Collaborative Research: Quantum Monte Carlo Algorithms and Quantum Circuit Complexity

合作研究：量子蒙特卡罗算法和量子电路复杂性

• 批准号: 0218563
• 负责人: Cristopher Moore
• 金额: $ 17.5万
• 依托单位: University of New Mexico
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-08-01 至 2005-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0218563&HistoricalAwards=false
• 关键词: Collaborative; Research; Quantum; Monte; Carlo

EIA-0218563Christopher D. MooreUniversity of New MexicoCollaborative Research: Quantum Mote Carlo Algorithms and ComplexityThis collaborative project with the University of Connecticut is exploring both new quantum algorithmic techniques and tools for proving impossibility results for quantum computation. Specifically, the focus is on quantum Monte Carlo algorithms, which try to solve problems by doing a random walk in the space of possible solutions. In addition, fundamental limits on the power of quantum computation, developing impossibility results for quantum circuits is being explored and proving that simple generalizations of Shor's factoring algorithm will not work for the Graph Isomorphism problem, which along with Factoring is a likely candidate for a quantum algorithm.Specifically, quantum walks (unitary analogues of stochastic processes) on various combinatorial structures by employing both Fourier analysis for groups and new tools suited for less symmetric Spaces is being studied. Cases where quantum walks explore the space more quickly than their classical counterparts, and other cases where they become localized and mix more slowly than a classical walk are being explored. In addition Fourier analysis to develop lower bounds for shallow quantum circuits and information-theoretic bounds on the process of sampling from the quantum Fourier transform are being studied. In particular, the quantum Fourier transform over non-Abelian groups, and hidden subgroup and hidden subspace problems for such groups is being investigated. This is closely related to Graph Isomorphism, and explore tractable special cases while showing it is hard for a Shor-type algorithm in general.

EIA-0218563Christopher D. Moore 新墨西哥大学合作研究：量子莫特卡洛算法和复杂性这个与康涅狄格大学的合作项目正在探索新的量子算法技术和工具，以证明量子计算的不可能性结果。具体来说，重点是量子蒙特卡罗算法，该算法尝试通过在可能解决方案的空间中进行随机游走来解决问题。 此外，正在探索量子计算能力的基本限制，为量子电路开发不可能性结果，并证明肖尔因式分解算法的简单概括不适用于图同构问题，该问题与因式分解一起可能是量子计算的候选者。具体来说，正在研究通过采用群的傅里叶分析和适合不太对称空间的新工具在各种组合结构上进行量子行走（随机过程的酉类似物）。 正在探索量子行走比经典行走更快地探索空间的情况，以及量子行走比经典行走更本地化和混合更慢的情况。 此外，正在研究傅里叶分析，以开发浅量子电路的下界和量子傅里叶变换采样过程的信息论界限。特别是，正在研究非阿贝尔群上的量子傅里叶变换，以及此类群的隐藏子群和隐藏子空间问题。 这与图同构密切相关，并探索易于处理的特殊情况，同时表明一般的 Shor 型算法很难。