Related papers: Hermitian Matrix Definiteness from Quantum Phase E…
Let $H$ be a positive semi-definite matrix partitioned in $\beta\times \beta$ Hermitian blocks, $H=[A_{s,t}]$, $1\le s,t,\le \beta$. Then, for all symmetric norms, {equation*} \| H \| \le \| \sum_{s=1}^{\beta} A_{s,s} \|. {equation*} The…
Given a square, nonsingular matrix of univariate polynomials $\mathbf{F} \in \mathbb{K}[x]^{n \times n}$ over a field $\mathbb{K}$, we give a fast, deterministic algorithm for finding the Hermite normal form of $\mathbf{F}$ with complexity…
We consider a symmetric matrix, the entries of which depend linearly on some parameters. The domains of the parameters are compact real intervals. We investigate the problem of checking whether for each (or some) setting of the parameters,…
We analyze when an arbitrary matrix pencil is equivalent to a dissipative Hamiltonian pencil and show that this heavily restricts the spectral properties. In order to relax the spectral properties, we introduce matrix pencils with…
The matrix logarithm, when applied to Hermitian positive definite matrices, is concave with respect to the positive semidefinite order. This operator concavity property leads to numerous concavity and convexity results for other matrix…
Quantum phase estimation is one of the key algorithms in the field of quantum computing, but up until now, only approximate expressions have been derived for the probability of error. We revisit these derivations, and find that by ensuring…
Hermitian and unitary matrices are two representatives of the class of normal matrices whose full eigenvalue decomposition can be stably computed in quadratic computing com plexity. Recently, fast and reliable eigensolvers dealing with low…
Matrix powering is a fundamental computational primitive in linear algebra. It has widespread applications in scientific computing and engineering, and underlies the solution of time-homogeneous linear ordinary differential equations,…
We present a universal technique for quantum state estimation based on the maximum-likelihood method. This approach provides a positive definite estimate for the density matrix from a sequence of measurements performed on identically…
We discuss different cases of dissipative Hamiltonian differential-algebraic equations and the linear algebraic systems that arise in their linearization or discretization. For each case we give examples from practical applications. An…
The efficient implementation of matrix arithmetic operations underpins the speedups of many quantum algorithms. We develop a suite of methods to perform matrix arithmetics -- with the result encoded in the off-diagonal blocks of a…
A quantum state for being an eigenstate of some local Hamiltonian should be constraint by zero energy variance and consequently, the constraint is rather strong that a single eigenstate may uniquely determine the Hamiltonian. For…
Quantum phase estimation combined with Hamiltonian simulation is the most promising algorithmic framework to computing ground state energies on quantum computers. Its main computational overhead derives from the Hamiltonian simulation…
An important step in building a quantum computer is calibrating experimentally implemented quantum gates to produce operations that are close to ideal unitaries. The calibration step involves estimating the systematic errors in gates and…
It is proposed to map the quantum information qubit not to individual spin 1/2 states, but to the collective spin states being eigenfunctions of the Hamiltonian including spin-spin interactions, which may be not small. Such an approach…
Inspired by the quantum computing algorithms for Linear Algebra problems [HHL,TaShma] we study how the simulation on a classical computer of this type of "Phase Estimation algorithms" performs when we apply it to solve the Eigen-Problem of…
We propose a class of randomized quantum algorithms for the task of sampling from matrix functions, without the use of quantum block encodings or any other coherent oracle access to the matrix elements. As such, our use of qubits is purely…
Quantum mechanics features a variety of distinct properties such as coherence and entanglement, which could be explored to showcase potential advantages over classical counterparts in information processing. In general, legitimate quantum…
Natural frequencies and normal modes are basic properties of a structure which play important roles in analyses of its vibrational characteristics. As their computation reduces to solving eigenvalue problems, it is a natural arena for…
To develop a unitary quantum theory with probabilistic description for pseudo- Hermitian systems one needs to consider the theories in a different Hilbert space endowed with a positive definite metric operator. There are different…