Quantum signal processing and quantum singular value transformation are powerful tools to implement polynomial transformations of block-encoded matrices on quantum computers, and has achieved asymptotically optimal complexity in many prominent quantum algorithms. We propose a framework of quantum signal processing and quantum singular value transformation on $U(N)$, which realizes multiple polynomials simultaneously from a block-encoded input, as a generalization of those on $U(2)$ in the original frameworks. We provide a comprehensive characterization of achievable polynomial matrices and give recursive algorithms to construct the quantum circuits that realize desired polynomial transformations. As three example applications, we propose a framework to realize bi-variate polynomial functions, demonstrate $N$-interval decision achieving $O(d)$ query complexity with a $\log_2 N$ improvement over iterative $U(2)$-QSP requiring $O(d\log_2 N)$ queries, and present a quantum amplitude estimation algorithm achieving the Heisenberg limit without adaptive measurements.
Sequence and Image Transformations with Monarq: Quantum Implementations for NISQ Devices
National Energy Research Scientific Computing Center, Lawrence Berkeley National Laboratory（美国能源部先进科学计算研究中心，劳伦斯伯克利国家实验室）；Applied Mathematics and Computational Research Division, Lawrence Berkeley National Laboratory（应用数学与计算研究处，劳伦斯伯克利国家实验室）；Computer Science Department, San Francisco State University（旧金山州立大学计算机科学系）；Scientific Data Division, Lawrence Berkeley National Laboratory（科学数据处，劳伦斯伯克利国家实验室）
Generalized quantum singular value transformation with application in quantum conjugate gradient least squares algorithm
School of Mathematics and Statistics, Wuhan University（武汉大学数学与统计学院）；Department of Applied Physics, School of Science, Xi’an Jiaotong University（西安交通大学理学院应用物理系）；Hubei Key Laboratory of Computational Science, Wuhan University（武汉大学计算科学湖北省重点实验室）
Adaptive identification of low-degree polynomials in quantum singular value transformation: application to nonlinear quantum properties estimation
Mitsubishi UFJ Financial Group, Inc.（三菱UFJ金融集团）；MUFG Bank, Ltd.（三菱UFJ银行）；Quantum Computing Center, Keio University（庆应义塾大学量子计算中心）；Graduate School of Science and Technology, Keio University（庆应义塾大学理工学研究科）；International Center for Elementary Particle Physics, University of Tokyo（东京大学基础粒子物理国际中心）；Advanced Material Engineering Division, Toyota Motor Corporation（丰田汽车公司先进材料工程部门）
Generalized KS transformations, $ND$ singular oscillator and generalized MICZ-Kepler system
在广义Kustaanheimo-Stiefel（KS）变换的框架下，给出了对偶（拟）精确可解模型及其隐对称代数的个数在不同分析层次上的描述.证明了N维奇异振子与（n+1）维广义MICZ-Kepler系统是对偶的，其对偶变换是KS变换的推广.在二重坐标系、球坐标系和抛物坐标系下，用分离变量法给出了这些问题的薛定谔方程的可解性。在SU（1，1）代数加法规则的框架下，二次Hahn代数QH（3）作为一个隐对称，不随原实空间${\rm I \!R}^{N}$分解成不同分量而改变.在Howe对偶的意义下，用交换子方法也清楚地证明了Higgs/Hahn代数等隐对称代数。对二维奇异振子进行降维处理，其中每个变量都是n维超半径r。本文还讨论了（N=2n）维奇异振子与（n+1）维广义MICZ-Kepler系统在拟精确（QE）问题类中的对偶联系。在广义KS变换的框架下，给出并分析了各向异性和非线性非谐项对谐振子模型及其对偶模型的某些推广。本文讨论了上述问题的薛定谔方程在QE类中的精确解析解，并给出了四类对偶拟精确可解模型的精确解析解。特别地，给出了在低维情况下与类似结果的比较及其推广.
