We present a stochastic quantum computing algorithm that can prepare any eigenvector of a quantum Hamiltonian within a selected energy interval $[E-ε, E+ε]$. In order to reduce the spectral weight of all other eigenvectors by a suppression factor $δ$, the required computational effort scales as $O[|\log δ|/(p ε)]$, where $p$ is the squared overlap of the initial state with the target eigenvector. The method, which we call the rodeo algorithm, uses auxiliary qubits to control the time evolution of the Hamiltonian minus some tunable parameter $E$. With each auxiliary qubit measurement, the amplitudes of the eigenvectors are multiplied by a stochastic factor that depends on the proximity of their energy to $E$. In this manner, we converge to the target eigenvector with exponential accuracy in the number of measurements. In addition to preparing eigenvectors, the method can also compute the full spectrum of the Hamiltonian. We illustrate the performance with several examples. For energy eigenvalue determination with error $ε$, the computational scaling is $O[(\log ε)^2/(p ε)]$. For eigenstate preparation, the computational scaling is $O(\log Δ/p)$, where $Δ$ is the magnitude of the orthogonal component of the residual vector. The speed for eigenstate preparation is exponentially faster than that for phase estimation or adiabatic evolution.
Rodeo Filtering for Direct Steady-State Estimation in Open Quantum Systems
Seoul National University（首尔大学）；Pusan National University（釜山国立大学）；University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）；Kyung Hee University（庆熙大学）
Unraveling Rodeo Algorithm Through the Zeeman Model
Universidade Federal da Integração Latino-Americana（拉丁美洲一体化联邦大学）；Universidade Federal de Ouro Preto - UFOP（欧罗普雷托联邦大学）；Universidade Federal de Juiz de Fora - UFJF（茹伊斯迪福拉联邦大学）
Persistent State Method for Resonances on Classical and Quantum Computers
Fudan University（复旦大学）；Ruhr-Universität Bochum（波鸿鲁尔大学）；Shanghai Research Center for Theoretical Nuclear Physics, NSFC and Fudan University（国家自然科学基金委与复旦大学理论核物理上海研究中心）；School of Physics, East China Normal University（华东师范大学物理系）；Michigan State University（密歇根州立大学）；Tohoku University（东北大学）；Institute for Advanced Simulation (IAS-4), Forschungszentrum Jülich（于利希研究中心先进模拟研究所（IAS-4））
Computing the Classical-Quantum channel capacity: experiments on a Blahut-Arimoto type algorithm and an approximate solution for the binary inputs, two-dimensional outputs channel
A quantum algorithm for computing the Carmichael function
量子计算机可以有效地解决许多数论问题。本文利用有效的量子算法作为预言机，给出了一个计算任意整数$N$的Carmichael函数的算法，该算法的概率尽可能接近1.该算法需要$O((\log n )^3n^3)$量子运算，或使用快速乘法的$O(\log\log n (\log n)^4 n^2)$运算。验证，量子优化和应用RSA和素性测试进行了讨论。
