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The cost of data input can dominate the run-time of quantum algorithms. Here, we consider data input of arithmetically structured matrices via block encoding circuits, the input model for the quantum singular value transform and related…
Matrix product states provide a natural entanglement basis to represent a quantum register and operate quantum gates on it. This scheme can be materialized to simulate a quantum adiabatic algorithm solving hard instances of a NP-Complete…
Here is discussed the Hamiltonian approach to construction of deterministic universal (in approximate sense) programmable quantum circuits with qubits or any other quantum systems with dimension of Hilbert space is $n \ge 2$.
We present an efficient decoding algorithm for constant rate quantum hypergraph-product LDPC codes which provably corrects adversarial errors of weight $\Omega(\sqrt{n})$ for codes of length $n$. The algorithm runs in time linear in the…
Efficient methods for the representation and simulation of quantum states and quantum operations are crucial for the optimization of quantum circuits. Decision diagrams (DDs), a well-studied data structure originally used to represent…
An algorithm for first-principles electronic structure calculations having a computational cost which scales linearly with the system size is presented. Our method exploits the real-space localization of the density matrix, and in this…
We introduce the Block-Recurrent Transformer, which applies a transformer layer in a recurrent fashion along a sequence, and has linear complexity with respect to sequence length. Our recurrent cell operates on blocks of tokens rather than…
Recently, data-enabled predictive control (DeePC) schemes based on Willems' fundamental lemma have attracted considerable attention. At the core are computations using Hankel-like matrices and their connection to the concept of persistency…
We present an inequality for tensor product of positive operators on Hilbert spaces by considering the tensor product of operators as words on certain alphabets (i.e., a set of letters). As applications of the operator inequality and by a…
Many matrices associated with fast transforms posess a certain low-rank property characterized by the existence of several block partitionings of the matrix, where each block is of low rank. Provided that these partitionings are known,…
In a recent work arXiv:2201.07655v2 we showed that there is a constant $\lambda >0$ such that it is possible to efficiently classically simulate a quantum system in which (i) qudits are placed on the nodes of a graph, (ii) each qudit…
Tensor network methods as presented in our open source Matrix Product States software have opened up the possibility to study many-body quantum physics in one and quasi-one-dimensional systems in an easily accessible package similar to…
Factorization of large dense matrices are ubiquitous in engineering and data science applications, e.g. preconditioners for iterative boundary integral solvers, frontal matrices in sparse multifrontal solvers, and computing the determinant…
This paper addresses an investigation on a factorization method for difference equations. It is proved that some classes of second order linear difference operators, acting in Hilbert spaces, can be factorized using a pair of mutually…
The computation of determinants plays a central role in diagrammatic Monte Carlo algorithms for strongly correlated systems. The evaluation of large numbers of determinants can often be the limiting computational factor determining the…
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…
This manuscript describes a technique for computing partial rank-revealing factorizations, such as, e.g, a partial QR factorization or a partial singular value decomposition. The method takes as input a tolerance $\varepsilon$ and an…
Various numerical linear algebra problems can be formulated as evaluating bivariate function of matrices. The most notable examples are the Fr\'echet derivative along a direction, the evaluation of (univariate) functions of…
An efficient low-order scaling method is presented for large-scale electronic structure calculations based on the density functional theory using localized basis functions, which directly computes selected elements of the density matrix by…
N-fold integer programming is a fundamental problem with a variety of natural applications in operations research and statistics. Moreover, it is universal and provides a new, variable-dimension, parametrization of all of integer…