Non-commutative polynomial optimization S. Pironio 1, M. Navascuès 2, A. Acín 3 1 Laboratoire d Information Quantique (Brussels) 2 Department of Mathematics (Bristol) 3 ICFO-The Institute of Photonic Sciences (Barcelona) MEGA 2011, Stockholm University, June 2, 2011
Our motivation: solve optimization problems arising in quantum physics. They can be vieed as polynomial optimization problems in non-commuting variables (i.e. operators or matrices). We introduced a hierarchy of semidefinite programming (SDP) relaxations hich generate a sequence of loer bounds that converges to the optimum. P., Navascuès, Acín, SIAM J. Opt. 20, 2157 (2010) The method can be seen as a non-commutative generalization of the SDP relaxations introduced by Lasserre and Parrilo for polynomial optimization. The method is very effective on some instances. For some class of problems, it is the only approach at our disposal. The non-commutative setting is very natural for polynomial optimization ; it contains the commutative case, but gives more intuitive proofs of many results (asymptotic convergence, optimality certificates, optimizer extraction).
Quantum correlations beteen separated systems X Y *E a + *E b + ψ H a X b Y Experiment characterized by joint probabilities P(ab XY) = ψ, E a E b ψ In many problems, quantities of interest are linear functions c ab P(ab XY) ab max ψ,e,h ψ, ab c ab E a E b ψ subject to ψ, ψ = 1 E a E a = δ aa E a a, a X, X (idem for B) a X E a = I Note: dim(h) is not fixed E a, E b = 0 a, b X (idem for B)
Quantum chemistry Fundamental problem: Compute ground-state energy of atom or molecule comprised of N electrons that can occupy M orbitals. min ψ, ψ,a,a ijkl a,h i a j a k a l ψ ijkl subject to ψ, ψ = 1 *a i, a j + = 0 *a i, a j + = 0 i, j = 1,, M i, j = 1,, M a i, a j = δ ij i, j = 1,, M i a i a i N ψ = 0 Note: H completely fixed by anti-commutation relations
Non-commutative polynomial optimization problem min ψ,x,h ψ, p X ψ subject to ψ, ψ = 1 q i X 0 i = 1,, m q r j X ψ = 0 j = 1,, m r ψ, s k X ψ 0 k = 1,, m s X = X 1, X 2,, X n is a set of bounded operators in a separable Hilbert space H. p X, q i X, r j X, s k (X) are (hermitian) polynomials of bounded degree d in the variables X Example: p X = 1 + 3X 1 2 + 4X 2 X 1 X 2 X 1 X 3 4 X 3 X 1 (+ c.t.) The aim is to find a vector ψ and a set of operators (i.e., matrices) X satisfying the constraints and minimizing the objective function. The size of these objects, i.e., the dimension of the underlying Hilbert space, is not fixed.
Why non-commutative polynomial optimization? min ψ,x,h ψ, p X, ψ subject to ψ, ψ = 1 q i X 0 i = 1,, m q X j, X k = 0 j, k = 1,, n Since the operators X commute, they generate an abelian algebra that is unitarily equivalent to an algebra of diagonal operators. It is then not difficult to sho that the above problem reduces to the standard polynomial optimization problem min p x x subject to q i x 0 i = 1,, m q in the scalar variables x = x 1,, x n R n.
Moments and moment matrices Let (X) be a monomial, i.e., a orld built out of the 2n letters X 1,, X n, X 1,, X n. Example: X = 1, X 1, X 2 X 3 X 1, Any polynomial p(x) of degree d can then be expanded as p X = p (X) d, here p R, C. Given a vector ψ and a set of operators X in some Hilbert space, define for each orld the moment y = ψ, X ψ R, C. Let y = y 2t be a finite sequence of moments corresponding to ords of length 2t. Define M t (y) as the moment matrix ith ros and columns labeled by ords of length t, and ith entries M t (y) v, = y v
Moment matrices are positive semidefinite Proof: M t y 0 z M t y z 0 for all vectors z R t, C t. z M t y z = z v M t y v, z v, = z v y v z v, = z v ψ, v X X ψ z v, = ψ, z X z X ψ 0, here z X = z X.
