Applications of Number Theory to the Quantum Gates Model

数论在量子门模型中的应用

基本信息

批准号：
2015305
负责人：
Naser Talebizadeh Sardari
金额：
$ 7.43万
依托单位：
American Institute of Mathematics
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2019
资助国家：
美国
起止时间：
2019-08-19 至 2022-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2015305&HistoricalAwards=false
关键词：
Applications Number Theory Quantum Gates

项目摘要

Both the research and broader activities in this award include current developments in number theory and their applications in quantum computing and quantum chaos. In terms of practical applications of the project, the PI expects that his refined version of Ross and Selinger algorithm will be used if a physical quantum computer is built. The hope is that quantum computers will eventually be able to efficiently simulate quantum physics and study many (important) computationally difficult problems inaccessible to modern-day computers. There are models such as the Quantum Gates Model that give theoretical constructions of efficient circuits to be used in quantum computers. This model is connected to the study of integral solutions to Diophantine equations, an ancient subject of interest to mathematicians. A question of interest to both quantum computer scientists as well as mathematicians is the optimal approximation of real solutions of special Diophantine equations by integral solutions. The PI has proved new (optimal) results in this direction. Furthermore, he has proved that this task is computationally hard (NP-complete) for generic inputs.On a more technical level, one of the central problems in the Quantum Gates Model is the approximation of an arbitrary qubit using a fixed set of generators called universal quantum gates. In the single-qubit case, this amounts to navigating the unitary group SU(2) by a specific set of topological generators (e.g. V-gates or the Lubotzky-Phillips-Sarnak generators) that are carefully chosen such that the associated transition matrix has the optimal spectral gap (e.g. the eigenvalues of the Hecke operators satisfy the Ramanujan bound). The PI proposes a refinement of the Ross and Selinger algorithm for approximating an arbitrary single-qubit that removes all heuristic assumptions from their algorithm. Among the new tools in this approach are the delta method, Sieve theory, and the spectral theory of modular forms and bounds on their Fourier coefficients. An objective of this project is to generalize the results of the PI to higher rank arithmetic groups which brings in the theory of the oscillator representation and the theory of automorphic representations. Motivated by Berry's conjecture in Quantum Chaos, the PI studies the statistical properties and the multiplicity of the eigenvalues of the transition matrix of the quantum gates (the Hecke operators). So far, the PI has proved power saving upper bounds as well as absolute upper bound on the multiplicity of the eigenvalues of the Hecke operators. Furthermore, the PI has proved lower bounds on the discrepancy of the spectral measure with respect to the Plancherel measure. The project brings together the deformation theory of Galois representations, Iwasawa theory, the Taylor-Wiles method, trace formulae, and other tools from the algebraic and analytic number theorists' toolbox in order to answer questions of interest to computer scientists as well as mathematicians.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

该奖项中的研究和更广泛的活动包括数量理论的当前发展及其在量子计算和量子混乱中的应用。就项目的实际应用而言，PI预计如果构建了物理量子计算机，将使用其精致版本的Ross和Selinger算法。希望是量子计算机最终将能够有效地模拟量子物理学并研究现代计算机无法访问的许多（重要）计算困难问题。有一些模型，例如量子门模型，这些模型提供了用于量子计算机中的有效电路的理论结构。该模型与对数学方程的积分解决方案的研究有关，这是数学家感兴趣的古老主题。量子计算机科学家和数学家的一个兴趣问题是通过积分解决方案对特殊二磷剂方程的实际解决方案的最佳近似。 PI已证明了这一方向的新（最佳）结果。此外，他已经证明，对于通用输入而言，此任务在计算上是硬（NP完整）。一个更具技术性的水平，量子门模型中的一个核心问题之一是使用称为固定的生成器的近似值通用量子门。在单一Qubit情况下，这相当于通过一组特定的拓扑发电机（例如V-Gates或Lubotzky-Phillips-Sarnak Generator）浏览单一组SU（2）最佳光谱差距（例如，Hecke操作员的特征值满足Ramanujan结合）。 PI提出了对Ross和Selinger算法的改进，以近似于任意的单量，该单量从其算法中删除了所有启发式假设。在这种方法中，新工具包括Delta方法，筛理论及其傅立叶系数上的模块化形式和边界的光谱理论。该项目的一个目的是将PI的结果推广到更高等级的算术群，这些算法群体带来了振荡器表示理论和自动形式理论。 PI是由Berry在量子混乱中的猜想的动机，研究了量子门（Hecke Operators）的过渡矩阵的特征值的统计特性和多样性。到目前为止，PI已证明了节能上限以及在Hecke操作员特征值的多重性上的绝对上限。此外，PI已证明了相对于Plancherel度量的光谱度量差异的下限。该项目汇集了Galois表示的变形理论，Iwasawa理论，Taylor-Wiles方法，痕量公式以及来自代数和分析数理论家工具箱的其他工具，以便回答计算机科学家以及数学家感兴趣的问题。该奖项反映了NSF的法定使命，并通过使用基金会的知识分子和更广泛的影响审查标准进行评估而被认为值得支持。