Leonid Gurvits - Profile on Academia.edu (original) (raw)

Papers by Leonid Gurvits

Proceedings of the thirty-fifth ACM symposium on Theory of computing - STOC '03

Constructive Approximation

gFoundations of Non. holonomic Motion Planning

Trace inequalities and linear programming (with applications to markov chains

Linear and Multilinear Algebra, 1998

Using a trace inequality for M-matrices prove that where P(row ) stochastic matrix and π is its s... more Using a trace inequality for M-matrices prove that where P(row ) stochastic matrix and π is its stationary probabilistic distribution Motivated by this result we introduce a cotion of Π superstochastic matrix ,I,e,a square matrix Q with nonnegative entries is called Π superstochastic iff We study when for a Π superstochastic matrix Q A maong other results we prove that ifΠ1 ≤ Π2≤⋯≤Πn/Π1ethen the inequality above holds for all Π-superstochastic matrices Q and e is the smallest contant of this type .Our solution is based on a passage to a dual problem of linear programming.We also give alternative linear programming -based proof for the inequality above for stochastic matrices Finally,we prove the following result:ifM+M is positive definite the eigenvalues of both matrices M (M − M :)and M (M − M :)have nonnegative real parts.As a direct corollaryu of this result we prove one inqulity above for symmetric matrices Using the idea of this proof we prove entropic inquality for symmetric matrices with nonegative entries

Sc ence Los Alamos

... 226 Raymond Laflamme, Emanuel Knill, David G. Cory, Evan M. Fortunato, Timothy F. Havel, Cesa... more ... 226 Raymond Laflamme, Emanuel Knill, David G. Cory, Evan M. Fortunato, Timothy F. Havel, Cesar Miquel, Rudy Martinez, Camille J. Negrevergne, Gerardo Ortiz, Marco A. Pravia, Yehuda Sharf, Suddhasattwa Sinha, Rolanda Somma, and Lorenza Viola ...

algebraic statistical tools for the study of some dyadic random graph models, including Markov ba... more algebraic statistical tools for the study of some dyadic random graph models, including Markov bases, that have important implications for the existence of maximum likelihood estimation and other statistical problems. These tools do not extend in a simple fashion to more complex models in the class of exponential random graph models. In this presentation, I explain why there are difficulties as we move away from dyadic models and I describe some of the challenges for algebraic statistics in this area of research.

2 Quantum Information 4 2.1 The Quantum Bit . . . . . . . . . . . . . . . . . . . . . . . . . . .... more 2 Quantum Information 4 2.1 The Quantum Bit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 Processing One Qubit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.3 Two Quantum Bits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.4 Processing Two Qubits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.5 Using Many Quantum Bits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.6 Qubit Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.7 Mixtures and Density Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.8 Quantum Computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.9 Resource Accounting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.10 From Factoring to Phase Estimation . . . . . . . . . . . . . . . . . . . . ...

Classical matching theory can be defined in terms of matrices with nonnegative entries. The notio... more Classical matching theory can be defined in terms of matrices with nonnegative entries. The notion of Positive operator , central in Quantum Theory , is a natural generalization of matrices with non-negative entries. Based on this point of view , we introduce a definition of perfect Quantum (operator) matching. We show that the new notion inherits many " classical " properties , but not all of them. This new notion goes somewhere beyound matroids. For separable bipartite quantum states this new notion coinsides with the full rank property of the intersection of two corresponding geometric matroids. In the classical situation , permanents are naturally associated with perfects matchings. We introduce an analog of perma-nents for positive operators, called Quantum Permanent and show how this generalization of the permanent is related to the Quantum Entanglement. Besides many other things , Quantum Permanents provide new rational inequalities necessary for the separability of...

We show that the common symbolic manipulation tasks of computing multiple partial derivatives, de... more We show that the common symbolic manipulation tasks of computing multiple partial derivatives, definite integration, and definite summation, are #P-hard, i.e., at least as hard as counting the accepting input strings for any Turing machine that halts in polynomial time. (The “multiple partial derivatives” part was previously known.)

Physical Review A, 2020

We study the computational complexity of quantum-mechanical expectation values of singleparticle ... more We study the computational complexity of quantum-mechanical expectation values of singleparticle operators in bosonic and fermionic multi-particle product states. Such expectation values appear, in particular, in full-counting-statistics problems. Depending on the initial multi-particle product state, the expectation values may be either easy to compute (the required number of operations scales polynomially with the particle number) or hard to compute (at least as hard as a permanent of a matrix). However, if we only consider full counting statistics in a finite number of final single-particle states, then the full-counting-statistics generating function becomes easy to compute in all the analyzed cases. We prove the latter statement for the general case of the fermionic product state and for the single-boson product state (the same as used in the boson-sampling proposal). This result may be relevant for using multi-particle product states as a resource for quantum computing.

