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Partial Degeneration of Tensors
... In the references just given the problem is formulated differently. The author believes that the solution to Problem XXXVII yields a polynomial upper bound on the approximation degree for tensors, a classical important direction in complexity theory [93,23,24,34], but the precise argument on why it is so is nontrivial and well beyond the scope of the current work. ...
We review the open problems in the theory of deformations of zero-dimensional objects, such as algebras, modules or tensors. We list both the well-known ones and some new ones that emerge from applications. In view of many advances in recent years, we can hope that all of them are in the range of current methods.
Tensor networks provide descriptions of strongly correlated quantum systems based on an underlying entanglement structure given by a graph of entangled states along the edges that identify the indices of the local tensors to be contracted. Considering a more general setting, where entangled states on edges are replaced by multipartite entangled states on faces, allows us to employ the geometric properties of multipartite entanglement in order to obtain representations in terms of superpositions of tensor network states with smaller effective dimension, leading to computational savings.
We prove that the border rank of the Kronecker square of the little Coppersmith–Winograd tensor T c w , q is the square of its border rank for q > 2 and that the border rank of its Kronecker cube is the cube of its border rank for q > 4 . This answers questions raised implicitly by Coppersmith & Winograd (1990, §11) and explicitly by Bläser (2013, Problem 9.8) and rules out the possibility of proving new upper bounds on the exponent of matrix multiplication using the square or cube of a little Coppersmith–Winograd tensor in this range.
In the positive direction, we enlarge the list of explicit tensors potentially useful for Strassen's laser method, introducing a skew-symmetric version of the Coppersmith–Winograd tensor, T s k e w c w , q . For q = 2 , the Kronecker square of this tensor coincides with the 3 × 3 determinant polynomial, det 3 ∈ C 9 ⊗ C 9 ⊗ C 9 , regarded as a tensor. We show that this tensor could potentially be used to show that the exponent of matrix multiplication is two.
We determine new upper bounds for the (Waring) rank and the (Waring) border rank of det 3 , exhibiting a strict submultiplicative behaviour for T s k e w c w , 2 which is promising for the laser method.
We establish general results regarding border ranks of Kronecker powers of tensors, and make a detailed study of Kronecker squares of tensors in C 3 ⊗ C 3 ⊗ C 3 .
In this note, we show that the method of Croot, Lev, and Pach can be used to bound the size of a subset of with no three terms in arithmetic progression by with . For q=3, the problem of finding the largest subset with no three terms in arithmetic progression is called the `cap problem'. Previously the best known upper bound for the cap problem, due to Bateman and Katz, was .
The relation betweenAPA-algorithms (i. e. approximating the result with an arbitrarily small error) andEC-algorithms (i. e. computing exactly the result) is analyzed. The existence of anAPA-algorithm of complexityt
0 and degreed implics the existence of anEC-algorithm of complexity (1+d)t
0. An application is given for problems associated to tensorial powers of a tensor, such as matrix product.
is is the first book to present an up-to-date and self-contained account of Algebraic Complexity Theory that is both comprehensive and unified. Requiring of the reader only some basic algebra and offering over 350 exercises, it is well- suited as a textbook for beginners at graduate level. With its extensive bibliography covering about 500 research papers, this text is also an ideal reference book for the professional researcher. The subdivision of the contents into 21 more or less independent chapters enables readers to familiarize themselves quickly with a specific topic, and facilitates the use of this book as a basis for complementary courses in other areas such as computer algebra.
We apply methods from algebraic geometry to study uniform matrix product states. Our main results concern the topology of the locus of tensors expressed as uMPS, their defining equations and identifiability. By an interplay of theorems from algebra, geometry and quantum physics we answer several questions and conjectures posed by Critch, Morton and Hackbusch.
The tensor network variety is a variety of tensors associated to a graph and a set of positive integer weights on its edges, called bond dimensions. We determine an upper bound on the dimension of the tensor network variety. A refined upper bound is given in cases relevant for applications such as varieties of matrix product states and projected entangled pairs states. We provide a range (the “supercritical range”) of the parameters where the upper bound is sharp.
We present three families of minimal border rank tensors: they come from highest weight vectors, smoothable algebras, and monomial algebras. We analyse them using Strassen's laser method and obtain an upper bound 2.431 on ω. We also explain how in certain monomial cases using the laser method directly is less profitable than first degenerating. Our results form possible paths in the search for valuable tensors for the laser method away from Coppersmith-Winograd tensors.
The theory of entanglement provides a fundamentally new language for describing interactions and correlations in many-body systems. Its vocabulary consists of qubits and entangled pairs, and the syntax is provided by tensor networks. How matrix product states and projected entangled pair states describe many-body wave functions in terms of local tensors is reviewed. These tensors express how the entanglement is routed, act as a novel type of nonlocal order parameter, and the manner in which their symmetries are reflections of the global entanglement patterns in the full system is described. The manner in which tensor networks enable the construction of real-space renormalization group flows and fixed points is discussed, and the entanglement structure of states exhibiting topological quantum order is examined. Finally, a summary of the mathematical results of matrix product states and projected entangled pair states, highlighting the fundamental theorem of matrix product vectors and its applications, is provided.
