无条件非均匀量子傅里叶与切比雪夫变换
Conditioning-Free Non-Uniform Quantum Fourier and Chebyshev Transforms
- University of Cincinnati（辛辛那提大学）
- Pennsylvania State University（宾夕法尼亚州立大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
提出一种无条件的非均匀量子切比雪夫变换算法，通过改进NUQFT消除几何依赖，利用受控电路实现，达到O(1)归一化及O(L)量子比特和O~(L^2)门的高效复杂度。
AI中文摘要：
我们提出了一种高效的非均匀切比雪夫变换量子算法。该变换定义为将函数投影到在给定节点处采样的切比雪夫多项式上，这些节点在$x\in[-1,1]$上是均匀的，因此在角度$\theta=\arccos x$上非均匀，这是基于QFT的量子切比雪夫变换无法处理的设置。我们的构造基于对现有非均匀量子傅里叶变换（NUQFT）的改进，通过该改进我们移除了非均匀节点采样的条件性。因此，误差界与先前工作中依赖于几何的参数$\kappa$无关。我们利用切比雪夫变换矩阵是两次第二类非均匀离散傅里叶变换的平均值这一事实，并通过单个受控NUQFT电路实现。我们提供了显式的预言机构造，包括先前仅作为假设保留的行访问预言机。所得的$\varepsilon$精确块编码具有$O(1)$归一化，并使用$O(L)$个量子比特和$\widetilde O(L^2)$个门，其中$L=\log N+\log(1/\varepsilon)$。我们给出了端到端的实现及复杂度分析，包括成功概率和输出态误差。
英文摘要：
We present an efficient quantum algorithm for the non-uniform Chebyshev transform. It is defined as the projection of a function onto Chebyshev polynomials sampled at given nodes that are uniform in $x\in[-1,1]$, and hence non-uniform in the angle $θ=\arccos x$, a setting that QFT-based quantum Chebyshev transforms cannot handle. Our construction is based on an improvement of an existing Non-uniform Quantum Fourier Transform (NUQFT) whereby we remove the conditioning from non-uniform node sampling. Hence, error bounds are independent of the geometry-dependent parameter $κ$ of prior work. We use the fact that Chebyshev transform matrix is the average of two Type-II non-uniform DFTs, which we implement with a single controlled NUQFT circuit. We provide explicit oracle constructions, including the row-access oracle previously left as an assumption. The resulting $\varepsilon$-accurate block encoding has $O(1)$ normalization and uses $O(L)$ qubits and $\widetilde O(L^2)$ gates, where $L=\log N+\log(1/\varepsilon)$. We give an end-to-end implementation with complexity analysis, including the success probability and output-state error.
