Randomized Quantum Singular Value Transformation

2025-10-08

AI Summary

Conventional quantum singular value transformation (QSVT) relies on Hamiltonian block-encoding, resulting in high auxiliary qubit overhead, intricate control logic, and circuit depth scaling linearly with the number of Hamiltonian terms. Method: We propose the first randomized QSVT algorithm that completely bypasses block-encoding, requiring only a single ancillary qubit. Our approach integrates importance sampling, qDRIFT, generalized quantum signal processing, and a novel classical extrapolation technique. Contribution/Results: We rigorously prove that the gate complexity is independent of the number of Hamiltonian terms and exhibits an optimal quadratic dependence on the polynomial degree. The algorithm achieves polynomial speedups for quantum linear systems solving and ground-state property estimation. Crucially, its circuit depth is reduced by several orders of magnitude compared to prior state-of-the-art methods, substantially enhancing resource efficiency on near-term fault-tolerant quantum hardware.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Sampling/Simulation-based Search
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines
• Security and Privacy: Data transparency and provenance
• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs

Abstract

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.

Problem

• Developing randomized QSVT algorithms without costly block encodings
• Achieving gate complexity independent of Hamiltonian terms count
• Providing efficient ground-state estimation for early quantum devices

Innovation

• Replaces block encodings with importance sampling
• Integrates qDRIFT into quantum signal processing
• Achieves gate complexity independent of Hamiltonian terms