相关论文: Lower Bounds on Matrix Rigidity via a Quantum Argu…
In nonadaptive group testing, the main research objective is to design an efficient algorithm to identify a set of up to $t$ positive elements among $n$ samples with as few tests as possible. Disjunct matrices and separable matrices are two…
Hardness magnification reduces major complexity separations (such as $\mathsf{\mathsf{EXP}} \nsubseteq \mathsf{NC}^1$) to proving lower bounds for some natural problem $Q$ against weak circuit models. Several recent works [OS18, MMW19,…
To witness quantum advantages in practical settings, substantial efforts are required not only at the hardware level but also on theoretical research to reduce the computational cost of a given protocol. Quantum computation has the…
We establish a connection between problems studied in rigidity theory and matroids arising from linear algebraic constructions like tensor products and symmetric products. A special case of this correspondence identifies the problem of…
In recent years, finding new satisfiability algorithms for various circuit classes has been a very active line of research. Despite considerable progress, we are still far away from a definite answer on which circuit classes allow fast…
We show that the maximal determinant D(n) for $n \times n$ ${\pm 1}$-matrices satisfies $R(n) := D(n)/n^{n/2} \ge \kappa_d > 0$. Here $n^{n/2}$ is the Hadamard upper bound, and $\kappa_d$ depends only on $d := n-h$, where $h$ is the maximal…
We comment on two randomized algorithms for constructing low-rank matrix decompositions. Both algorithms employ the Subsampled Randomized Hadamard Transform [14]. The first algorithm appeared recently in [9]; here, we provide a novel…
To any complex Hadamard matrix we associate a quantum permutation group. The correspondence is not one-to-one, but the quantum group encapsulates a number of subtle properties of the matrix. We investigate various aspects of the…
The density matrices are positively semi-definite Hermitian matrices of unit trace that describe the state of a quantum system. The goal of the paper is to develop minimax lower bounds on error rates of estimation of low rank density…
A novel lower bound is introduced for the full rank probability of random finite field matrices, where a number of elements with known location are identically zero, and remaining elements are chosen independently of each other, uniformly…
Errors occurring on noisy hardware pose a key challenge to reliable quantum computing. Existing techniques such as error correction, mitigation, or suppression typically separate the error handling from the algorithm analysis and design. In…
Rough sets were proposed to deal with the vagueness and incompleteness of knowledge in information systems. There are may optimization issues in this field such as attribute reduction. Matroids generalized from matrices are widely used in…
Optimal control of closed quantum systems is a well studied geometrically elegant set of computational theory and techniques that have proven pivotal in the implementation and understanding of quantum computers. The design of a circuit…
The radius of regularity sometimes spelled as the radius of nonsingularity is a measure providing the distance of a given matrix to the nearest singular one. Despite its possible application strength this measure is still far from being…
Matrices are typically considered over fields or rings. Motivated by applications in parametric differential equations and data-driven modeling, we suggest to study matrices with entries from a Hilbert space and present an elementary theory…
A matrix-compression algorithm is derived from a novel isogenic block decomposition for square matrices. The resulting compression and inflation operations possess strong functorial and spectral-permanence properties. The basic observation…
We analyze the set of real and complex Hadamard matrices with additional symmetry constrains. In particular, we link the problem of existence of maximally entangled multipartite states of $2k$ subsystems with $d$ levels each to the set of…
In this article we present new proofs for the boundedness and the compactness on $\ell^2$ of the Rhaly matrices, also known as terraced matrices. We completely characterize when such matrices belong to the Schatten class…
An interesting example of the deep interrelation between Physics and Mathematics is obtained when trying to impose mathematical boundary conditions on physical quantum fields. This procedure has recently been re-examined with care. Comments…
We give an elementary proof of a generalization of Bourgain and Tzafriri's Restricted Invertibility Theorem, which says roughly that any matrix with columns of unit length and bounded operator norm has a large coordinate subspace on which…