Quantum singular value transformation without block encodings: Near-optimal complexity with minimal ancilla

2025-04-03

AI Summary

Existing quantum singular value transformation (QSVT) frameworks rely on block-encoding, incurring substantial ancilla-qubit overhead and high circuit depth. Method: We propose the first block-encoding-free QSVT paradigm, requiring only a single ancilla qubit. Our approach constructs Hamiltonian evolution sequences via Trotterization and qDRIFT, systematically cancels errors from interleaved unitaries and Hamiltonian dynamics using Richardson extrapolation, and introduces two randomized QSVT algorithms. Contribution/Results: We rigorously prove that these algorithms achieve a tight quadratic query complexity lower bound. Circuit depth is reduced to $widetilde{O}(L(dlambda_{mathrm{comm}})^{1+o(1)})$, with no multi-controlled gates. The framework enables oracle-free quantum linear systems solving and ground-state property estimation, significantly lowering hardware resource requirements while preserving near-optimal asymptotic complexity.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Non-convex Optimization
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Security and Privacy: Data transparency and provenance
• Systems and Infrastructure for Web, Mobile and WoT: Virtualization and resource management in Web systems and infrastructures

Abstract

We develop new algorithms for Quantum Singular Value Transformation (QSVT), a unifying framework underlying a wide range of quantum algorithms. Existing implementations of QSVT rely on block encoding, incurring $O(log L)$ ancilla overhead and circuit depth $widetilde{O}(dlambda L)$ for polynomial transformations of a Hamiltonian $H=sum_{k=1}^L lambda_k H_k$, where $d$ is polynomial degree, and $lambda=sum_k |lambda_k|$. We introduce a new approach that eliminates block encoding, needs only a single ancilla qubit, and maintains near-optimal complexity, using only basic Hamiltonian simulation methods such as Trotterization. Our method achieves a circuit depth of $widetilde{O}(L(dlambda_{mathrm{comm}})^{1+o(1)})$, without any multi-qubit controlled gates. Here, $lambda_{mathrm{comm}}$ depends on the nested commutators of the $H_k$'s and can be much smaller than $lambda$. Central to our technique is a novel use of Richardson extrapolation, enabling systematic error cancellation in interleaved sequences of arbitrary unitaries and Hamiltonian evolution operators, establishing a broadly applicable framework beyond QSVT. Additionally, we propose two randomized QSVT algorithms for cases with only sampling access to Hamiltonian terms. The first uses qDRIFT, while the second replaces block encodings in QSVT with randomly sampled unitaries. Both achieve quadratic complexity in $d$, which we establish as a lower bound for any randomized method implementing polynomial transformations in this model. Finally, as applications, we develop end-to-end quantum algorithms for quantum linear systems and ground state property estimation, achieving near-optimal complexity without oracular access. Our results provide a new framework for quantum algorithms, reducing hardware overhead while maintaining near-optimal performance, with implications for both near-term and fault-tolerant quantum computing.

Problem

• Eliminates block encoding in QSVT with minimal ancilla
• Achieves near-optimal complexity using basic Hamiltonian simulation
• Develops randomized QSVT algorithms for sampling-based Hamiltonian access

Innovation

• Eliminates block encoding with single ancilla
• Uses Richardson extrapolation for error cancellation
• Randomized QSVT algorithms with quadratic complexity