Efficient Algorithms for Tensor Scaling, Quantum Marginals, and Moment Polytopes

Abstract

We present a polynomial time algorithm to approximately scale tensors of any format to arbitrary prescribed marginals (whenever possible). This unifies and generalizes a sequence of past works on matrix, operator and tensor scaling. Our algorithm provides an efficient weak membership oracle for the associated moment polytopes, an important family of implicitly-defined convex polytopes with exponentially many facets and a wide range of applications. These include the entanglement polytopes from quantum information theory (in particular, we obtain an efficient solution to the notorious one-body quantum marginal problem) and the Kronecker polytopes from representation theory (which capture the asymptotic support of Kronecker coefficients). Our algorithm can be applied to succinct descriptions of the input tensor whenever the marginals can be efficiently computed, as in the important case of matrix product states or tensor-train decompositions, widely used in computational physics and numerical mathematics.Beyond these applications, the algorithm enriches the arsenal of "numerical" methods for classical problems in invariant theory that are significantly faster than "symbolic" methods which explicitly compute invariants or covariants of the relevant action. We stress that (like almost all past algorithms) our convergence rate is polynomial in the approximation parameter; it is an intriguing question to achieve exponential convergence rate, beating symbolic algorithms exponentially, and providing strong membership and separation oracles for the problems above.We strengthen and generalize the alternating minimization approach of previous papers by introducing the theory of highest weight vectors from representation theory into the numerical optimization framework. We show that highest weight vectors are natural potential functions for scaling algorithms and prove new bounds on their evaluations to obtain polynomial-time convergence. Our techniques are general and we believe that they will be instrumental to obtain efficient algorithms for moment polytopes beyond the ones consider here, and more broadly, for other optimization problems possessing natural symmetries.

Spectral Analysis of Matrix Scaling and Operator Scaling

“…where the partial trace operations tr n and tr m are linear functions that satisfy tr n (X ⊗ Y ) := tr(Y ) · X and tr m (X ⊗ Y ) = tr(X) · Y for X ∈ R m×m and Y ∈ R n×n . This phrasing of the operator scaling problem is in line with the more general quantum marginal problem [11].…”

Operator scaling with specified marginals

“…We should not expect to be able to compute each step of Algorithm 3 exactly, but rather to polynomially many bits of precision. The previous version of this paper had a rather messy analysis of the rounding; [5] contains a much more pleasant analysis, which we now sketch.…”

“…The author views this paper as a step towards solving this problem -however, we have only two specified marginals and can produce only a very inefficient membership oracle for our polytope. The work [5], which came shortly after this one, remedies the former issue but not the latter.…”

Two-Qubit Circuit Depth and the Monodromy Polytope

“…Cole Franks et al [17,9] have described effective numerical methods for solving the additive analogue of the eigenvalue problem. One might explore multiplicative variations on their methods (especially those with the orthogonality constraint kept in mind) which would then adapt to solve the problem posed here.…”