Quantum Information Processing and Quantum Error Correction, 2012

Geometric and Functional Analysis, 2018

The celebrated Brascamp-Lieb (BL) inequalities [BL76, Lie90], and their reverse form of Barthe [B... more The celebrated Brascamp-Lieb (BL) inequalities [BL76, Lie90], and their reverse form of Barthe [Bar98], are an important mathematical tool, unifying and generalizing numerous inequalities in analysis, convex geometry and information theory, with many used in computer science. While their structural theory is very well understood, far less is known about computing their main parameters (which we later define below). Prior to this work, the best known algorithms for any of these optimization tasks required at least exponential time. In this work, we give polynomial time algorithms to compute: (1) Feasibility of BL-datum, (2) Optimal BLconstant, (3) Weak separation oracle for BL-polytopes. What is particularly exciting about this progress, beyond the better understanding of BLinequalities, is that the objects above naturally encode rich families of optimization problems which had no prior efficient algorithms. In particular, the BL-constants (which we efficiently compute) are solutions to non-convex optimization problems, and the BL-polytopes (for which we provide efficient membership and separation oracles) are linear programs with exponentially many facets. Thus we hope that new combinatorial optimization problems can be solved via reductions to the ones above, and make modest initial steps in exploring this possibility. Our algorithms are obtained by a simple efficient reduction of a given BL-datum to an instance of the Operator Scaling problem defined by [Gur04]. To obtain the results above, we utilize the two (very recent and different) algorithms for the operator scaling problem [GGOW16, IQS15]. Our reduction implies algorithmic versions of many of the known structural results on BL-inequalities, and in some cases provide proofs that are different or simpler than existing ones. Further, the analytic properties of the [GGOW16] algorithm provide new, effective bounds on the magnitude and continuity of BL-constants; prior work relied on compactness, and thus provided no bounds. On a higher level, our application of operator scaling algorithm to BL-inequalities further connects analysis and optimization with the diverse mathematical areas used so far to motivate and solve the operator scaling problem, which include commutative invariant theory, non-commutative algebra, computational complexity and quantum information theory.

Many "hard" combinatorial and geometric quantities (such as the number of perfect matchings in bi... more Many "hard" combinatorial and geometric quantities (such as the number of perfect matchings in bipartite graphs (the permanent), number of perfect matchings in general graphs (the hafnian) , the number of matching of the fixed size, the number of Hamiltonian cycles, the number of exact 3-coverings, the number of common bases in the intersection of unimodular geometric matroid and the matroid of transversals (the Mixed Discriminant), the Mixed Volume (responsible for the number of isolated solutions of systems of polynomial equations) etc.) can be expressed as the coefficient a 1,.

We show that the permanent of a doubly stochastic n × n matrix A = (a ij) is at least as large as... more We show that the permanent of a doubly stochastic n × n matrix A = (a ij) is at least as large as i,j (1 − a ij) 1−aij and at most as large as 2 n times this number. Combined with previous work, this improves on the deterministic approximation factor for the permanent, giving 2 n instead of e n-approximation. We also give a combinatorial application of the lower bound, proving S. Friedland's "Asymptotic Lower Matching Conjecture" for the monomer-dimer problem.

2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS), 2017

We design a deterministic polynomial time c n approximation algorithm for the permanent of positi... more We design a deterministic polynomial time c n approximation algorithm for the permanent of positive semidefinite matrices where c = e γ+1 ≃ 4.84. We write a natural convex relaxation and show that its optimum solution gives a c n approximation of the permanent. We further show that this factor is asymptotically tight by constructing a family of positive semidefinite matrices.

