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arXiv:2602.20781

Toward speedup without quantum coherent access

Nhat A. Nghiem

quant-phSubmitted Feb 24, 2026• Announced Feb 25, 2026• 0 citations

Abstract

Along with the development of quantum technology, finding useful applications of quantum computers has been a central pursuit. Despite various quantum algorithms have been developed, many of them often require strong input assumptions, which is hardware demanding. In particular, recent advances on dequantization have revealed that the quantum advantage is more of a mere artifact of strong input assumptions. In this work, we propose a variant of these algorithms, leveraging both classical and quantum resources. Provided the classical knowledge (the entries) of the matrix/vector of interest, a classical procedure is used to pre-process this information. Then they are fed into a quantum circuit which is shown to be a block encoding of the matrix of interest. From this block-encoding, we show how to use it to tackle a wide range of problems, including principal component analysis, linear equation solving, Hamiltonian simulation, preparing ground state, and data fitting. We also analyze our protocol, showing that both the classical and quantum procedure can achieve logarithmic complexity in the input dimension, thus implying its potential for near term realization. We then discuss several implications and corollaries of our result. First,, our results suggest there are certain matrices/Hamiltonians where our method can provide exponential improvement compared to the existing ones with respect to the sparsity. Regarding dense linear systems, our method achieves exponential speed-up with respect to the inverse of error tolerance, compared to the best previously known quantum algorithm for dense systems. Last, and most importantly, regarding quantum data fitting, we show how the output of our quantum algorithms can be leveraged to predict unseen data. Thus, it provides an end-to-end application, which has been an open aspect of the previous quantum data fitting algorithm.

Comments: arXiv admin note: substantial text overlap with arXiv:2504.00833

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Along with the development of quantum technology, finding useful applications of quantum computers has been a central pursuit. Despite various quantum algorithms have been developed, many of them often require strong input assumptions, which is hardware demanding. In particular, recent advances on dequantization have revealed that the quantum advantage is more of a mere artifact of strong input assumptions. In this work, we propose a variant of these algorithms, leveraging both classical and quantum resources. Provided the classical knowledge (the entries) of the matrix/vector of interest, a classical procedure is used to pre-process this information. Then they are fed into a quantum circuit which is shown to be a block encoding of the matrix of interest. From this block-encoding, we show how to use it to tackle a wide range of problems, including principal component analysis, linear equation solving, Hamiltonian simulation, preparing ground state, and data fitting. We also analyze our protocol, showing that both the classical and quantum procedure can achieve logarithmic complexity in the input dimension, thus implying its potential for near term realization. We then discuss several implications and corollaries of our result. First,, our results suggest there are certain matrices/Hamiltonians where our method can provide exponential improvement compared to the existing ones with respect to the sparsity. Regarding dense linear systems, our method achieves exponential speed-up with respect to the inverse of error tolerance, compared to the best previously known quantum algorithm for dense systems. Last, and most importantly, regarding quantum data fitting, we show how the output of our quantum algorithms can be leveraged to predict unseen data. Thus, it provides an end-to-end application, which has been an open aspect of the previous quantum data fitting algorithm.

We outline refined versions of two major quantum algorithms for performing principal component analysis and solving linear equations. Our methods are exponentially faster than their classical counterparts and even previous quantum algorithms/dequantization algorithms. Oracle/black-box access to classical data is not required, thus implying great capacity for near-term realization. Several applications and implications of these results are discussed. First, we show that a Hamiltonian

H

with classically known rows/columns can be efficiently simulated, adding another model in addition to the well-known sparse access and linear combination of unitaries models. Second, we provide a simpler proof of the known result that quantum matrix inversion cannot achieve sublinear complexity

κ^{1-γ}

where

κ

is the conditional number of the inverted matrix.

Demonstrating quantum advantage has been a pressing challenge in the field. Most claimed quantum speedups rely on a subroutine in which classical information can be accessed in a coherent quantum manner, which imposes a crucial constraint on the implementability of these quantum algorithms. It has even been shown that without such an access, the quantum computer cannot be stronger than the classical counterparts. Thus, whether a quantum computer can be useful for practical applications is still open. In this work, we develop several variants of quantum algorithms. Our key framework employs a classical preprocessing step and a quantum procedure to perform gradient descent. We then translate such algorithm into an algorithm for solving linear systems, performing least-square fitting, building a support vector machine, performing supervised cluster assignment, training neural network, and solving for ground-state/excited-state energy, performing principle component analysis, with end-to-end applications of quantum algorithms. The classical preprocessing and the quantum procedure of our framework are shown to have logarithmic complexity in the dimension of input data, and quantum coherent access to input data is not required. Thus, our framework suggests an alternatively efficient route for quantum computers to handle real-world problems.

