205 research outputs found
Multi-copy and stochastic transformation of multipartite pure states
Characterizing the transformation and classification of multipartite
entangled states is a basic problem in quantum information. We study the
problem under two most common environments, local operations and classical
communications (LOCC), stochastic LOCC and two more general environments,
multi-copy LOCC (MCLOCC) and multi-copy SLOCC (MCSLOCC). We show that two
transformable multipartite states under LOCC or SLOCC are also transformable
under MCLOCC and MCSLOCC. What's more, these two environments are equivalent in
the sense that two transformable states under MCLOCC are also transformable
under MCSLOCC, and vice versa. Based on these environments we classify the
multipartite pure states into a few inequivalent sets and orbits, between which
we build the partial order to decide their transformation. In particular, we
investigate the structure of SLOCC-equivalent states in terms of tensor rank,
which is known as the generalized Schmidt rank. Given the tensor rank, we show
that GHZ states can be used to generate all states with a smaller or equivalent
tensor rank under SLOCC, and all reduced separable states with a cardinality
smaller or equivalent than the tensor rank under LOCC. Using these concepts, we
extended the concept of "maximally entangled state" in the multi-partite
system.Comment: 8 pages, 1 figure, revised version according to colleagues' comment
Efficient algorithms for tensor scaling, quantum marginals and moment polytopes
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.
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
Effective constructions in plethysms and Weintraub's conjecture
We give a short proof of Weintraub's conjecture by constructing explicit
highest weight vectors in the symmetric power of an even exterior power
Log-concavity and lower bounds for arithmetic circuits
One question that we investigate in this paper is, how can we build
log-concave polynomials using sparse polynomials as building blocks? More
precisely, let be a
polynomial satisfying the log-concavity condition a\_i^2 \textgreater{} \tau
a\_{i-1}a\_{i+1} for every where \tau
\textgreater{} 0. Whenever can be written under the form where the polynomials have at most
monomials, it is clear that . Assuming that the
have only non-negative coefficients, we improve this degree bound to if \tau \textgreater{} 1,
and to if .
This investigation has a complexity-theoretic motivation: we show that a
suitable strengthening of the above results would imply a separation of the
algebraic complexity classes VP and VNP. As they currently stand, these results
are strong enough to provide a new example of a family of polynomials in VNP
which cannot be computed by monotone arithmetic circuits of polynomial size
Alternating Minimization, Scaling Algorithms, and the Null-Cone Problem from Invariant Theory
Alternating Minimization, Scaling Algorithms, and the Null-Cone Problem from Invariant Theory
Alternating minimization heuristics seek to solve a (difficult) global optimization task through iteratively solving a sequence of (much easier) local optimization tasks on different parts (or blocks) of the input parameters. While popular and widely applicable, very few examples of this heuristic are rigorously shown to converge to optimality, and even fewer to do so efficiently.
In this paper we present a general framework which is amenable to rigorous analysis, and expose its applicability. Its main feature is that the local optimization domains are each a group of invertible matrices, together naturally acting on tensors, and the optimization problem is minimizing the norm of an input tensor under this joint action. The solution of this optimization problem captures a basic problem in Invariant Theory, called the null-cone problem.
This algebraic framework turns out to encompass natural computational problems in combinatorial optimization, algebra, analysis, quantum information theory, and geometric complexity theory. It includes and extends to high dimensions the recent advances on (2-dimensional) operator scaling.
Our main result is a fully polynomial time approximation scheme for this general problem, which may be viewed as a multi-dimensional scaling algorithm. This directly leads to progress on some of the problems in the areas above, and a unified view of others. We explain how faster convergence of an algorithm for the same problem will allow resolving central open problems.
Our main techniques come from Invariant Theory, and include its rich non-commutative duality theory, and new bounds on the bitsizes of coefficients of invariant polynomials. They enrich the algorithmic toolbox of this very computational field of mathematics, and are directly related to some challenges in geometric complexity theory (GCT)
Polynomial Time Algorithms in Invariant Theory for Torus Actions
An action of a group on a vector space partitions the latter into a set of orbits. We consider three natural and useful algorithmic "isomorphism" or "classification" problems, namely, orbit equality, orbit closure intersection, and orbit closure containment. These capture and relate to a variety of problems within mathematics, physics and computer science, optimization and statistics. These orbit problems extend the more basic null cone problem, whose algorithmic complexity has seen significant progress in recent years.
In this paper, we initiate a study of these problems by focusing on the actions of commutative groups (namely, tori). We explain how this setting is motivated from questions in algebraic complexity, and is still rich enough to capture interesting combinatorial algorithmic problems. While the structural theory of commutative actions is well understood, no general efficient algorithms were known for the aforementioned problems. Our main results are polynomial time algorithms for all three problems. We also show how to efficiently find separating invariants for orbits, and how to compute systems of generating rational invariants for these actions (in contrast, for polynomial invariants the latter is known to be hard). Our techniques are based on a combination of fundamental results in invariant theory, linear programming, and algorithmic lattice theory
Numerically stable computation of CreditRisk+
The CreditRisk+ model launched by CSFB in 1997 is widely used by practitioners in the banking sector as a simple means for the quantification of credit risk, primarily of the loan book. We present an alternative numerical recursion scheme for CreditRisk+, equivalent to an algorithm recently proposed by Giese, based on well-known expansions of the logarithm and the exponential of a power series. We show that it is advantageous to the Panjer recursion advocated in the original CreditRisk+ document, in that it is numerically stable. The crucial stability arguments are explained in detail. Furthermore, the computational complexity of the resulting algorithm is stated
Balancing Bounded Treewidth Circuits
Algorithmic tools for graphs of small treewidth are used to address questions
in complexity theory. For both arithmetic and Boolean circuits, it is shown
that any circuit of size and treewidth can be
simulated by a circuit of width and size , where , if , and otherwise. For our main construction,
we prove that multiplicatively disjoint arithmetic circuits of size
and treewidth can be simulated by bounded fan-in arithmetic formulas of
depth . From this we derive the analogous statement for
syntactically multilinear arithmetic circuits, which strengthens a theorem of
Mahajan and Rao. As another application, we derive that constant width
arithmetic circuits of size can be balanced to depth ,
provided certain restrictions are made on the use of iterated multiplication.
Also from our main construction, we derive that Boolean bounded fan-in circuits
of size and treewidth can be simulated by bounded fan-in
formulas of depth . This strengthens in the non-uniform setting
the known inclusion that . Finally, we apply our
construction to show that {\sc reachability} for directed graphs of bounded
treewidth is in
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