Further properties of moments Let y = ψ, X ψ be the moments associated to a feasible solution of the problem Then min ψ,x,h ψ, p X, ψ subject to ψ, ψ = 1 q i X 0 i = 1,, m q r j X ψ = 0 j = 1,, m r ψ, s k X ψ 0 k = 1,, m s. ψ, p X, ψ = p ψ, X, ψ ψ, ψ = y 1 = 1 = p y ψ, v (X)r j X ψ = r j, y v = 0 ψ, s k X ψ = s k, y 0
Localizing matrices The condition q i X 0 implies that the localizing matrix M t (q i y) ith ros and columns labeled by ords of length k, and ith entries M t q i y v, = q i,u y v u is positive semidefinite. u Proof: M t q i y 0 z M t q i y z 0 for all vectors z R t, C t. z M t q i y z = z v M t q i y v, z v, = z v q i,u y v u z v, u = z v ψ, v X q i X X ψ z v, = ψ, z X q i X z X ψ 0, here z X = z X.
Putting everything together To any feasible solution ψ, X, H of our optimization problem e can associate finite sequences y = y 2t of moments (one for each value of 2t d) such that. min ψ,x,h ψ, p X, ψ p y M t y 0 subject to ψ, ψ = 1 y 1 = 1 q i X 0 M t d (q i y) 0 r j X ψ = 0 r j, y v = 0 ψ, s k X ψ 0 s k, y 0
Hierachy of SDP relaxations p = min ψ,x,h ψ, p X, ψ subject to ψ, ψ = 1 q i X 0 r j X ψ = 0 ψ, s k X ψ 0 p t = min y p y subject to M t y 0 y 1 = 1 M t d (q i y) 0 r j, y v = 0 s k, y 0 p p t+1 p t
Asymptotic convergence Archmimedean assumption Suppose that the polynomials q i (X) are such that C X k X k = f i X f i X + g ij X q i X g ij (X) k for some C 0 and some polynomials f i (X), g ij X. i This implies that any set of operators X satisfying q i X 0 (i.e. any feasible ij solution of the optimization problem) must be bounded: Then lim t p t = p k X k X k C. The proof is constructive. From the sequence of optimal solutions of the SDP relaxations, e sho ho to construct optimal ψ and X achieving p. The optimal ψ and X are infinite-dimensional.
Optimality at a finite relaxation step Rank-loop condition If the moment matrix at relaxation step t satisfies rank M t y = rank M k d (y) here d is the maximal degree of the polynomials q i X 0, then p t = p Furthermore, there is an explicit procedure to build the optimal ψ and X out of the moment matrix M t y. They are defined in a vector space H of dimension dim H = rank M t (y)
Dealing ith equality constraints Suppose that the problems contains polynomial equality constraints e i X = 0. A clever ay to deal ith such constraints is to express every polynomial modulo the ideal I = the quotient ring P/I. i f i e i g i, that is, to ork using a monomial basis B for All results still hold if the relaxations are build from such a monomial basis B.
Link ith commutative polynomial optimization min ψ,x,h ψ, p X, ψ subject to ψ, ψ = 1 q i X 0 X j, X k = 0 min p x x subject to q i x 0 Equality constraints: use commutative monomial basis: 1, X 1, X 2, X 1 2, X 1 X 2, X 2 2, SDP relaxations Rank condition for optimality Optimizer extraction = = = Lasserre- Parrilo SDP relaxations Curto Fialko flat extension Henrion - Lasserre
Dual approach and sums of squares min ψ,x,h ψ, p X, ψ subject to ψ, ψ = 1 q i X 0 max λ,x,h subject to p X λ 0 q i X 0 λ Helton and McCullough positivstenllensatz for non commutative polynomials : p X λ 0 on q i X 0 SDP relaxations: p X λ = i f i X f i X + ij g ij X q i X g ij (X) = S.O.S Dual of the moment-based SDP relaxations λ t λ t+1.. p and lim t λ t = p λ t = max λ.f i,g ij λ subject to p X λ = S.O.S deg S.O.S 2t,
Applications Quantum correlations beteen separate subsystems: method already introduced in Navascuès, P., Acín 07, Navascuès, P. Acín 08, Doherty et al 08. Effective: in practice, convergences observed at lo-order relaxations Vertesi-Pal: tested 241 Bell inequalities, 3 rd relaxation yields optimum for 225 The fact that it provides loer-bounds is very important (quantum crypto). Basically, only algorithm that e have. Quantum chemistry: lo-order relaxations coincide ith «reduced-densitymatrix methods» Good: robustness, high-accuracy. Bad: computational time, memory consumption. Other problems (many-body physics, Weyl algebra, ) : to explore!
References P., Navascuès, Acín, SIAM J. Opt. 20, 2157 (2010). Navascuès, P., Acín, in Handbook of Semidefinite, Cone and Polynomial Optimization: Theory, Algorithms, Softare and Applications, edited by M. Anjos and J. Lasserre, forthcoming.