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1. Quadratic Quantum Variational Monte Carlo

   Su, B; Liu, Q (December 2024, Advances in Neural Information Processing Systems 37 (NeurIPS 2024))

   This paper introduces the Quadratic Quantum Variational Monte Carlo (Q2 VMC) algorithm, an innovative algorithm in quantum chemistry that significantly enhances the efficiency and accuracy of solving the Schrödinger equation. Inspired by the discretization of imaginary-time Schrödinger evolution, Q2 VMC employs a novel quadratic update mechanism that integrates seamlessly with neural network-based ansatzes. Our extensive experiments showcase Q2 VMC's superior performance, achieving faster convergence and lower ground state energies in wavefunction optimization across various molecular systems, without additional computational cost. This study not only advances the field of computational quantum chemistry but also highlights the important role of discretized evolution in variational quantum algorithms, offering a scalable and robust framework for future quantum research. 

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2. On the benefits of memory for modeling time-dependent PDEs

   Buitrago, R; Marwah, T; Gu, A; Risteski, A (April 2025, International Conference on Learning Representations (ICLR), 2025)

   Data-driven techniques have emerged as a promising alternative to traditional numerical methods for solving PDEs. For time-dependent PDEs, many approaches are Markovian -- the evolution of the trained system only depends on the current state, and not the past states. In this work, we investigate the benefits of using memory for modeling time-dependent PDEs: that is, when past states are explicitly used to predict the future. Motivated by the Mori-Zwanzig theory of model reduction, we theoretically exhibit examples of simple (even linear) PDEs, in which a solution that uses memory is arbitrarily better than a Markovian solution. Additionally, we introduce Memory Neural Operator (MemNO), a neural operator architecture that combines recent state space models (specifically, S4) and Fourier Neural Operators (FNOs) to effectively model memory. We empirically demonstrate that when the PDEs are supplied in low resolution or contain observation noise at train and test time, MemNO significantly outperforms the baselines without memory -- with up to 6x reduction in test error. Furthermore, we show that this benefit is particularly pronounced when the PDE solutions have significant high-frequency Fourier modes (e.g., low-viscosity fluid dynamics) and we construct a challenging benchmark dataset consisting of such PDEs. 

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3. The Vehicle Monitoring and Collection Technology Era

   Elvy, StacyAnn (November 2024, Iowa law review)

   ABSTRACT: Vehicle monitoring and collection (“VMC”) technology, including starter interrupt devices that remotely disable vehicles, and other GPS tracking devices are used in consumer vehicle agreements in subprime transactions. Subscription-based models supported by VMC features, such as over-the-air software updates that remotely disable or enable vehicular functions, have also appeared in the non-subprime automobile context as well. This Article contends that the rise in VMC technology and features raises several alarming privacy, electronic subjugation, and cybersecurity risks. The Article makes an express link between the consumer risks and harms associated with the use of VMC technology in subprime lending transactions and the broader technological shift towards a subscription-based service model supported by VMC features in non-subprime vehicle transactions. The Article’s evaluation and critique of VMC technology and features is conducted simultaneously through the lens of commercial law, state VMC technology statutes, state privacy laws, such as the California Consumer Privacy Act (“CCPA”), and federal frameworks governing transactions involving consumer vehicles. I argue that these legal regimes do not consistently protect consumer interests and are insufficient to comprehensively meet the challenges associated with the VMC age. The Article concludes by offering a detailed path forward. 

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4. Solving Multidimensional Partial Differential Equations Using Efficient Quantum Circuits

   https://doi.org/10.3390/a18030176  

   Chaudhary, Manu; El-Araby, Kareem; Nobel, Alvir; Jha, Vinayak; Kneidel, Dylan; Islam, Ishraq; Singh, Manish; Ogundele, Sunday; Phillips, Ben; Egan, Kieran; et al (March 2025, Algorithms)

   Quantum computing has the potential to solve certain compute-intensive problems faster than classical computing by leveraging the quantum mechanical properties of superposition and entanglement. This capability can be particularly useful for solving Partial Differential Equations (PDEs), which are challenging to solve even for High-Performance Computing (HPC) systems, especially for multidimensional PDEs. This led researchers to investigate the usage of Quantum-Centric High-Performance Computing (QC-HPC) to solve multidimensional PDEs for various applications. However, the current quantum computing-based PDE-solvers, especially those based on Variational Quantum Algorithms (VQAs) suffer from limitations, such as low accuracy, long execution times, and limited scalability. In this work, we propose an innovative algorithm to solve multidimensional PDEs with two variants. The first variant uses Finite Difference Method (FDM), Classical-to-Quantum (C2Q) encoding, and numerical instantiation, whereas the second variant utilizes FDM, C2Q encoding, and Column-by-Column Decomposition (CCD). We evaluated the proposed algorithm using the Poisson equation as a case study and validated it through experiments conducted on noise-free and noisy simulators, as well as hardware emulators and real quantum hardware from IBM. Our results show higher accuracy, improved scalability, and faster execution times in comparison to variational-based PDE-solvers, demonstrating the advantage of our approach for solving multidimensional PDEs. 

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5. Solving PDEs in space-time: 4D tree-based adaptivity, mesh-free and matrix-free approaches

   https://doi.org/10.1145/3295500.3356198  

   Ishii, Masado; Fernando, Milinda; Saurabh, Kumar; Khara, Biswajit; Ganapathysubramanian, Baskar; Sundar, Hari (November 2019, SC '19: Proceedings of the International Conference for High Performance Computing, Networking, Storage and Analysis)

   Numerically solving partial differential equations (PDEs) remains a compelling application of supercomputing resources. The next generation of computing resources - exhibiting increased parallelism and deep memory hierarchies - provide an opportunity to rethink how to solve PDEs, especially time dependent PDEs. Here, we consider time as an additional dimension and simultaneously solve for the unknown in large blocks of time (i.e. in 4D space-time), instead of the standard approach of sequential time-stepping. We discretize the 4D space-time domain using a mesh-free kD tree construction that enables good parallel performance as well as on-the-fly construction of adaptive 4D meshes. To best use the 4D space-time mesh adaptivity, we invoke concepts from PDE analysis to establish rigorous a posteriori error estimates for a general class of PDEs. We solve canonical linear as well as non-linear PDEs (heat diffusion, advection-diffusion, and Allen-Cahn) in space-time, and illustrate the following advantages: (a) sustained scaling behavior across a larger processor count compared to sequential time-stepping approaches, (b) the ability to capture "localized" behavior in space and time using the adaptive space-time mesh, and (c) removal of any time-stepping constraints like the Courant-Friedrichs-Lewy (CFL) condition, as well as the ability to utilize spatially varying time-steps. We believe that the algorithmic and mathematical developments along with efficient deployment on modern architectures shown in this work constitute an important step towards improving the scalability of PDE solvers on the next generation of supercomputers. 

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