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:
- 金额:$ 7.43万
- 依托单位:
- 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:
- 关键词:
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.
该奖项的研究和更广泛的活动包括数论的当前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预计,如果构建物理量子计算机,将使用他的Ross和Selinger算法的改进版本。人们希望量子计算机最终能够有效地模拟量子物理,并研究许多现代计算机无法解决的(重要的)计算困难问题。有一些模型,如量子门模型,给出了量子计算机中使用的有效电路的理论构造。这个模型与丢番图方程的积分解的研究有关,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特殊丢番图方程的真实的解的积分解的最佳逼近。PI已经证明了这一方向的新(最佳)结果。此外,他还证明了这个任务对于一般输入是计算困难的(NP完全)。在更技术的层面上,量子门模型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,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),这些生成器被仔细选择,使得相关的过渡矩阵具有最佳的谱隙(例如Hecke算子的本征值满足Ramanujan界)。PI提出了Ross和Selinger算法的改进,用于近似任意单量子比特,从他们的算法中删除了所有启发式假设。在这种方法中的新工具是三角洲方法,筛理论,和频谱理论的模块化形式和边界上的傅立叶系数。这个项目的一个目标是推广PI的结果,以更高的秩算术群带来的理论的振子表示和理论的自守表示。受量子混沌中Berry猜想的启发,PI研究了量子门(Hecke算子)的转移矩阵的统计特性和本征值的多重性。到目前为止,PI已经证明了Hecke算子特征值重数的省电上界和绝对上界。此外,PI已经证明了关于Plancherel测度的谱测度的差异的下界。该项目汇集了伽罗瓦表示的变形理论,岩泽理论,泰勒-怀尔斯方法,迹公式,该奖项反映了NSF的法定使命,并被认为值得通过使用基金会的智力价值和更广泛的影响进行评估来支持审查标准。
项目成果
期刊论文数量(4)
专著数量(0)
科研奖励数量(0)
会议论文数量(0)
专利数量(0)
Vanishing Fourier coefficients of Hecke eigenforms
Hecke 特征函数的消失傅立叶系数
- DOI:10.1007/s00208-021-02178-7
- 发表时间:2021
- 期刊:
- 影响因子:1.4
- 作者:Calegari, Frank;Talebizadeh Sardari, Naser
- 通讯作者:Talebizadeh Sardari, Naser
Ramanujan graphs and exponential sums over function fields
拉马努金图和函数域上的指数和
- DOI:10.1016/j.jnt.2020.05.010
- 发表时间:2020
- 期刊:
- 影响因子:0.7
- 作者:Sardari, Naser T.;Zargar, Masoud
- 通讯作者:Zargar, Masoud
Asymptotic trace formula for the Hecke operators
Hecke 算子的渐近迹公式
- DOI:10.1007/s00208-020-02054-w
- 发表时间:2020
- 期刊:
- 影响因子:1.4
- 作者:Jung, Junehyuk;Talebizadeh Sardari, Naser
- 通讯作者:Talebizadeh Sardari, Naser
The least prime number represented by a binary quadratic form
用二进制二次形式表示的最小质数
- DOI:10.4171/jems/1031
- 发表时间:2021
- 期刊:
- 影响因子:2.6
- 作者:Talebizadeh Sardari, Naser
- 通讯作者:Talebizadeh Sardari, Naser
{{
item.title }}
{{ item.translation_title }}
- DOI:
{{ item.doi }} - 发表时间:
{{ item.publish_year }} - 期刊:
- 影响因子:{{ item.factor }}
- 作者:
{{ item.authors }} - 通讯作者:
{{ item.author }}
数据更新时间:{{ journalArticles.updateTime }}
{{ item.title }}
- 作者:
{{ item.author }}
数据更新时间:{{ monograph.updateTime }}
{{ item.title }}
- 作者:
{{ item.author }}
数据更新时间:{{ sciAawards.updateTime }}
{{ item.title }}
- 作者:
{{ item.author }}
数据更新时间:{{ conferencePapers.updateTime }}
{{ item.title }}
- 作者:
{{ item.author }}
数据更新时间:{{ patent.updateTime }}
Naser Talebizadeh Sardari其他文献
Optimal strong approximation for quadrics over math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg" class="math"msubmrowmi mathvariant="double-struck"F/mi/mrowmrowmiq/mi/mrow/msubmo stretchy="false"[/momit/mimo stretchy="false"]/mo/math
