arXiv subjects

Hua Xiang

Publications and source records attributed to Hua Xiang.

36 records · Page 2Linked to original sources

arXiv2021

Quantum Algorithm for Matrix Logarithm by Integral Formula

The matrix logarithm is one of the important matrix functions. Recently, a quantum algorithm that computes the state $|f\rangle$ corresponding to matrix-vector product $f(A)b$ is proposed in [Takahira, et al. Quantum algorithm for matrix functions by Cauchy's integral formula, QIC, Vol.20, No.1\&2, pp.14-36, 2020]. However, it can not be applied to matrix logarithm. In this paper, we propose a quantum algorithm, which uses LCU method and block-encoding technique as subroutines, to compute the state $|f\rangle = \log(A)|b\rangle / \|\log(A)|b\rangle\|$ corresponding to $\log(A)b$ via the integral representation of $\log(A)$ and the Gauss-Legendre quadrature rule.

math.NA↗

arXiv2019

Incompatibility probability of random quantum measurements

Incompatibility of quantum measurements is of fundamental importance in quantum mechanics. It is closely related to many nonclassical phenomena such as Bell nonlocality, quantum uncertainty relations, and quantum steering. We study the necessary and sufficient conditions of quantum compatibility for a given collection of $n$ measurements in $d$-dimensional space. From the compatibility criterion for two-qubit measurements, we compute the incompatibility probability of a pair of independent random measurements. For a pair of unbiased random qubit measurements, we derive that the incompatibility probability is exactly $\frac35$. Detailed results are also presented in figures for pairs of general qubit measurements.

quant-ph↗

arXiv2019

A variant of Horn's problem and derivative principle

Identifying the spectrum of the sum of two given Hermitian matrices with fixed eigenvalues is the famous Horn's problem.In this note, we investigate a variant of Horn's problem, i.e., we identify the probability density function (abbr. pdf) of the diagonals of the sum of two random Hermitian matrices with given spectra. We then use it to re-derive the pdf of the eigenvalues of the sum of two random Hermitian matrices with given eigenvalues via \emph{derivative principle}, a powerful tool used to get the exact probability distribution by reducing to the corresponding distribution of diagonal entries.We can recover Jean-Bernard Zuber's recent results on the pdf of the eigenvalues of two random Hermitian matrices with given eigenvalues. Moreover, as an illustration, we derive the analytical expressions of eigenvalues of the sum of two random Hermitian matrices from $\rG\rU\rE(n)$ or Wishart ensemble by derivative principle, respectively.We also investigate the statistics of exponential of random matrices and connect them with Golden-Thompson inequality, and partly answer a question proposed by Forrester. Some potential applications in quantum information theory, such as uniform average quantum Jensen-Shannon divergence and average coherence of uniform mixture of two orbits,are discussed.

quant-ph↗

arXiv2019

Quantum algorithm for total least squares data fitting

The total least squares~(TLS) method is widely used in data-fitting. Compared with the least squares fitting method, the TLS fitting takes into account not only observation errors, but also errors from the measurement matrix of the variables. In this work, the TLS problem is transformed to finding the ground state of a Hamiltonian matrix. We propose quantum algorithms for solving this problem based on quantum simulation of resonant transitions. Our algorithms can achieve at least polynomial speedup over the known classical algorithms.

quant-ph↗

arXiv2019

Randomized Row and Column Iterative Methods with a Quantum Computer

We consider the quantum implementations of the two classical iterative solvers for a system of linear equations, including the Kaczmarz method which uses a row of coefficient matrix in each iteration step, and the coordinate descent method which utilizes a column instead. These two methods are widely applied in big data science due to their very simple iteration schemes. In this paper we use the block-encoding technique and propose fast quantum implementations for these two approaches, under the assumption that the quantum states of each row or each column can be efficiently prepared. The quantum algorithms achieve exponential speed up at the problem size over the classical versions, meanwhile their complexity is nearly linear at the number of steps.

quant-ph↗

arXiv2018

Quantum Regularized Least Squares Solver with Parameter Estimate

In this paper we propose a quantum algorithm to determine the Tikhonov regularization parameter and solve the ill-conditioned linear equations, for example, arising from the finite element discretization of linear or nonlinear inverse problems. For regularized least squares problem with a fixed regularization parameter, we use the HHL algorithm and work on an extended matrix with smaller condition number. For the determination of the regularization parameter, we combine the classical L-curve and GCV function, and design quantum algorithms to compute the norms of regularized solution and the corresponding residual in parallel and locate the best regularization parameter by Grover's search. The quantum algorithm can achieve a quadratic speedup in the number of regularization parameters and an exponential speedup in the dimension of problem size.

