Complete Upper Bound Hierarchies for Spectral Minimum in Noncommutative Polynomial Optimization
Abstract: This work focuses on finding the spectral minimum (ground state energy) of a noncommutative polynomial subject to a finite number of noncommutative polynomial constraints. Based on the Helton-McCullough Positivstellensatz, the Navascu\'es-Pironio-Ac\'in (NPA) hierarchy is the noncommutative analog of Lasserre's moment-sum of squares hierarchy and provides a sequence of lower bounds converging to the spectral minimum, under mild assumptions on the constraint set. Each lower bound can be obtained by solving a semidefinite program. This paper derives complementary complete hierarchies of upper bounds for the spectral minimum. They are noncommutative analogues of the upper bound hierarchies due to Lasserre for minimizing commutative polynomials over compact sets. Each upper bound is obtained by solving a generalized eigenvalue problem. The derived hierarchies apply to optimization problems in bounded and unbounded operator algebras, as demonstrated on a variety of examples.
- Bell, J.S. (1964). On the einstein podolsky rosen paradox. Physics Physique Fizika, 1(3), 195.
- Optimization of polynomials in non-commuting variables. Springer.
- Choi, M.D. (1980). The full C∗superscript𝐶∗C^{\ast}italic_C start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT-algebra of the free group on two generators. Pacific J. Math., 87(1), 41–48.
- Proposed experiment to test local hidden-variable theories. Physical review letters, 23(15), 880.
- Integration with respect to the Haar measure on unitary, orthogonal and symplectic group. Comm. Math. Phys., 264(3), 773–795.
- Improved convergence rates for Lasserre-type hierarchies of upper bounds for box-constrained polynomial optimization. SIAM Journal on Optimization, 27(1), 347–367.
- Convergence analysis of a Lasserre hierarchy of upper bounds for polynomial minimization on the sphere. Mathematical Programming, 1–21.
- Convergence analysis for Lasserre’s measure-based hierarchy of upper bounds for polynomial optimization. Mathematical Programming A, 1–30.
- The quantum moment problem and bounds on entangled multi-prover games. In 2008 23rd Annual IEEE Conference on Computational Complexity, 199–210. IEEE.
- Dykema, K.J. (1998). Faithfulness of free product states. J. Funct. Anal., 154(2), 323–329.
- A positivstellensatz for non-commutative polynomials. Transactions of the American Mathematical Society, 356(9), 3721–3737.
- Certifying optimality of bell inequality violations: Noncommutative polynomial optimization through semidefinite programming and local optimization. Accepted for publication in SIAM Journal on Optimization.
- Null- and Positivstellensätze for rationally resolvable ideals. Linear Algebra Appl., 527, 260–293.
- Lasserre, J.B. (2011). A new look at nonnegativity on closed sets and polynomial optimization. SIAM Journal on Optimization, 21(3), 864–885.
- Lasserre, J.B. (2001). Global optimization with polynomials and the problem of moments. SIAM Journal on optimization, 11(3), 796–817.
- Lasserre, J.B. (2021). Connecting optimization with spectral analysis of tri-diagonal matrices. Mathematical Programming, 190(1), 795–809.
- The Christoffel–Darboux Kernel for Data Analysis, volume 38. Cambridge University Press.
- Laurent, M. (2009). Sums of squares, moment matrices and optimization over polynomials. Emerging applications of algebraic geometry, 157–270.
- Sparse polynomial optimization: theory and practice. World Scientific.
- A convergent hierarchy of semidefinite programs characterizing the set of quantum correlations. New Journal of Physics, 10(7), 073013.
- Quantum Monte Carlo methods in physics and chemistry. 525. Springer Science & Business Media.
- Convergent relaxations of polynomial optimization problems with noncommuting variables. SIAM Journal on Optimization, 20(5), 2157–2180.
- Symbolic integration with respect to the haar measure on the unitary group. arXiv preprint arXiv:1109.4244.
- Putinar, M. (1993). Positive polynomials on compact semi-algebraic sets. Indiana University Mathematics Journal, 42(3), 969–984.
- Ricou, A. (2020). Necessary conditions for nonnegativity on*-algebras and ground state problem. Master’s thesis, National University of Singapore (Singapore).
- Slot, L. (2022). Sum-of-squares hierarchies for polynomial optimization and the christoffel–darboux kernel. SIAM Journal on Optimization, 32(4), 2612–2635.
- Takesaki, M. (2002). Theory of operator algebras. I, volume 124 of Encyclopaedia of Mathematical Sciences. Springer-Verlag, Berlin. Reprint of the first (1979) edition, Operator Algebras and Non-commutative Geometry, 5.
- Takesaki, M. (2003). Theory of operator algebras. III, volume 127 of Encyclopaedia of Mathematical Sciences. Springer-Verlag, Berlin.
- Semidefinite programming. SIAM review, 38(1), 49–95.
- White, S.R. (1992). Density matrix formulation for quantum renormalization groups. Physical review letters, 69(19), 2863.
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