发表机构
Tsinghua University; TraverseQuantum Co., Ltd.; Yau Mathematical Sciences Center, Tsinghua University; Qiuzhen College, Tsinghua University; Institute for AI Industry Research (AIR), Tsinghua University; Academy of Mathematics and Systems Science, Chinese Academy of Sciences; Institute for Applied Mathematics, Tsinghua University; Beijing Institute of Mathematical Sciences and Applications; Department of Computer Science and Technology, Tsinghua University（清华大学; TraverseQuantum有限公司; 清华大学丘成桐数学科学中心; 清华大学邱士泽学院; 清华大学人工智能产业研究院; 中国科学院数学与系统科学研究院; 清华大学应用数学中心; 北京国际数学研究中心; 清华大学计算机科学与技术系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出量子加权Riesz方法统一求解四类Riccati问题，实现近最优查询复杂度，并证明相关承诺问题的BQP完全性，应用于控制与量子化学。
AI 中文摘要
我们开发了量子算法，用于构造连续时间和离散时间代数Riccati方程（CAREs和DAREs）、微分Riccati方程（DREs）以及有限Riccati递推的解矩阵的块编码。量子加权Riesz方法统一了四类Riccati问题，利用加权Riesz分支算子构造解图上的投影算子，并恢复解矩阵。在所陈述的访问和归一化假设下，我们针对CAREs和DAREs的算法具有近最优的查询复杂度$\Theta(\mathcal R\alpha)$（直至对数因子），其中广义奇异性因子$\mathcal R$反映谱分离，$\alpha$为输出归一化。对于DREs和有限时域问题，查询复杂度额外包含初始化条件因子$\kappa_{\rm init}$。我们针对指定的输入预言机族建立了乘积查询下界，证明了对谱分离和初始化或输出尺度的必要联合依赖。我们还证明了在显式电路生成的单控制离散时间代数和零终端微分Riccati方程族上，选定值承诺问题是BQP完全的。应用包括线性二次控制的经典反馈评估、量子化学中稳定随机相位近似Riccati方程的查询复杂度比较，以及带数值验证的加热边界控制。
英文摘要
We develop quantum algorithms that construct block-encodings of solution matrices for continuous- and discrete-time algebraic Riccati equations (CAREs and DAREs), differential Riccati equations (DREs), and finite Riccati recursions. The Quantum Weighted Riesz Method, unifying four kinds of Riccati problems, constructs projectors onto the solution graphs using weighted Riesz branching operators and recovers the solution matrices. Under the stated access and normalization assumptions, our algorithms for CAREs and DAREs have near-optimal query complexity $Θ(\mathcal Rα)$ up to logarithmic factors, with the generalized singularity factor $\mathcal R$ that reflects spectral separation, and the output normalization $α$. For DREs and finite-horizon problems, the query complexity has an additional initialization conditioning factor $κ_{\rm init}$. We establish product query lower bounds for specified input-oracle families, demonstrating necessary joint dependence on spectral separation and initialization or output scale. We also prove that selected-value promise problems are BQP-complete on explicit circuit-generated families of single-control discrete-time algebraic and zero-terminal differential Riccati equations. Applications include classical feedback evaluation for linear-quadratic control, a query-complexity comparison for stable random-phase approximation Riccati equations in quantum chemistry, and heated boundary control with numerical validation.
Comments128 pages (including appendices), 7 figures, 3 tables
