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arXiv:2608.22470

Sample-Query Interconversion of Block Encoding of Unknown Quantum States

Manaki Arihara, Mio Murao

quant-phSubmitted Aug 23, 2026• Announced Aug 25, 2026• 0 citations
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Abstract

Block encoding embeds a matrix as a sub-block of a unitary matrix and serves as a fundamental input model for quantum algorithms based on quantum singular value transformation, enabling polynomial transformations of matrices encoded in unitary operators. Block encoding of unknown quantum states can be useful for quantum learning; however, the fundamental limits on converting between unknown quantum states and their block-encoding unitary channels remain poorly understood. In this paper, we investigate this convertibility in both directions. First, we prove that implementing an $\varepsilon$-approximate block-encoding unitary channel of an unknown quantum state requires $Ω(1/\varepsilon)$ copies of the state, matching known upper bounds up to logarithmic factors. Second, we show that recovering a rank-$r$, $d$-dimensional quantum state $ρ$ given query access to its block-encoding unitary channel generally requires $Ω((1/λ_{\max}(ρ))\sqrt{d/r})$ queries, where $λ_{\max}(ρ)$ is the maximum eigenvalue of $ρ$, revealing an unavoidable dependence on the dimension of the state. Our results identify inherent limitations of block encoding as a representation of unknown quantum states and reveal a separation between learning properties of a quantum state and generating the state itself. Using our techniques, we further establish lower bounds for specific state-generation tasks, including ground-state preparation and Gibbs-state preparation.

Comments
19 pages, 3 figures

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Quantum algorithms for basic numerical linear algebraic tasks have proven essential for translating diverse problems to a unified quantum computational context. Many of these tasks---e.g., applying a function to the spectrum of a matrix embedded in a unitary process (a so-called block encoding), or taking linear combinations of block encodings---are well-addressed by techniques like quantum singular value transformation or linear combination of unitaries. However, there exist useful matrix transforms whose realization is unclear or inefficient. In this work we construct improved quantum algorithms for some of these transforms, the simplest of which is a polynomial function applied element-wise. We show the space required to compute quantum element-wise transforms (QEWTs) can be reduced exponentially in the degree of the applied function compared to prior work, and rectify errors in previous constructions. We prove novel space and query lower bounds for QEWTs, construct useful function families nearly saturating these bounds, and establish the hardness of QEWT dequantization. This represents the first sound theoretical treatment of QEWTs, and brings their resource requirements on par with other practical block encoding techniques. We present our algorithms alongside concrete applications to machine learning inference and signal processing.
We introduce a quantum algorithmic framework based on probabilistic mixtures of unitary channels that, similar to the framework of quantum singular value transformations, enables the application of arbitrary polynomials of hermitian operators onto arbitrary initial states. We show that our framework supports a flexible tradeoff between sample- and query complexity ranging from optimal query complexity, meaning logarithmic in the error, and exponentially scaling sample complexity to sub-polynomial query complexity in the error and polynomial sample complexity. Combined with the considerably lower quantum circuit complexity, compared to quantum singular value transformations with a linear combination of unitaries block encoding, we argue that our framework can be seamlessly scaled from NISQ to fault-tolerant quantum computing.
In quantum computing, every operation must be unitary to ensure reversibility and preserve probability amplitudes, which is a fundamental requirement for any quantum process. However, many real-world computations in areas such as machine learning, signal processing, and dynamical modeling inherently involve non-unitary matrices. Since these cannot be directly implemented on quantum hardware, a practical workaround is required. One promising solution is block-encoding, a framework that embeds a non-unitary matrix A in C^{n x n} into a larger unitary matrix U_A in C^{2n x 2n}, enabling the simulation of non-unitary actions within quantum circuits through ancilla preparation and post-selection. This thesis presents a novel and efficient method for block-encoding based on Gram-Schmidt orthogonalization. The algorithm constructs a unitary U_A with A embedded as the top-left block and remaining components filled to ensure global unitarity. Compared to traditional Singular Value Decomposition (SVD)-based approaches, the proposed method achieves at least 20% improvement in execution time, without compromising output fidelity or numerical stability. A corresponding quantum circuit is developed to extract A|psi> (that is, the transformed version of the input quantum state |psi> after applying the linear operator A) using post-selection on ancilla qubits, with the success probability determined by
Modern quantum algorithms are increasingly formulated as coherent procedures for implementing polynomial transformations of operators and singular values. This perspective provides a powerful and unifying language for quantum algorithm design, connecting a wide range of distinct problems through five closely related key tools: block-encoding, qubitization, QSP, QSVT and GQSP. Block-encoding embeds non-unitary matrices into larger unitaries; qubitization converts block-encodings into structured operators; QSP, QSVT and GQSP enable polynomial transformations with near-optimal query complexity. Together, these techniques form a general toolkit for transforming matrix functions into implementable quantum circuits. This paper develops these techniques from first principles as a unified framework for constructing quantum algorithms. We apply this framework to representative applications to highlight design principles and demonstrate how distinct algorithms can be constructed from a unified sequence of operator transformations. A central contribution is a systematic decision workflow for selecting the appropriate approach according to the operator structure and the desired transformation polynomial. This perspective clarifies when direct GQSP or through qubitization, or Laurent expansion, or QSVT is most appropriate. We organize algorithmic design into an end-to-end pipeline: identifying the target matrix function, constructing an appropriate block-encoding, determining the relevant spectral domain, designing a polynomial or Laurent-polynomial approximation, synthesizing the phase factors, and translating the transformation into an executable quantum circuit. By applying this unified framework to example applications, we showcase a practical methodology for reasoning, designing, and implementing quantum algorithms based on polynomial transformations.