发表机构
Rice University（莱斯大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出阴影量子奇异值变换（Shadow QSVT），通过三种算法利用输入态和可观测量结构，降低QSVT电路深度，实现可观测量高效估计。
AI 中文摘要
我们引入了阴影量子奇异值变换（Shadow QSVT）：给定初始态$|{\psi}\rangle$、厄米矩阵$H$、多项式$f$以及一组可观测量$\{O_1,\dots,O_m\}$，目标是估计所有$j\in\{1,\dots,m\}$的$\langle{\psi}|f(H)^{\dagger}O_j f(H)|{\psi}\rangle$。Shadow QSVT提供了一条系统性的途径来减少标准QSVT所需的量子资源，标准QSVT需要构造$f(H)$的酉块编码。它利用了输入态和可观测量中的结构，以及许多应用仅需要可观测量估计而非合成完整酉算子这一事实。我们提出了三种利用初始态和可观测量中的结构来降低量子电路深度的算法。首先，我们开发了一种态感知的QSVT算法，当与$H$和$|\psi\rangle$相关的Krylov子空间是低维的或允许精确的低维近似时，该算法能以低电路深度制备目标态。其次，我们引入了一种可观测量感知的Shadow QSVT算法，该算法将新的可观测量感知Krylov子空间与历史态相结合，以进一步降低电路深度和门复杂度。最后，我们开发了经典Shadow QSVT，它从$H$、$f$和$|\psi\rangle$构造经典表示，无需预先知道可观测量或显式制备与$f(H)|\psi\rangle$成比例的目标态。该表示使得能够在量子计算之后对指定的可观测量估计目标量。这三种算法共同为在各种设置中降低基于QSVT的计算的电路深度提供了工具。
英文摘要
We introduce shadow quantum singular value transformation (Shadow QSVT): given an initial state $|ψ\rangle$, a Hermitian matrix $H$, a polynomial $f$, and a set of observables $\{O_1,\dots,O_m\}$, the goal is to estimate $\langleψ|f(H)^{\dagger}O_j f(H)|ψ\rangle$ for all $j\in\{1,\dots,m\}$. Shadow QSVT provides a systematic route to reduce the quantum resources required by standard QSVT, which constructs a unitary block-encoding of $f(H)$. It uses structure in the input state and observables, together with the fact that many applications require only observable estimates rather than synthesizing the full unitary. We present three algorithms that exploit structure in the initial state and observables to reduce quantum circuit depth. First, we develop a state-aware QSVT algorithm that prepares the target state with low circuit depth when the Krylov subspace associated with $H$ and $|ψ\rangle$ is low-dimensional or admits an accurate low-dimensional approximation. Second, we introduce an observable-aware Shadow QSVT algorithm that combines a new observable-aware Krylov subspace with history states to further reduce circuit depth and gate complexity. Finally, we develop Classical Shadow QSVT, which constructs a classical representation from $H$, $f$, and $|ψ\rangle$ without prior knowledge of the observables or explicit preparation of the target state proportional to $f(H)|ψ\rangle$. This representation enables estimation of the target quantities for observables specified after the quantum computation. Together, these three algorithms provide tools for reducing the circuit depth of QSVT-based computations across a range of settings.
