Applications of Number Theory to the Quantum Gates Model

该奖项的研究和更广泛的活动包括数论及其在量子计算和量子混沌中的应用的最新发展。在该项目的实际应用方面，PI预计，如果建立物理量子计算机，他的改进版本的罗斯和塞林格算法将被使用。人们希望量子计算机最终能够有效地模拟量子物理，并研究现代计算机无法解决的许多(重要的)计算难题。有一些模型，如量子门模型，给出了用于量子计算机的有效电路的理论结构。这个模型与丢番图方程的整体解的研究有关，丢番图方程是数学家感兴趣的一个古老课题。量子计算机科学家和数学家都感兴趣的一个问题是积分解对特殊丢番图方程实数解的最佳逼近。PI已经在这个方向上证明了新的(最优)结果。此外，他还证明了这一任务对于一般输入来说是计算困难(NP-完全)的。在更技术的层面上，量子门模型中的中心问题之一是使用一组固定的生成器来近似任意量子比特，该集合被称为通用量子门。在单量子比特的情况下，这相当于通过一组特定的拓扑生成器(例如，V门或Lubotzky-Phillips-Sarnak生成器)来导航酉群SU(2)，这些生成器经过精心选择，使得相关的转移矩阵具有最优的谱间隙(例如，Hecke算符的本征值满足Ramanujan界)。PI提出了对Ross和Selinger算法的改进，以近似任意单个量子比特，去掉了他们算法中的所有启发式假设。这种方法中的新工具包括Delta方法、筛子理论和模形式的谱理论以及它们的傅里叶系数的界限。这个项目的一个目标是将PI的结果推广到高阶算术群，从而引入振子表示理论和自同构表示理论。在量子混沌中Berry猜想的启发下，PI研究了量子门转移矩阵(Hecke算符)的统计性质和本征值的重数。到目前为止，PI已经证明了Hecke算子本征值的重数的节能上界和绝对上界。此外，PI还证明了谱测度相对于Plancerel测度的偏差的下界。该项目汇集了伽罗瓦表示的形变理论、岩泽理论、泰勒-威尔斯方法、迹公式以及代数和解析数学家工具箱中的其他工具，以回答计算机科学家和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智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.