arXiv subjects

Hoang Ta

Publications and source records attributed to Hoang Ta.

21 records · Page 2Linked to original sources

arXiv2022

Symmetric Subrank of Tensors and Applications

Strassen (Strassen, J. Reine Angew. Math., 375/376, 1987) introduced the subrank of a tensor as a natural extension of matrix rank to tensors. Subrank measures the largest diagonal tensor that can be obtained by applying linear operations to the different indices (legs) of the tensor (just like the matrix rank measures the largest diagonal matrix that can be obtained using row and column operations). Motivated by problems in combinatorics and complexity theory we introduce the new notion of symmetric subrank of tensors by restricting these linear operations to be the same for each index. We prove precise relations and separations between subrank and symmetric subrank. We prove that for symmetric tensors the subrank and the symmetric subrank are asymptotically equal. This proves the asymptotic subrank analogon of a conjecture known as Comon's conjecture in the theory of tensors. This result allows us to prove a strong connection between the general and symmetric version of an asymptotic duality theorem of Strassen. We introduce a representation-theoretic method to asymptotically bound the symmetric subrank called the symmetric quantum functional in analogy with the quantum functionals (Christandl, Vrana, Zuiddam, J. Amer. Math. Soc., 2021), and we study the relations between these functionals.

cs.CC↗

arXiv2022

A hierarchy of efficient bounds on quantum capacities exploiting symmetry

Optimal rates for achieving an information processing task are often characterized in terms of regularized information measures. In many cases of quantum tasks, we do not know how to compute such quantities. Here, we exploit the symmetries in the recently introduced $D^{\#}$ in order to obtain a hierarchy of semidefinite programming bounds on various regularized quantities. As applications, we give a general procedure to give efficient bounds on the regularized Umegaki channel divergence as well as the classical capacity and two-way assisted quantum capacity of quantum channels. In particular, we obtain slight improvements for the capacity of the amplitude damping channel. We also prove that for fixed input and output dimensions, the regularized sandwiched Rényi divergence between any two quantum channels can be approximated up to an $ε$ accuracy in time that is polynomial in $1/ε$.

quant-ph↗

arXiv2021

Larger Corner-Free Sets from Combinatorial Degenerations

There is a large and important collection of Ramsey-type combinatorial problems, closely related to central problems in complexity theory, that can be formulated in terms of the asymptotic growth of the size of the maximum independent sets in powers of a fixed small (directed or undirected) hypergraph, also called the Shannon capacity. An important instance of this is the corner problem studied in the context of multiparty communication complexity in the Number On the Forehead (NOF) model. Versions of this problem and the NOF connection have seen much interest (and progress) in recent works of Linial, Pitassi and Shraibman (ITCS 2019) and Linial and Shraibman (CCC 2021). We introduce and study a general algebraic method for lower bounding the Shannon capacity of directed hypergraphs via combinatorial degenerations, a combinatorial kind of "approximation" of subgraphs that originates from the study of matrix multiplication in algebraic complexity theory (and which play an important role there) but which we use in a novel way. Using the combinatorial degeneration method, we make progress on the corner problem by explicitly constructing a corner-free subset in $F_2^n \times F_2^n$ of size $Ω(3.39^n/poly(n))$, which improves the previous lower bound $Ω(2.82^n)$ of Linial, Pitassi and Shraibman (ITCS 2019) and which gets us closer to the best upper bound $4^{n - o(n)}$. Our new construction of corner-free sets implies an improved NOF protocol for the Eval problem. In the Eval problem over a group $G$, three players need to determine whether their inputs $x_1, x_2, x_3 \in G$ sum to zero. We find that the NOF communication complexity of the Eval problem over $F_2^n$ is at most $0.24n + O(\log n)$, which improves the previous upper bound $0.5n + O(\log n)$.

cs.CC↗