Quantum Singular Value Transformation (QSVT) provides a unified framework for applying polynomial functions to the singular values of a block-encoded matrix. QSVT prepares a state proportional to $\bA^{-1}\bb$ with circuit depth $O(d\cdot\mathrm{polylog}(N))$, where $d$ is the polynomial degree of the $1/x$ approximation and $N$ is the size of $\bA$. Current polynomial approximation methods are over the continuous interval $[a,1]$, giving $d = O(\sqrt{\kap}\log(1/\varepsilon))$, and make no use of any properties of $\bA$. We observe here that QSVT solution accuracy depends only on the polynomial accuracy at the eigenvalues of $\bA$. When all $N$ eigenvalues are known exactly, a pure spectral polynomial $p_{S}$ can interpolate $1/x$ at these eigenvalues and achieve unit fidelity at reduced degree. But its practical applicability is limited. To address this, we propose a spectral correction that exploits prior knowledge of $K$ eigenvalues of $\bA$. Given any base polynomial $p_0$, such as Remez, of degree $d_0$, a $K\times K$ linear system enforces exact interpolation of $1/x$ only at these $K$ eigenvalues without increasing $d_0$. The spectrally corrected polynomial $p_{SC}$ preserves the continuous error profile between eigenvalues and inherits the parity of $p_0$. QSVT experiments on the 1D Poisson equation demonstrate up to a $5\times$ reduction in circuit depth relative to the base polynomial, at unit fidelity and improved compliance error. The correction is agnostic to the choice of base polynomial and robust to eigenvalue perturbations up to $10\%$ relative error. Extension to the 2D Poisson equation suggests that correcting a small fraction of the spectrum may suffice to achieve fidelity above $0.999$.

翻译：量子奇异值变换（QSVT）为将多项式函数应用于块编码矩阵的奇异值提供了一个统一框架。QSVT以电路深度 $O(d\cdot\mathrm{polylog}(N))$ 制备与 $\bA^{-1}\bb$ 成比例的状态，其中 $d$ 是 $1/x$ 逼近的多项式次数，$N$ 是 $\bA$ 的尺寸。当前的多项式逼近方法是在连续区间 $[a,1]$ 上进行的，得到 $d = O(\sqrt{\kap}\log(1/\varepsilon))$，并且没有利用 $\bA$ 的任何性质。我们在此观察到，QSVT解的精度仅取决于多项式在 $\bA$ 的特征值处的逼近精度。当所有 $N$ 个特征值都精确已知时，一个纯谱多项式 $p_{S}$ 可以在这些特征值处插值 $1/x$，并以降低的次数实现单位保真度。但其实际适用性有限。为了解决这个问题，我们提出了一种谱修正方法，它利用了 $\bA$ 的 $K$ 个特征值的先验知识。给定任意一个基多项式 $p_0$（例如 Remez 多项式），其次数为 $d_0$，一个 $K\times K$ 线性系统强制 $1/x$ 仅在这 $K$ 个特征值处精确插值，而不增加 $d_0$。谱修正多项式 $p_{SC}$ 保持了特征值之间的连续误差分布，并继承了 $p_0$ 的奇偶性。在一维泊松方程上的 QSVT 实验表明，相对于基多项式，在单位保真度和改善的合规误差下，电路深度最多可减少 $5$ 倍。该修正对基多项式的选择是无关的，并且对于高达 $10\%$ 相对误差的特征值扰动具有鲁棒性。扩展到二维泊松方程表明，修正一小部分谱可能足以实现高于 $0.999$ 的保真度。