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的最新发展及其在量子计算和量子混沌中的应用。在该项目的实际应用方面，PI希望如果建造物理量子计算机，他的罗斯和塞林格算法的改进版本将被使用。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。这个模型与丢芬图方程的积分解的研究有关，丢芬图方程是数学家感兴趣的古老课题。量子计算机科学家和数学家都感兴趣的一个问题是用积分解对特殊丢芬图方程的实解进行最优逼近。PI在这个方向上证明了新的（最优的）结果。此外，他还证明了对于一般输入，这个任务在计算上是困难的（np完全）。在更技术性的层面上，量子门模型的核心问题之一是使用一组称为通用量子门的固定生成器来近似任意量子位。在单量子位的情况下，这相当于通过一组特定的拓扑生成器（例如v门或lubotzky - philips - sarnak生成器）导航酉群SU(2)，这些拓扑生成器经过精心选择，使得相关的跃迁矩阵具有最佳的谱间隙（例如Hecke算子的特征值满足Ramanujan界）。PI提出了罗斯和塞林格算法的改进，以近似任意的单量子位，从他们的算法中删除所有启发式假设。这种方法中的新工具包括delta方法、筛理论和模形式及其傅立叶系数边界的谱理论。本课题的目的是将PI的结果推广到高阶算术群，从而引入振荡表示理论和自同构表示理论。基于量子混沌中的Berry猜想，PI研究了量子门跃迁矩阵（Hecke算子）的统计性质和特征值的多重性。到目前为止，PI已经证明了Hecke算子特征值多重性的省电上界和绝对上界。此外，PI还证明了光谱测度相对于Plancherel测度差异的下界。该项目汇集了伽罗瓦表示的变形理论、岩川理论、泰勒-怀尔斯方法、轨迹公式以及其他代数和解析数论学家工具箱中的工具，以回答计算机科学家和数学家感兴趣的问题。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识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.