability Limitations Conclusions Fast Monomial Substitution Noncommutative symbolic libraries come with substitution routines. They are prepared for all eventualities. We know what monomials look like. Substitution routine can be tuned accordingly. Peter Wittek Noncommuting Polynomial Optimization Problems to SDPs
  19. 19. Noncommuting Polynomials SDP Relaxations The Problem of Translation Sparsity Scalability Limitations Conclusions Ground-state Energy The constraints are simple: † {cr , cs } = δrs Ir {cr , cs } = 0. Most of these are monomial substitutions. The number of actual equalities is linear. Peter Wittek Noncommuting Polynomial Optimization Problems to SDPs
  20. 20. Noncommuting Polynomials SDP Relaxations The Problem of Translation Sparsity Scalability Limitations Conclusions Multipartite Correlations AB|C AB|C [Mc|z , Mc |z ] = 0. Notice the independence of algebras. Reduce the size of the moment matrix. AB|C AC|B BC|A Peter Wittek Noncommuting Polynomial Optimization Problems to SDPs
  21. 21. Noncommuting Polynomials SDP Relaxations The Problem of Translation Sparsity Scalability Limitations Conclusions Mutually Unbiased Bases [Πx O1 Πx , Πx O2 Πx ] = 0 implies over 34 billion constraints i i i i for D = 6, K = 4. Constant number of equalities: d x i=1 Πi = I. The rest are all monomial substitutions. Peter Wittek Noncommuting Polynomial Optimization Problems to SDPs
  22. 22. Noncommuting Polynomials SDP Relaxations The Problem of Translation Sparsity Scalability Limitations Conclusions We Cannot Change the Complexity We can make wise decisions in the implementation. SDP wrapper libraries will not generate sufficiently sparse SDPs: Yalmip, CvxOpt, PICOS. Re-implementation from symbolic manipulations. Choose your interpreter wisely: Pypy. Peter Wittek Noncommuting Polynomial Optimization Problems to SDPs
  23. 23. Noncommuting Polynomials SDP Relaxations The Problem of Translation Sparsity Scalability Limitations Conclusions Memory Use Memory use (MBytes) 100000 10000 Python Python monom. subst. C++ C++ monom. subst. C++ multicore monom. subst. 1000 100 10 1 4 9 16 25 36 49 64 81 100 Number of variables Figure : Memory use of different implementations (log log scale). Peter Wittek Noncommuting Polynomial Optimization Problems to SDPs
  24. 24. Noncommuting Polynomials SDP Relaxations The Problem of Translation Sparsity Scalability Limitations Conclusions Running time 100000 Running time (s) 10000 Python Python monom. subst. C++ C++ monom. subst. C++ multicore monom. subst. 1000 100 10 1 4 9 16 25 36 49 64 81 100 Number of variables Figure : Running time of different implementations (log log scale). Peter Wittek Noncommuting Polynomial Optimization Problems to SDPs
  25. 25. Noncommuting Polynomials SDP Relaxations The Problem of Translation Sparsity Scalability Limitations Conclusions Parallelization Not all programming languages parallelize easily. Internal symmetries of the moment matrix imply data dependencies. Race conditions exist. Only calculating the moment matrix is hard to parallelize. Peter Wittek Noncommuting Polynomial Optimization Problems to SDPs
  26. 26. Noncommuting Polynomials SDP Relaxations The Problem of Translation Sparsity Scalability Limitations Conclusions Solving the SDPs Target solver: SDPA. Accepts sparse file format. Parallel and distributed. Accelerated by GPUs: Fujisawa, K.; Sato, H.; Matsuoka, S.; Endo, T.; Yamashita, M. & Nakata, M. High-performance general solver for extremely large-scale semidefinite programming problems. Proceedings of SC-12, International Conference on High Performance Computing, Networking, Storage and Analysis, 2012, 93:1–93:11. Peter Wittek Noncommuting Polynomial Optimization Problems to SDPs
  27. 27. Noncommuting Polynomials SDP Relaxations The Problem of Translation Sparsity Scalability Limitations Conclusions Where the SDPs Go Wrong Generating the SPDs is not flawless. Toy example works. Calculating the ground-state energy of systems of harmonic oscillators works. Multipartite entanglement, MUBs, more involved Hamiltonians still have problems. Peter Wittek Noncommuting Polynomial Optimization Problems to SDPs
  28. 28. Noncommuting Polynomials SDP Relaxations The Problem of Translation Sparsity Scalability Limitations Conclusions Summary http://arxiv.org/abs/1308.6029 GitHub repositories: ncpol2sdpa, multipartite entanglement, ncpol2sdpa-cpp. Problems in this domain imply sparse structures. Scalability to real-world problems: hundred noncommuting variables with quadratic numbers of constraints. Limitations apply. Peter Wittek Noncommuting Polynomial Optimization Problems to SDPs
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