quant-ph↗

arXiv2018

Randomized QLP algorithm and error analysis

In this paper, we describe the randomized QLP (RQLP) algorithm and its enhanced version (ERQLP) for computing the low rank approximation to $A$ of size $m\times n$ efficiently such that $A\approx QLP$, where $L$ is the rank-$k$ lower-triangular matrix, $Q$ and $P$ are column orthogonal matrices. The theoretical cost of the implementation of RQLP and ERQLP only needs $\mathcal{O}(mnk)$. Moreover, we derive the upper bounds of the expected approximation error $\mathbb{E}\left [ (σ_{j}(A) - σ_{j} (L))/ σ_{j}(A) \right] $ for $j=1,\cdots, k$, and prove that the $L$-values of the proposed methods can track the singular values of $A$ accurately. These claims are supported by extensive numerical experiments.

math.NA↗

arXiv2018

M-eigenvalues of The Riemann Curvature Tensor

The Riemann curvature tensor is a central mathematical tool in Einstein's theory of general relativity. Its related eigenproblem plays an important role in mathematics and physics. We extend M-eigenvalues for the elasticity tensor to the Riemann curvature tensor. The definition of M-eigenproblem of the Riemann curvature tensor is introduced from the minimization of an associated function. The M-eigenvalues of the Riemann curvature tensor always exist and are real. They are invariants of the Riemann curvature tensor. The associated function of the Riemann curvature tensor is always positive at a point if and only if the M-eigenvalues of the Riemann curvature tensor are all positive at that point. We investigate the M-eigenvalues for the simple cases, such as the 2D case, the 3D case, the constant curvature and the Schwarzschild solution, and all the calculated M-eigenvalues are related to the curvature invariants.

math.DG↗

arXiv2018

Singular Values of Riemann Curvature Tensor

We introduce the concept of singular values for the Riemann curvature tensor, a central mathematical tool in Einstein's theory of general relativity. We study the properties related to the singular values, and investigate five typical cases to show its relationship to the Ricci scalar and other invariants.

math.DG↗

arXiv2018

Quantum Circulant Preconditioner for Linear System of Equations

We consider the quantum linear solver for $Ax=b$ with the circulant preconditioner $C$. The main technique is the singular value estimation (SVE) introduced in [I. Kerenidis and A. Prakash, Quantum recommendation system, in ITCS 2017]. However, some modifications of SVE should be made to solve the preconditioned linear system $C^{-1} Ax = C^{-1} b$. Moreover, different from the preconditioned linear system considered in [B. D. Clader, B. C. Jacobs, C. R. Sprouse, Preconditioned quantum linear system algorithm, Phys. Rev. Lett., 2013], the circulant preconditioner is easy to construct and can be directly applied to general dense non-Hermitian cases. The time complexity depends on the condition numbers of $C$ and $C^{-1} A$, as well as the Frobenius norm $\|A\|_F$.

quant-ph↗

arXiv2018

On the M-eigenvalues of elasticity tensor and the strong ellipticity condition

Strong ellipticity is an important property in the elasticity theory. In 2009, M-eigenvalues were introduced for the elasticity tensor. It was shown that M-eigenvalues are invariant under coordinate system choices, and the strong ellipticity condition holds if and only if all the M-eigenvalues of the elasticity tensor are positive. Thus, M-eigenvalues are some intrinsic parameters of the elasticity tensor. In this paper, we show that the M-eigenvalues of the elasticity tensor are closely related with some elastic moduli, such as the bulk modulus, the shear modulus, Lamé's first parameter, the P-wave modulus, etc, and the positiveness of the M-eigenvalues are corresponding to some existing conditions for strong ellipticity in some special cases, such as the isotropic case, the cubic case, the polar anisotropic case and the tetragonal case. We also present new sufficient conditions for the strong ellipticity of the orthotropic case. These, in a certain sense, further reveal the physical meanings of M-eigenvalues.

math.RA↗

arXiv2017

Randomized Iterative Methods with Alternating Projections

We use a unified framework to summarize sixteen randomized iterative methods including Kaczmarz method, coordinate descent method, etc. Some new iterative schemes are given as well. Some relationships with \textsc{mg} and \textsc{ddm} are also discussed. We analyze the convergence properties of the iterative schemes by using the matrix integrals associated with alternating projectors, and demonstrate the convergence behaviors of the randomized iterative methods by numerical examples.

math.NA↗

arXiv2017

Average entropy of a subsystem over a global unitary orbit of a mixed bipartite state