Tensor network states form a variational ansatz class widely used, both analytically and numerically, in the study of quantum many-body systems. It is known that if the underlying graph contains a cycle, e.g., as in projected entangled pair states, then the set of tensor network states of given bond dimension is not closed. Its closure is the tensor network variety. Recent work has shown that states on the boundary of this variety can yield more efficient representations for states of physical interest, but it remained unclear how to systematically find and optimize over such representations. We address this issue by defining an ansatz class of states that includes states at the boundary of the tensor network variety of given bond dimension. We show how to optimize over this class in order to find ground states of local Hamiltonians by only slightly modifying standard algorithms and code for tensor networks. We apply this method to different models and observe favorable energies and runtimes when compared with standard tensor network methods.
Two central problems in computer science are P vs NP and the complexity of matrix multiplication. The first is also a leading candidate for the greatest unsolved problem in mathematics. The second is of enormous practical and theoretical importance. Algebraic geometry and representation theory provide fertile ground for advancing work on these problems and others in complexity. This introduction to algebraic complexity theory for graduate students and researchers in computer science and mathematics features concrete examples that demonstrate the application of geometric techniques to real world problems. Written by a noted expert in the field, it offers numerous open questions to motivate future research. Complexity theory has rejuvenated classical geometric questions and brought different areas of mathematics together in new ways. This book will show the beautiful, interesting, and important questions that have arisen as a result.
Let T_1 be an k_1-tensor and let T_2 be a k_2-tensor and consider the tensor product of T_1 \otimes T_2, which is a (k_1 + k_2)-tensor. It has recently been shown ([Christandl, Jensen, Zuiddam]) that the tensor rank can be strictly submultiplicative under the tensor product. The necessary upper bounds were obtained with help of the border rank. It was left open whether border rank itself can be strictly submultiplicative. In the current work, we answer this question in the affirmative. In order to do so, we construct lines in projective space along which the border rank drops multiple times and use this result in conjunction with a previous construction for a tensor rank drop. Our results also imply strict submultiplicativity for cactus rank and border cactus rank.
The tensor rank of a tensor is the smallest number r such that the tensor can be decomposed as a sum of r simple tensors. Let s be a k-tensor and let t be an l-tensor. The tensor product of s and t is a (k + l)-tensor (not to be confused with the "tensor Kronecker product" used in algebraic complexity theory, which multiplies two k-tensors to get a k-tensor). Tensor rank is sub-multiplicative under the tensor product. We revisit the connection between restrictions and degenerations. It is well-known that tensor rank is not in general multiplicative under the tensor Kronecker product. A result of our study is that tensor rank is also not in general multiplicative under the tensor product. This answers a question of Draisma and Saptharishi. It remains an open question whether border rank and asymptotic rank are multiplicative under the tensor product. Interestingly, all lower bounds on border rank obtained from Young flattenings are multiplicative under the tensor product.
Let || denote the matrix multiplication tensor for || matrices. We use the border substitution method [2, 3, 6] combined with Koszul flattenings [8] to prove the border rank lower bound ||.
Two theorems on the canonical Kronecker form of a perturbed matrix pencil and the characterization of the closure of the set of all matrix pencils with a fixed Kronecker canonical form are given.
The possibility of a new kind of electronic state is pointed out, corresponding roughly to Pauling's idea of “resonating valence bonds” in metals. As observed by Pauling, a pure state of this type would be insulating; it would represent an alternative state to the Néel antiferromagnetic state for S = 1/2. An estimate of its energy is made in one case.
A large class of multiplication problems in arithmetic complexity can be viewed as the simultaneous evaluation of a set of bilinear forms. This class includes the multiplication of matrices, polynomials, quaternions, Cayley and complex numbers. Considering bilinear algorithms, the optimal number of nonscalar multiplications can be described as the rank of a three-tensor or as the smallest member of rank one matrices necessary to include a given set of matrices in their span.
In this paper, we attack a rather large subclass of three-tensors, namely that of (p,q,2)-tensors, for arbitrary p and q, and solve it completely in the case where the field of constants contains the roots of a polynomial associated with the given tensor. In all other cases, we prove that, in general, our bounds cannot be improved. The complexity of a general pair of bilinear forms is determined explicitly in terms of parameters related to Kronecker’s theory of pencils and to the theory of invariant polynomials. This reveals unexpected results and shows explicitly the dependence on the algebraic structure of the constants; we display, for example, a pair of bilinear forms whose complexity is 3 over the field and which, however, requires exactly 4 nonscalar multiplications over the fields or . Corresponding optimal algorithms are described and several applications are considered.
Let G be a connected linear algebraic group, and p a rational representation of G on a finite-dimensional vector space V , all defined over the complex number field C .
We call such a triplet ( G, p, V ) a prehomogeneous vector space if V has a Zariski-dense G -orbit. The main purpose of this paper is to classify all prehomogeneous vector spaces when p is irreducible, and to investigate their relative invariants and the regularity.
Invertible local transformations of a multipartite system are used to define equivalence classes in the set of entangled states. This classification concerns the entanglement properties of a single copy of the state. Accordingly, we say that two states have the same kind of entanglement if both of them can be obtained from the other by means of local operations and classical communcication (LOCC) with nonzero probability. When applied to pure states of a three-qubit system, this approach reveals the existence of two inequivalent kinds of genuine tripartite entanglement, for which the GHZ state and a W state appear as remarkable representatives. In particular, we show that the W state retains maximally bipartite entanglement when any one of the three qubits is traced out. We generalize our results both to the case of higher dimensional subsystems and also to more than three subsystems, for all of which we show that, typically, two randomly chosen pure states cannot be converted into each other by means of LOCC, not even with a small probability of success. Comment: 12 pages, 1 figure; replaced with revised version; terminology adapted to earlier work; reference added; results unchanged