We introduce the first randomized algorithms for Quantum Singular Value Transformation (QSVT), a unifying framework for many quantum algorithms. Standard implementations of QSVT rely on block encodings of the Hamiltonian, which are costly to construct, requiring a logarithmic number of ancilla qubits, intricate multi-qubit control, and circuit depth scaling linearly with the number of Hamiltonian terms. In contrast, our algorithms use only a single ancilla qubit and entirely avoid block encodings. We develop two methods: (i) a direct randomization of QSVT, where block encodings are replaced by importance sampling, and (ii) an approach that integrates qDRIFT into the generalized quantum signal processing framework, with the dependence on precision exponentially improved through classical extrapolation. Both algorithms achieve gate complexity independent of the number of Hamiltonian terms, a hallmark of randomized methods, while incurring only quadratic dependence on the degree of the target polynomial. We identify natural parameter regimes where our methods outperform even standard QSVT, making them promising for early fault-tolerant quantum devices. We also establish a fundamental lower bound showing that the quadratic dependence on the polynomial degree is optimal within this framework. We apply our framework to two fundamental tasks: solving quantum linear systems and estimating ground-state properties of Hamiltonians, obtaining polynomial advantages over prior randomized algorithms. Finally, we benchmark our ground-state property estimation algorithm on electronic structure Hamiltonians and the transverse-field Ising model with long-range interactions. In both cases, our approach outperforms prior work by several orders of magnitude in circuit depth, establishing randomized QSVT as a practical and resource-efficient alternative for early fault-tolerant quantum devices.

翻译：我们首次提出了量子奇异值变换（QSVT）的随机化算法，该框架统一了多种量子算法。传统的QSVT实现依赖于哈密顿量的块编码，其构建成本高昂，需要对数数量的辅助量子比特、复杂的多量子比特控制，以及随哈密顿量项数线性增长的电路深度。相比之下，我们的算法仅使用单个辅助量子比特，并完全避免了块编码。我们开发了两种方法：（i）对QSVT的直接随机化，其中块编码被重要性采样所取代；（ii）将qDRIFT集成到广义量子信号处理框架中的方法，通过经典外推将精度依赖关系指数级提升。两种算法均实现了与哈密顿量项数无关的门复杂度——这是随机化方法的标志性特征，同时仅对目标多项式的次数产生二次依赖。我们确定了若干自然参数区域，在这些区域中我们的方法甚至优于标准QSVT，使其在早期容错量子设备中具有应用前景。我们还建立了一个基本下界，证明在该框架内对多项式次数的二次依赖是最优的。我们将该框架应用于两个基本任务：求解量子线性系统和估计哈密顿量的基态性质，相比先前的随机化算法获得了多项式优势。最后，我们在电子结构哈密顿量和具有长程相互作用的横向场伊辛模型上对基态性质估计算法进行了基准测试。在这两种情况下，我们的方法在电路深度上均优于先前工作数个数量级，从而确立了随机化QSVT作为早期容错量子设备实用且资源高效替代方案的潜力。