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Workshop on quantum stochastic differential equations for the quantum simulation of physical systems

George Siopsis

Published Sep 22, 2016• 0 citations
OpenAlex

Abstract

Abstract : This is a report on the Workshop on quantum stochastic differential equations for the quantum simulation of physical systems held at theAdelphi Laboratory Center, Adelphi, MD, on Monday, August 22, 2016. It attracted mathematicians, computer scientists, logicians, andphysicists (both theorists and experimentalists) who discussed their research and participated in stimulating discussions on how to applymathematical tools to the quantum simulation of physical systems of interest to the Army. There were participants from US Governmentagencies, industry, and academia. The Workshop was financially supported by ARO via a grant to the University of Tennessee. The web sitefor the Workshop is http://aesop.phys.utk.edu/QI/Workshop.html.

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Abstract : This is a report on the Workshop on quantum stochastic differential equations for the quantum simulation of physical systems held at theAdelphi Laboratory Center, Adelphi, MD, on Monday, August 22, 2016. It attracted mathematicians, computer scientists, logicians, andphysicists (both theorists and experimentalists) who discussed their research and participated in stimulating discussions on how to applymathematical tools to the quantum simulation of physical systems of interest to the Army. There were participants from US Governmentagencies, industry, and academia. The Workshop was financially supported by ARO via a grant to the University of Tennessee. The web sitefor the Workshop is http://aesop.phys.utk.edu/QI/Workshop.html.
This grant supported a workshop, held at Caltech in August 2004, that brought together experts in quantum mechanics, control theory, and dynamical systems for an exchange of ideas on the analysis, control, and design of quantum systems. One of the major goals of the workshop was to establish substantive technical interactions between physicists working on quantum 'applications' and control and dynamical systems researchers who have insights and tools but lack a concrete sense of what problems are of primary interest. The workshop highlighted anticipated applications of quantum control in quantum information science, control of molecular reactions, and design of electronic and photonic nanostructures. The workshop results are available on the web and documented in a paper that has been submitted to the International Journal of Robust and Nonlinear Control.
This workshop focused on the development of prototype quantum information processors, quantum algorithms used for physical modeling applications, and efficient numerical simulations. The Air Force Research Laboratory hosted the Quantum Computation for Physical Modeling (QCPM) Workshop 2004 on 12-15 September 2004, our third bi-annual workshop on quantum computation for physical modeling held at The Harbor View Hotel on Martha's Vineyard. The workshop was hosted by our laboratory and sponsored by the Air Force Office of Scientific Research (AFOSR). The three day workshop comprised six sessions spanning the following research areas: (1) quantum algorithms; (2) quantum effects and devices; (3) superconductive quantum information processing; (4) nuclear magnetic resonance quantum information processing; (5) quantum information, quantum cryptography and quantum communication. We reviewed the research progress carried out by our extramural university research programs following our laboratory's in-house basic research mission and funded through confederated AFOSR grants. Our workshop focused on the development of prototype quantum information processors and quantum algorithms used for physical modeling applications and efficient numerical simulations. The workshop was an outstanding and unqualified success and represents a new milestone in our progress toward achieving a practical quantum computer using state-of-the-art nanotechnology.
In the last years, substantial progress has been made in many areas of mathematical physics. The goal of this workshop was to bring together researchers working on analytic and probabilistic aspects of many-body quantum systems and quantum statistical mechanics, to discuss recent developments, exchange ideas and propose new challenges and research directions. Among the questions addressed during the workshop were the derivation of effective equations, the analysis of physically interesting nonlinear partial differential equations emerging from microscopic theories, the study of open quantum systems in and out of equilibrium, and the investigation of the ground state properties and of the dynamics of quantum spin systems.
Stochastic nonlinear dynamics underlie many models in engineering and computational physics, yet accurate high-dimensional simulation remains challenging. We present a quantum algorithm for a broad class of NNN-dimensional stochastic differential equations with dissipation and quadratic drift. The algorithm applies to strongly nonlinear systems with all-to-all interactions, thereby extending the scope of previously known quantum algorithms that were limited to weak nonlinearity and sparse systems. For norm-preserving drifts, a condition satisfied by key fluid dynamics discretizations, our method approximates expectation values of low-order correlation functions with rigorous error bounds at a cost polynomial in log⁡(N)\log{(N)}log(N) and linear in the evolution time. Our main technical advance is a subroutine for simulating an auxiliary system of NNN interacting quantum harmonic oscillators with cost polylogarithmic in NNN. Finally, we formulate turbulence models, including Navier-Stokes and damped Euler equations, within this framework, opening a route to quantum simulation of strongly nonlinear SDEs governing turbulence and nonlinear wave dynamics.
This paper introduces Quantum-Inspired Stochastic Differential Equations (QSDEs), a novel approach to modeling stochastic systems by leveraging core concepts from quantum mechanics. Traditional Stochastic Differential Equations (SDEs) often rely on Gaussian noise, which can be a significant limitation when accurately representing complex, real-world phenomena. QSDEs propose a fundamental shift by incorporating superposition and entanglement principles to model noise, moving beyond Gaussian distributions. The core mechanism involves introducing a "quantum noise" term, denoted as *η(t, x, P)*, where *η* is a superposition of probability distributions dynamically adjusted based on the system's state *x*. This allows for a more nuanced representation of uncertainty, potentially leading to vastly improved simulations across diverse fields. We outline the theoretical framework, explore the key mathematical formulations, and discuss the potential impact of QSDEs on areas such as financial modeling, physics simulations, and control systems. The central claim of this work is that classical SDEs can be significantly enhanced by incorporating quantum mechanical principles.
Quantum-Inspired Stochastic Differential Equation (Q-SDE) Optimization addresses the challenges of optimizing complex, stochastic systems by integrating quantum-inspired stochastic differential equation modeling. This research presents a novel optimization algorithm, the Q-SDE-based method, designed to effectively handle highly non-linear, stochastic optimization problems characterized by significant noise. We develop a framework that utilizes the inherent stochasticity of Q-SDEs, coupled with differential equation principles, to provide a robust and adaptive approach to finding optimal solutions. The core mechanism involves a dynamic adjustment of the Q-SDE's parameters to minimize the error function across a wider range of solutions, mitigating the impact of noise and enhancing the algorithm's performance in challenging scenarios. This work demonstrates the effectiveness of this approach through extensive numerical simulations and comparisons with established optimization techniques, highlighting its potential for significant improvements in the optimization of stochastic systems.