References Publications referenced by this paper. Solution of systems of linear equations Vladimir G. Direct and iterative methods for the solution of linear operator equations in Hilbert space Wolodymyr V. A general method of steepest ortho-descent," Bull. The convergence of the method of steepest descents for nonlinear equations with variational or quasi-variational operators," J.

Iterative methods for the solutions of nonlinear operator equations in Hubert space," Ph. Todd , Ed. Related Papers. By clicking accept or continuing to use the site, you agree to the terms outlined in our Privacy Policy , Terms of Service , and Dataset License. Acquiring the low-lying eigenpairs or singular triplets of D is in general computationally expensive and requires the use of additional iterative methods that also suffer from critical slowing down.

Solutions to address this issue include EigCG 6 , 7 , inexact deflation 8 , and adaptive multigrid 9 , 10 , 11 , Quantum annealing offers an alternative approach to solving a general system of equations.

We map each variable x j using R number of qubits such that. Defining the vectors. The ground state of Eq. For current commercial quantum annealers, auxiliary qubits are required to reduce multi-linear terms down to bilinear interactions through quadratization 14 , 15 , 16 , 17 , 18 , 19 , 20 , We provide the details of quadratization through reduction-by-substitution on a system of second-order polynomials in Methods.

It follows that the entire energy spectrum is positive, and if the exact solution is recovered, then the ground state energy must be zero.

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A system of linear equations simplifies Eq. In addition, constant terms that arise from the substitution of Eq. Restriction to linear least squares demands that the fitting function is linear in the unknown parameters, and therefore may be written in the form. The solution for linear regression is obtained by expanding Eq. The extrema of the objective function can be determined by taking the derivative of Eq. We demonstrate the validity of the algorithm on a system of two second order polynomial equations.

The problem is chosen to be small such that the solution can be confirmed by an explicit search over the entire Hilbert space, and evaluated onto a D-Wave annealer. Consider the following system of equations,. For the sake of discussion, we set up Eq. The tensors P n are obtained by inspection,. After transforming to the qubit-basis, direct search of the ground state of the 4-body Hamiltonian yields.

## Atkinson : A Survey of Numerical Methods for Solving Nonlinear Integral Equations

The coefficient C should be chosen large enough such that the constraint is satisfied under optimization. Additional details of the quadratization used is given in Methods. We repeat the exercise of solving the same system of polynomial equations on the D-Wave annealer with the symmetrized and quadratized representation, and successfully reproduce the solution,. The software implementation of various many-body and 2-body quadratized representations for a system of second-order polynomial equations, the brute force solver, and D-Wave solver are made publicly available The Toeplitz correlation matrix 23 is chosen to simulate a correlated time-series dataset, where the correlations decay exponentially as a function of x.

Following the notation in Eq. Using Eq. These correct coefficients for the generating function in Eq. We next test the algorithm by solving the objective function using quantum annealing. The target Hamiltonian of Eq. The correct result is reproduced in 0. In this section we show results and scaling of a classical method and the quantum annealer. The condition number of a matrix P 1 is defined as the ratio of maximum and minimum singular values. In the case of symmetric matrices, this is equivalent to the ratio of largest and smallest eigenvalues.

We vary our test matrices in two ways: 1 vary the problem size while holding the condition number fixed, 2 the problem size is held constant with varying condition number. The accurately of the solution will be judged by the relative residual sum of squares,. For conjugate gradient, a tolerance for Eq. For quantum annealing the role of the relative residual is more subtle. The annealer is run many times and the lowest energy eigenpair is returned. The eigenvector from this set is substituted for x approx , allowing a relative residual to be defined for the total anneal. Although conjugate gradient is not the optimal choice for classically solving such systems, the scaling comparison in condition number with the quantum algorithm is informative.

The same right-hand side is solved for all cases. The matrices from this and subsequent results are constructed by creating a random unitary matrix of rank N , denoted as U. In the following section, we demonstrate the scaling of the annealing algorithm under varying problem size, condition number, and precision of the search space.

- Fractional Calculus and Applied Analysis!
- Phoenix Rising (Ministry of Peculiar Occurrences, Book 1).
- Iterative Methods for the Solution of a Linear Operator Equation in Hilbert Space. A Survey!

We conclude by applying the algorithm iteratively on a fixed problem and study the convergence of the relative residual. Problem size : In Fig. Due to prior knowledge of the conjugate-gradient solution, the search space for all N parameters are fixed for the set of problems, and encompass the minimum and maximum results of the solution vector x. Additionally, knowledge of the result allows us to identify the ground-state QUBO solution by minimizing the difference between the conjugate-gradient and QUBO results the forward error , and studies the theoretical scaling of the algorithm absent of current hardware limitations.

Due to the small condition number of this study, minimizing the forward error is equivalent to minimizing the backwards error.

The red crosses are results of the lowest energy state from , quantum annealing measurements. Physical measurements deviate from the theoretical minimum as problem size grows. Bottom The corresponding percentage of measurements in the minimum energy state are shown in blue. The vertical axis is shown in a logarithmic scale.

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The forward error prediction dashed black line is omitted and not a reliable measure of the relative residual for larger condition numbers. Note that the vertical axis of the bottom plot showing the percentage of measurements observed in the lowest-lying state is on a linear scale. Note that both vertical axes are on a logarithmic scale. By the ninth iteration the result reaches single precision. Bottom The percentage of quantum annealing solutions in the lowest-lying state. The algorithm successfully resolves the solution at single precision for this example without issue.

With increased problem size, we observe that the percentage of annealed solutions which return the ground state decreases exponentially.

## On general iterative methods for the solutions of a class of nonlinear operator equations

This indicates the solution for a dense matrix may require exponentially more evaluations to obtain for current quantum annealers. The observed scaling is consistent with the assumption that the energy gap exponentially vanishes with increasing size for a dense Hamiltonian. Condition number : Fig. The condition number affects the solution vector x , and therefore for this study we restrict the search range to span exactly the minimum and maximum values of x.

The chosen search range keeps the resulting relative residual approximately constant under varying condition number. For linear systems with larger condition numbers, minimizing the forward error is no longer a reliable estimate of the residual of the backwards error, and is therefore dropped from this study. With increasing condition number, we observe that the percentage of solutions that converge to the lowest-lying state is a relatively constant value as demonstrated by a less than one order-of-magnitude change between the different examples.

This behavior is in stark contrast with the scaling observed in Fig. This is in amazing contrast to the classical result from Fig.