AI 中文总结
研究通过厄米矩阵线性组合公式实现非正规矩阵特征值变换\(g(A)\)，给出矩阵幂等离散公式，无截断和角积分误差。基于此的QET算法能实现最优电路深度和后选择重复次数，统一多种量子线性代数问题，门复杂度接近最优。
AI 中文摘要
我们发现了两种互补的厄米矩阵线性组合（LCHM）公式，以实现一般的非正规矩阵特征值变换\(g(A)\)。首先，对于\(A = L + \mathrm{i}H\)（\(L\)和\(H\)为厄米矩阵），普通LCHM公式将\(g(A)\)表示为\(g(\mathrm{i}(H + kL))\)的核积分，其中包含哈密顿量模拟线性组合（LCHS）作为矩阵指数的特殊情况。其次，对于角厄米矩阵\(X_\theta = \cos\theta L + \sin\theta H\)，外尔LCHM公式通过对\(g(\mathrm{e}^{\mathrm{i}\theta}(X_\theta \pm \mathrm{i}(I - X_\theta^2)^{1/2}))\)积分来表示\(g(A)\)。对于矩阵幂\(g(A)=A^m\)，外尔LCHM的傅里叶投影给出了精确的离散公式，无截断和角积分误差，并提供\(\mathcal{O}(1)\)的后选择权重。LCHM公式导致了新的量子特征值变换（QET）算法。对于\(|\psi\rangle\)上的\(d\)次多项式\(p_d(A)\)，我们的QET算法可以实现最优的\(\Theta(d)\)电路深度和最优的\(\mathcal{O}(||p_d||_{\infty}/||p_d(A)|\psi\rangle||)\)后选择重复次数。基于LCHM的QET统一了各种量子线性代数问题，具有接近最优的\(\mathcal{\widetilde O}(d\log(d/\epsilon))\) Clifford\(+T\)门。
英文摘要
We discover two complementary linear-combination-of-Hermitian-matrices (LCHM) formulations to achieve a general non-normal matrix eigenvalue transformation $g(A)$. Firstly, for $A=L+\mathrm{i} H$ with Hermitian $L$ and $H$, the vanilla LCHM formula represents $g(A)$ as a kernel integral of $g(\mathrm{i}(H+kL))$, and it contains linear-combination-of-Hamiltonian-simulation (LCHS) [An, Liu, Lin, Phys. Rev. Lett. 2023] as the special case for matrix exponentials. Secondly, for the angular Hermitian $X_θ= \cosθL+\sinθH$, the Weyl LCHM formula expresses $g(A)$ via integrating $g(\mathrm{e}^{\mathrm{i}θ} (X_θ\pm\mathrm{i}(I-X_θ^2)^{1/2}))$. For the matrix power $g(A)=A^m$, the Fourier projection of Weyl LCHM gives \[ A^m=\frac{2}π\int_0^π\text{e}^{\text{i} mθ}T_m(X_θ) \text{d}θ= \frac{2}{N}\sum_{j=0}^{N-1} \text{e}^{\text{i} mθ_j}T_m(X_{θ_j}),\quadθ_j=\frac{πj}{N},\quad \text{for every } N>m \] with Chebyshev polynomial of Hermitian $T_m(X_θ)$ and $N$ samples. The discrete formula is exact, introduces no truncation and angular quadrature error, and offers $\mathcal{O}(1)$ post-selection weights. LCHM formulas lead to new quantum eigenvalue transformation (QET) algorithms. For a degree-$d$ polynomial $p_d(A)$ on $|ψ\rangle$, our QET algorithm can achieve optimal $Θ(d)$ circuit depth and optimal $\mathcal{O}(||p_d||_{\infty}/||p_d(A)|ψ\rangle||)$ post-selection repetitions. LCHM-based QETs unify various quantum linear algebraic problems with near-optimal $\mathcal{\widetilde O}(d\log(d/ε))$ Clifford$+T$ gates, including driven ODEs (reduced to standard LCHS), iterative methods, resolvents, $\log(I+A)$, $(λI+A)^ν$, Sign and ReLU transforms, and Faber approximation on noncircular domains.
Comments60 pages, 3 tables