Foundations of Computational Mathematics, 2019

In this paper we present a deterministic polynomial time algorithm for testing if a symbolic matr... more In this paper we present a deterministic polynomial time algorithm for testing if a symbolic matrix in non-commuting variables over Q is invertible or not. The analogous question for commuting variables is the celebrated polynomial identity testing (PIT) for symbolic determinants. In contrast to the commutative case, which has an efficient probabilistic algorithm, the best previous algorithm for the non-commutative setting required exponential time [IQS17] (whether or not randomization is allowed). The algorithm efficiently solves the "word problem" for the free skew field, and the identity testing problem for arithmetic formulae with division over noncommuting variables, two problems which had only exponential-time algorithms prior to this work. The main contribution of this paper is a complexity analysis of an existing algorithm due to Gurvits [Gur04], who proved it was polynomial time for certain classes of inputs. We prove it always runs in polynomial time. The main component of our analysis is a simple (given the necessary known tools) lower bound on central notion of capacity of operators (introduced by Gurvits [Gur04]). We extend the algorithm to actually approximate capacity to any accuracy in polynomial time, and use this analysis to give quantitative bounds on the continuity of capacity (the latter is used in a subsequent paper on Brascamp-Lieb inequalities). We also extend the algorithm to compute not only singularity, but actually the (non-commutative) rank of a symbolic matrix, yielding a factor 2 approximation of the commutative rank. This naturally raises a relaxation of the commutative PIT problem to achieving better deterministic approximation of the commutative rank. Symbolic matrices in non-commuting variables, and the related structural and algorithmic questions, have a remarkable number of diverse origins and motivations. They arise independently in (commutative) invariant theory and representation theory, linear algebra, optimization, linear system theory, quantum information theory, approximation of the permanent and naturally in non-commutative algebra. We provide a detailed account of some of these sources and their interconnections. In particular we explain how some of these sources played an important role in the development of Gurvits' algorithm and in our analysis of it here.

Electron. Colloquium Comput. Complex., 2007

Let K = (K1...Kn) be a n-tuple of convex compact subsets in the Euclidean space R , and let V (·)... more Let K = (K1...Kn) be a n-tuple of convex compact subsets in the Euclidean space R , and let V (·) be the Euclidean volume in R. The Minkowski polynomial VK is defined as VK(x1, ..., xn) = V (λ1K1 + ... + λnKn) and the mixed volume V (K1, ..., Kn) as V (K1...Kn) = ∂ ∂λ1...∂λn VK(λ1K1 + · · ·λnKn). We study in this paper randomized algorithms to approximate the mixed volume of wellpresented convex compact sets. Our main result is a poly-time algorithm which approximates V (K1, ..., Kn) with multiplicative error e n and with better rates if the affine dimensions of most of the sets Ki are small. Our approach is based on the particular convex relaxation of log(V (K1, ..., Kn)) via the geometric programming. We prove the mixed volume analogues of the Van der Waerden and the Schrijver/Valiant conjectures on the permanent. These results , interesting on their own, allow to ”justify” the above mentioned convex relaxation, which is solved using the ellipsoid method and a randomized poly-time...

Consider a homogeneous polynomial p(z1,...,zn)p(z_1,...,z_n)p(z1,...,zn) of degree nnn in nnn complex variables . Assum... more Consider a homogeneous polynomial p(z1,...,zn)p(z_1,...,z_n)p(z1,...,zn) of degree nnn in nnn complex variables . Assume that this polynomial satisfies the property : \\ ∣p(z1,...,zn)∣geqprod1leqileqnRe(zi)|p(z_1,...,z_n)| \geq \prod_{1 \leq i \leq n} Re(z_i)∣p(z1,...,zn)∣geqprod1leqileqnRe(zi) on the domain (z1,...,zn):Re(zi)geq0,1leqileqn\{(z_1,...,z_n) : Re(z_i) \geq 0, 1 \leq i \leq n \}(z1,...,zn):Re(zi)geq0,1leqileqn . \\ We prove that ∣fracpartialnpartialz1...partialznp∣geqfracn!nn|\frac{\partial^n}{\partial z_1...\partial z_n} p | \geq \frac{n!}{n^n}∣fracpartialnpartialz1...partialznp∣geqfracn!nn . Our proof is relatively short and self-contained (i.e. we only use basic properties of hyperbolic polynomials). As the van der Waerden conjecture for permanents, proved by D.I. Falikman and G.P. Egorychev, as well Bapat's conjecture for mixed discriminants, proved by the author, are particular cases of this result. We also prove so called "small rank" lower bound (in the permanents context it corresponds to sparse doubly-stochastic matrices, i.e. with small number of non-zero entries in each column). The later lower bound generalizes (with simpler proofs) recent lower bounds by A.Schrijver for the number of ...

A Deterministic Algorithm for Approximating the Mixed Discriminant and Mixed Volume, and a Combinatorial Corollary

Discrete & Computational Geometry