Improving the Variational Quantum Eigensolver Using Variational Adiabatic Quantum Computing

The variational quantum eigensolver (VQE) is a hybrid quantum-classical algorithm for finding the minimum eigenvalue of a Hamiltonian that involves the optimization of a parameterized quantum circuit. Since the resulting optimization problem is in general nonconvex, the method can converge to suboptimal parameter values that do not yield the minimum eigenvalue. In this work, we address this shortcoming by adopting the concept of variational adiabatic quantum computing (VAQC) as a procedure to improve VQE. In VAQC, the ground state of a continuously parameterized Hamiltonian is approximated via a parameterized quantum circuit. We discuss some basic theory of VAQC to motivate the development of a hybrid quantum-classical homotopy continuation method. The proposed method has parallels with a predictor-corrector method for numerical integration of differential equations. While there are theoretical limitations to the procedure, we see in practice that VAQC can successfully find good initial circuit parameters to initialize VQE. We demonstrate this with two examples from quantum chemistry. Through these examples, we provide empirical evidence that VAQC, combined with other techniques (an adaptive termination criteria for the classical optimizer and a variance-based resampling method for the expectation evaluation), can provide more accurate solutions than “plain” VQE, for the same amount of effort.

Buckling Analysis of Piles in Multi-Layered Soils

Pile buckling is infrequent, but sometimes it can occur in slender piles (i.e., micropiles) driven into soils with soft layers and/or voids. Buckling analysis of piles becomes more complex if the pile is surrounded by multi-layered soil. In this case, the well-known Timoshenko’s solution for pile buckling is of no use because it refers to single-layered soils. A variational approach for buckling analysis of piles in multi-layered soils is herein proposed. The proposed method allows for the estimation of the critical buckling load of piles in any multi-layered soil and for any boundary condition, provided that the distribution of the soil coefficient of the subgrade reaction is available. An eigenvalue-eigenvector problem is defined, where each eigenvector is the set of coefficients of a Fourier series describing the second-order displaced shape of the pile, and the related buckling load is the eigenvalue, thus obtaining the effective buckling load as the minimum eigenvalue. Besides the pile deformed shape, the stiffness distribution in the multi-layered soil is also described through a Fourier series. The Rayleigh–Ritz direct method is used to identify the Fourier development coefficients describing the pile deformation. For validation, buckling analysis results were compared with those obtained from an experimental test and a finite element analysis available in the literature, which confirmed this method’s reliability.

A Descriptor-Based Advanced Feature Detector for Improved Visual Tracking

Despite years of work, a robust, widely applicable generic “symmetry detector” that can paral-lel other kinds of computer vision/image processing tools for the more basic structural charac-teristics, such as a “edge” or “corner” detector, remains a computational challenge. A new symmetry feature detector with a descriptor is proposed in this paper, namely the Simple Robust Features (SRF) algorithm. A performance comparison is made among SRF with SRF, Speeded-up Robust Features (SURF) with SURF, Maximally Stable Extremal Regions (MSER) with SURF, Harris with Fast Retina Keypoint (FREAK), Minimum Eigenvalue with FREAK, Features from Accelerated Segment Test (FAST) with FREAK, and Binary Robust Invariant Scalable Keypoints (BRISK) with FREAK. A visual tracking dataset is used in this performance evaluation in terms of accuracy and computational cost. The results have shown that combining the SRF detector with the SRF descriptor is preferable, as it has on average the highest accuracy. Additionally, the computational cost of SRF with SRF is much lower than the others.

The spectra of finite 3-transposition groups

We calculate the spectrum of the diagram for each finite 3-transposition group. Such graphs with a given minimum eigenvalue have occurred in the context of compact Griess subalgebras of vertex operator algebras.

On the lower bounds for the minimum eigenvalue of the Hadamard product of an M-matrix and its inverse

Recently, some authors have established a number of inequalities involving the minimum eigenvalue for the Hadamard product of M-matrices. In this paper, we improve these results and give some new lower bounds on the minimum eigenvalue for the Hadamard product of an M-matrix A and its inverse {A^{-1}}. Finally, it is shown by the numerical examples that our bounds are also better than some previous results.