We investigate the average entropy of a subsystem within a global unitary orbit of a given mixed bipartite state in the finite-dimensional space. Without working out the closed-form expression of such average entropy for the mixed state case, we provide an analytical lower bound for this average entropy. In deriving this analytical lower bound, we get some useful by-products of independent interest. We also apply these results to estimate average correlation along a global unitary orbit of a given mixed bipartite state. When the notion of von Neumann entropy is replaced by linear entropy, the similar problem can be considered also, and moreover the exact average linear entropy formula is derived for a subsystem over a global unitary orbit of a mixed bipartite state.

quant-ph↗

arXiv2016

Preconditioners and Their Analyses for Edge Element Saddle-point Systems Arising from Time-harmonic Maxwell Equations

We shall propose and analyze some new preconditioners for the saddle-point systems arising from the edge element discretization of the time-harmonic Maxwell equations in three dimensions. We will first consider the saddle-point systems with vanishing wave number, for which we present an important relation between the solutions of the singular curl-curl system and the non-singular saddle-point system, then demonstrate that the saddle-point system can be efficiently solved by the Hiptmair-Xu solver. For the saddle-point systems with non-vanishing wave numbers, we will show that the PCG with a new preconditioner can apply for the non-singular system when wave numbers are small, while the methods like preconditioned MINRES may apply for some existing and new preconditioners when wave numbers are large. The spectral behaviors of the resulting preconditioned systems for the existing and new preconditioners are analyzed and compared, and numerical experiments are presented to demonstrate and compare the efficiencies of these preconditioners.

math.NA↗

arXiv2014

Randomized Algorithms for Large-scale Inverse Problems with General Regularizations

We shall investigate randomized algorithms for solving large-scale linear inverse problems with general regularizations. We first present some techniques to transform inverse problems of general form into the ones of standard form, then apply randomized algorithms to reduce large-scale systems of standard form to much smaller-scale systems and seek their regularized solutions in combination with some popular choice rules for regularization parameters. Then we will propose a second approach to solve large-scale ill-posed systems with general regularizations. This involves a new randomized generalized SVD algorithm that can essentially reduce the size of the original large-scale ill-posed systems. The reduced systems can provide approximate regularized solutions with about the same accuracy as the ones by the classical generalized SVD, and more importantly, the new approach gains obvious robustness, stability and computational time as it needs only to work on problems of much smaller size. Numerical results are given to demonstrated the efficiency of the algorithms.

math.NA↗

arXiv2014

Perturbation Analysis and Randomized Algorithms for Large-Scale Total Least Squares Problems

In this paper, we present perturbation analysis and randomized algorithms for the total least squares (TLS) problems. We derive the perturbation bound and check its sharpness by numerical experiments. Motivated by the recently popular probabilistic algorithms for low-rank approximations, we develop randomized algorithms for the TLS and the truncated total least squares (TTLS) solutions of large-scale discrete ill-posed problems, which can greatly reduce the computational time and still keep good accuracy.

math.NA↗

arXiv2014

An Inexact Uzawa Algorithm for Generalized Saddle-Point Problems and Its Convergence

We propose an inexact Uzawa algorithm with two variable relaxation parameters for solving the generalized saddle-point system. The saddle-point problems can be found in a wide class of applications, such as the augmented Lagrangian formulation of the constrained minimization, the mixed finite element method, the mortar domain decomposition method and the discretization of elliptic and parabolic interface problems. The two variable parameters can be updated at each iteration, requiring no a priori estimates on the spectrum of two preconditioned subsystems involved. The convergence and convergence rate of the algorithm are analysed. Both symmetric and nonsymmetric saddle-point systems are discussed, and numerical experiments are presented to demonstrate the robustness and effectiveness of the algorithm.

math.NA↗

arXiv2014

Adaptive Uzawa algorithm for nonsymmetric generalized saddle point problem

In this paper, we extend the inexact Uzawa algorithm in [Q. Hu, J. Zou, SIAM J. Matrix Anal., 23(2001), pp. 317-338] to the nonsymmetric generalized saddle point problem. The techniques used here are similar to those in [Bramble \emph{et al}, Math. Comput. 69(1999), pp. 667-689], where the convergence of Uzawa type algorithm for solving nonsymmetric case depends on the spectrum of the preconditioners involved. The main contributions of this paper focus on a new linear Uzawa type algorithm for nonsymmetric generalized saddle point problems and its convergence. This new algorithm can always converge without any prior estimate on the spectrum of two preconditioned subsystems involved, which may not be easy to achieve in applications. Numerical results of the algorithm on the Navier-Stokes problem are also presented.

math.NA↗