Quantum Algorithms Without Coherent Quantum Access

TL;DR

The paper presents a framework for quantum gradient descent that operates without coherent access to classical data by embedding classical variables into diagonal operators and manipulating them via block encoding and the quantum singular value transformation. This approach enables end-to-end quantum algorithms for solving linear systems, least-squares fitting, SVMs, supervised clustering, neural network training, ground/excited-state energy calculations, and PCA, with resource costs that scale logarithmically in the data dimension and do not rely on QRAM. By treating gradient steps as structured polynomials or products, the authors develop explicit algorithms for three function classes, with convergence guided by standard optimization results and explicit amplification techniques to remove constant factors. The framework promises practical quantum advantages for extremely high-dimensional problems and reduces the input/output bottlenecks that have hindered earlier quantum speedups. The work therefore provides a compelling route to realistic, scalable quantum computation on real-world data, stimulating further experimental and theoretical exploration of quantum-accelerated optimization across domains.

Abstract

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.

Figures (7)

• Figure 1: Illustration of gradient descent method applied to a multi-variate function. The function $f(x,y)$ drawn on the left exhibits many extrema, as indicated by the contour on the right figure. The initialization is critical in the performance of gradient descent, as different initialization may result in different point of extrema. The extreme case is when the function is convex, then the local minima is also global minima, and gradient descent is guaranteed to converge to global minima.
• Figure 2: Examples illustrating least-squares data fitting techniques applied to a linear function and a nonlinear function.
• Figure 3: Illustration of support vector machine applied to two different cases: linearly seperable and non-linearly seperable.
• Figure 4: Illustration of supervised cluster assignment, which is a typical supervised learning method. Data from different classes are colored red (R), blue (B) and green (G). $M$ representative samples $\Vec{r}_i \in R, \Vec{b}_i \in B$ , $\Vec{g}_i \in G$ (for $i=1,2,...,M$) from each class are provided. The centroid of each class is defined as $\mathscr{C}_1 = \frac{1}{|R|} \sum_i \Vec{r}_i$, $\mathscr{C}_2 = \frac{1}{|B|} \sum_i \Vec{b}_i$, and $\mathscr{C}_3 =\frac{1}{|G|} \sum_k \Vec{g}_k$.
• Figure 5: Description of a specific neural network. A layer consists of nodes (colored circles in the above figure). Input layer has $n$ nodes, hidden layer has $3m$ nodes and output layer has $k$ nodes. The input layer contains our dataset vectors, where each $x_i$ in the above figure is the corresponding coordinate of some feature vector $\textbf{x}$. In the context of Algorithm \ref{['algo: trainingneuralnetwork']}, each $\textbf{x}^i$ would be input layer. The output layer holds the output of such a neural network, which is the result to our problem of interest. Generally, the number of hidden layers and the value of $m$ are user-dependent. The figure above has 3 hidden layers. The number of nodes in output layer depends on purpose, for example, in binary classification problem, 1 node is sufficient to hold the label $\{0,1\}$. All nodes between different layers are connected (as indicated by the line), and there is associated weight between two connected nodes. The goal of training neural network is to obtain those weights that minimizes the loss function, via the backpropagation algorithm, for which we will describe in Algorithm \ref{['algo: trainingneuralnetwork']}.

Theorems & Definitions (25)

• Lemma 1: Efficient State Preparation
• Lemma 2: Theorem 2 in rattew2023non, guo2024nonlinear; Block Encoding into a Diagonal Matrix
• Theorem 1: Quantum Gradient Descent Algorithm (First Type)
• Lemma 3
• Theorem 2: Quantum Gradient Descent Algorithm (Second Type)
• Theorem 3: Quantum Gradient Descent Algorithm (Third Type)
• Theorem 4: Quantum Algorithm For Solving Linear Systems of Equations
• Theorem 5: Quantum Algorithm For Data Fitting
• Theorem 6: Quantum Algorithm for Support Vector Machine
• Lemma 4