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The time evolution operator plays a crucial role in the precise computation of chemical experiments on quantum computers and holds immense promise for advancing the fields of physical and computer sciences, with applications spanning quantum simulation and machine learning. However, the construction of large-scale quantum computers poses significant challenges, prompting the need for innovative and resource-efficient strategies. Traditional methods like phase estimation or variational algorithms come with certain limitations such as the use of classical optimization or complex quantum circuitry. One successful method is the Trotterization technique used for quantum simulation, specifically in atomic structure problems with a gate complexity of approximately O(n^2) for an n-qubit realization. In this work, we have proposed a new encoding method, namely quantum approximate time evolution (QATE) for the quantum implementation of the kinetic energy operator as a diagonal unitary operator considering the first quantization level. The theoretical foundations of our approach are discussed, and experimental results are obtained on an IBM quantum machine. Our proposed method offers gate complexity in sub-quadratic polynomial with qubit size $n$ which is an improvement over previous work. Further, the fidelity improvement for the time evolution of the Gaussian wave packet has also been demonstrated.

The exploration of potential energy operators in quantum systems holds paramount significance, offering profound insights into atomic behaviour, defining interactions, and enabling precise prediction of molecular dynamics. By embracing the Born-Oppenheimer picture, we delve into the intricate quantum evolution due to potential energy, facilitating accurate modelling and simulation of atomic phenomena with improved quantum fidelity. This research delves into time evolution operation due to potential energy functions for applications spanning quantum chemistry and condensed matter physics. Challenges in practical implementation, encompassing the formidable curse of dimensionality and intricate entangled interactions, are thoughtfully examined. Drawing upon seminal works, we lay a robust foundation for comprehensive investigations into potential energy landscapes with two proposed algorithms. In one methodology, we have shown a systematic decomposition of the potential energy function into Hadamard bases with composite construction of Pauli-Z, identity and RZ gates which can construct the unitary time evolution operator corresponding to the potential energy with a very high fidelity. The other method is a trade-off between complexity and fidelity, where we propose a novel quantum framework that can reduce the gate complexity from {\Theta}(2n) to {\Theta}(nCr ) (for some r < n). The proposed quantum algorithms are capable of efficiently simulating potential energy operators. The algorithms were implemented in simulators and IBM quantum hardware to prove their efficacy

The Corona KH-4 reconnaissance satellite missions from 1962-1972 acquired panoramic stereo imagery with high spatial resolution of 1.8-7.5 m. The potential of 800,000+ declassified Corona images has not been leveraged due to the complexities arising from handling of panoramic imaging geometry, film distortions and limited availability of the metadata required for georeferencing of the Corona imagery. This paper presents Corona Stereo Pipeline (CoSP): A pipeline for processing of Corona KH-4 stereo panoramic imagery. CoSP utlizes a deep learning based feature matcher SuperGlue to automatically match features point between Corona KH-4 images and recent satellite imagery to generate Ground Control Points (GCPs). To model the imaging geometry and the scanning motion of the panoramic KH-4 cameras, a rigorous camera model consisting of modified collinearity equations with time dependent exterior orientation parameters is employed. The results show that using the entire frame of the Corona image, bundle adjustment using well-distributed GCPs results in an average standard deviation (SD) of less than 2 pixels. The distortion pattern of image residuals of GCPs and y-parallax in epipolar resampled images suggest that film distortions due to long term storage as likely cause of systematic deviations. Compared to the SRTM DEM, the Corona DEM computed using CoSP achieved a Normalized Median Absolute Deviation (NMAD) of elevation differences of ~4 m over an area of approx. 4000 $km^2$. We show that the proposed pipeline can be applied to sequence of complex scenes involving high relief and glacierized terrain and that the resulting DEMs can be used to compute long term glacier elevation changes over large areas., Comment: 24 Pages, 16 Figures