Abstract Kernel principal component analysis (kernel PCA) is a nonlinear dimensionality reduction technique that employs kernel functions to map data into a high-dimensional feature space, thereby extending the applicability of linear PCA to nonlinear data and facilitating the extraction of informative principal components. However, kernel PCA necessitates the manipulation of large-scale matrices, leading to high computational complexity and posing challenges for efficient implementation in big data environments. Quantum computing has recently been integrated with kernel methods in machine learning, enabling effective analysis of input data within intractable feature spaces. Although existing quantum kernel PCA proposals promise exponential speedups, they impose stringent requirements on quantum hardware that are challenging to fulfill. In this work, we propose a quantum algorithm for kernel PCA by establishing a connection between quantum kernel methods and block encoding, thereby diagonalizing the centralized kernel matrix on a quantum computer. The query complexity is logarithmic with respect to the size of the data vector, D , and linear with respect to the size of the dataset. An exponential speedup could be achieved when the dataset consists of a few high-dimensional vectors, wherein the dataset size is polynomial in log ⁡ ( D ) , with D being significantly large. In contrast to existing work, our algorithm enhances the efficiency of quantum kernel PCA and reduces the requirements for quantum hardware. Furthermore, we have also demonstrated that the algorithm based on block encoding matches the lower bound of query complexity, indicating that our algorithm is nearly optimal. Our research has laid down new pathways for developing quantum machine learning algorithms aimed at addressing tangible real-world problems and demonstrating quantum advantages within machine learning.

While quantum algorithms for solving large scale systems of linear equations offer potentially exponential speedups, their application has largely been confined to sparse matrices. This work extends the scope of these algorithms to a broad class of structured dense matrices arise in potential theory, covariance modeling, and computational physics, namely, hierarchically block separable (HBS) matrices. We develop two distinct methods to make these systems amenable to quantum solvers. The first is a pre-processing approach that transforms the dense matrix into a larger but sparse format. The second is a direct block encoding scheme that recursively constructs the necessary oracles from the HBS structure. We provide a detailed complexity analysis and rigorous error bounds for both methods. Numerical experiments are presented to validate the effectiveness of our approaches.

Abstract. Many standard linear algebra problems can be solved on a quantum computer by using recently developed quantum linear algebra algorithms that make use of block encodings and quantum eigenvalue/singular value transformations. A block encoding embeds a properly scaled matrix of interest [Formula: see text] in a larger unitary transformation [Formula: see text] that can be decomposed into a product of simpler unitaries and implemented efficiently on a quantum computer. Although quantum algorithms can potentially achieve exponential speedup in solving linear algebra problems compared to the best classical algorithm, such a gain in efficiency ultimately hinges on our ability to construct an efficient quantum circuit for the block encoding of [Formula: see text], which is difficult in general, and not trivial even for well structured sparse matrices. In this paper, we give a few examples on how efficient quantum circuits can be explicitly constructed for some well structured sparse matrices and discuss a few strategies used in these constructions. We also provide implementations of these quantum circuits in MATLAB.

A central roadblock to analyzing quantum algorithms on quantum states is the lack of a comparable input model for classical algorithms. Inspired by recent work of the author [E. Tang, STOC 2019.], we introduce such a model, where we assume we can efficiently perform ℓ^{2}-norm samples of input data, a natural analog to quantum algorithms that assume efficient state preparation of classical data. Though this model produces less practical algorithms than the (stronger) standard model of classical computation, it captures versions of many of the features and nuances of quantum linear algebra algorithms. With this model, we describe classical analogs to Lloyd, Mohseni, and Rebentrost's quantum algorithms for principal component analysis [S. Lloyd, M. Mohseni, and P. Rebentrost, Nat. Phys. 10, 631 (2014).NPAHAX1745-247310.1038/nphys3029] and nearest-centroid clustering [S. Lloyd, M. Mohseni, and P. Rebentrost, Quantum algorithms for supervised and unsupervised machine learning]. Since they are only polynomially slower, these algorithms suggest that the exponential speedups of their quantum counterparts are simply an artifact of state preparation assumptions.