Kudo SHUHEI (工藤 周平)

ABSTRACT

Approximate block diagonalization is a problem of transforming a given symmetric matrix as close to block diagonal as possible by symmetric permutations of its rows and columns. This problem arises as a preprocessing stage of various scientific calculations and has been shown to be NP‐complete. In this paper, we consider solving this problem approximately using the D‐Wave Advantage quantum annealer. For this purpose, several steps are needed. First, we have to reformulate the problem as a quadratic unconstrained binary optimization (QUBO) problem. Second, the QUBO has to be embedded into the physical qubit network of the quantum annealer. Third, and optionally, reverse annealing for improving the solution can be applied. We propose two QUBO formulations and four embedding strategies for the problem and discuss their advantages and disadvantages. Through numerical experiments, it is shown that the combination of domain‐wall encoding and D‐Wave's automatic embedding is the most efficient in terms of usage of physical qubits, while the combination of one‐hot encoding and automatic embedding is superior in terms of the probability of obtaining a feasible solution. It is also shown that reverse annealing is effective in improving the solution for medium‐sized problems.

Abstract

The Palais matrix represents an n n -dimensional rotation between two vectors that is functionally equivalent to the Householder reflection. This study introduces a one-parameter family of unitary transforms, termed the θ \theta transform, which encompasses the transform by the Palais matrix, the Householder reflection, and their unitary extensions. Furthermore, we define the θ ∠ {\theta }_{\angle } transform, a variant of the θ \theta transform featuring bounded component norms. It is demonstrated that the θ ∠ {\theta }_{\angle } transform is computationally efficient and backward stable when one of the vectors has the “one-hot” structure, making it highly valuable for matrix decompositions such as the QR decomposition. In addition, the θ \theta transform exhibits additional characteristics, including its convergence to the identity and the rowwise structure of its backward error.

Summary

We consider automatic performance tuning of dense symmetric eigenvalue problems using ATMathCoreLib, which is a library to assist automatic tuning. We deal with two problems, namely, automatic code selection for the symmetric generalized eigenvalue problem in distributed‐memory parallel environments and automatic parameter tuning in tridiagonalization of dense symmetric matrices on multicore processors. As for the first problem, numerical experiments show that ATMathCoreLib can choose the fastest solver for a given computing environment and problem size quickly even if the fluctuation in the execution time is as high as 40%. As for the second problem, ATMathCoreLib was able to select nearly optimal combinations of the algorithm and its parameter reliably and efficiently for various computing environments and matrix sizes. The performance of auto‐tuning was further enhanced by incorporating a user‐provided execution‐time model into ATMathCoreLib.