在\(F\)上二次曲面的最优强逼近
- DOI:
10.1016/j.aim.2022.108852 - 发表时间:
2023-01-15 - 期刊:
- 影响因子:1.500
- 作者:
Naser Talebizadeh Sardari;Masoud Zargar - 通讯作者:
Masoud Zargar
Naser Talebizadeh Sardari的其他文献
{{
item.title }}
{{ item.translation_title }}
- DOI:
{{ item.doi }} - 发表时间:
{{ item.publish_year }} - 期刊:
- 影响因子:{{ item.factor }}
- 作者:
{{ item.authors }} - 通讯作者:
{{ item.author }}
{{ truncateString('Naser Talebizadeh Sardari', 18)}}的其他基金
Number Theory, Potential Theory, and Convex Optimization
数论、势论和凸优化
- 批准号:
2401242 - 财政年份:2024
- 资助金额:
$ 7.43万 - 项目类别:
Standard Grant
Applications of Number Theory to the Quantum Gates Model
数论在量子门模型中的应用
- 批准号:
1902185 - 财政年份:2019
- 资助金额:
$ 7.43万 - 项目类别:
Standard Grant
相似国自然基金
关于群上的短零和序列及其cross number的研究
- 批准号:11501561
- 批准年份:2015
- 资助金额:18.0 万元
- 项目类别:青年科学基金项目
相似海外基金
A1-Homotopy Theory and Applications to Enumerative Geometry and Number Theory
A1-同伦理论及其在枚举几何和数论中的应用
- 批准号:
2405191 - 财政年份:2024
- 资助金额:
$ 7.43万 - 项目类别:
Standard Grant
Probabilistic models of zeta-functions and applications to number theory
Zeta 函数的概率模型及其在数论中的应用
- 批准号:
22KJ2747 - 财政年份:2023
- 资助金额:
$ 7.43万 - 项目类别:
Grant-in-Aid for JSPS Fellows
Invariable generation in finite groups with applications to algorithmic number theory
有限群中的不变生成及其在算法数论中的应用
- 批准号:
EP/T017619/3 - 财政年份:2022
- 资助金额:
$ 7.43万 - 项目类别:
Fellowship
Applications of random matrix theory in analytic number theory
随机矩阵理论在解析数论中的应用
- 批准号:
RGPIN-2019-04888 - 财政年份:2022
- 资助金额:
$ 7.43万 - 项目类别:
Discovery Grants Program - Individual
Model theory with applications to algebra, geometry and number theory
模型理论及其在代数、几何和数论中的应用
- 批准号:
RGPIN-2021-02474 - 财政年份:2022
- 资助金额:
$ 7.43万 - 项目类别:
Discovery Grants Program - Individual
Applications of random matrix theory in analytic number theory
随机矩阵理论在解析数论中的应用
- 批准号:
RGPIN-2019-04888 - 财政年份:2021
- 资助金额:
$ 7.43万 - 项目类别:
Discovery Grants Program - Individual
Invariable generation in finite groups with applications to algorithmic number theory
有限群中的不变生成及其在算法数论中的应用
- 批准号:
EP/T017619/2 - 财政年份:2021
- 资助金额:
$ 7.43万 - 项目类别:
Fellowship
Model theory with applications to algebra, geometry and number theory
模型理论及其在代数、几何和数论中的应用
- 批准号:
RGPIN-2021-02474 - 财政年份:2021
- 资助金额:
$ 7.43万 - 项目类别:
Discovery Grants Program - Individual
CAREER: Decoupling Theory, Oscillatory Integral Theory, and Their Applications in Analytic Number Theory and Combinatorics
职业:解耦理论、振荡积分理论及其在解析数论和组合学中的应用
- 批准号:
2044828 - 财政年份:2021
- 资助金额:
$ 7.43万 - 项目类别:
Continuing Grant
Dirichlet L-functions, Erdos-Kac theorems, and applications to number theory
Dirichlet L 函数、Erdos-Kac 定理以及在数论中的应用
- 批准号:
RGPIN-2016-03756 - 财政年份:2021
- 资助金额:
$ 7.43万 - 项目类别:
Discovery Grants Program - Individual