Collaborative Research: Quantum Monte Carlo Algorithms and quantum circuit complexity

合作研究：量子蒙特卡罗算法和复杂性这个与新墨西哥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结果。具体来说，重点是量子蒙特卡罗算法，它试图通过在可能的解决方案的空间中进行随机漫步来解决问题。此外，量子计算能力的基本限制，正在探索量子电路的不可能结果，并证明Shor的分解算法的简单推广将不适用于图同构问题，图同构问题与分解算法一起可能是量子算法的候选问题。具体地说，量子行走（随机过程的酉类似物）在各种组合结构上通过使用傅里叶分析和适合于不对称空间的新工具进行研究。量子行走比经典行走更快地探索空间的情况，以及它们比经典行走更慢地本地化和混合的其他情况正在探索中。此外，还研究了利用傅里叶分析来建立浅量子电路的下界和量子傅里叶变换采样过程的信息理论边界。特别地，研究了非阿贝尔群上的量子傅里叶变换，以及这些群的隐子群和隐子空间问题。

EIA-0218443Alexander C. RussellUniversity of ConnecticutCollaborative Research: Quantum Mote Carlo Algorithms and ComplexityThis collaborative project with the University of New Mexico 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 are 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.