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AQ-Stacker: An Adaptive Quantum Matrix Multiplication Algorithm with Scaling via Parallel Hadamard Stacking

Wladimir Silva

Abstract

Matrix multiplication (MatMul) is the computational backbone of modern machine learning, yet its classical complexity remains a bottleneck for large-scale data processing. We propose a hybrid quantum-classical algorithm for matrix multiplication based on an adaptive configuration of Hadamard tests. By leveraging Quantum Random Access Memory (QRAM) for state preparation, we demonstrate that the complexity of computing the inner product of two vectors can be reduced to $O(\log N)$. We introduce an "Adaptive Stacking" framework that allows the algorithm to dynamically reconfigure its execution pattern from sequential horizontal stacking to massive vertical parallelism based on available qubit resources. This flexibility enables a tunable time-complexity range, theoretically reaching $O(N^2)$ on fault-tolerant systems while maintaining compatibility with near-term hardware. We validate the numerical stability of our approach through a Quantum Machine Learning (QML) simulation, achieving 96% accuracy on the MNIST handwritten digit dataset. Our results suggest that adaptive quantum MatMul provides a viable path toward super-classical efficiency in high-dimensional linear algebra operations.

AQ-Stacker: An Adaptive Quantum Matrix Multiplication Algorithm with Scaling via Parallel Hadamard Stacking

Abstract

Matrix multiplication (MatMul) is the computational backbone of modern machine learning, yet its classical complexity remains a bottleneck for large-scale data processing. We propose a hybrid quantum-classical algorithm for matrix multiplication based on an adaptive configuration of Hadamard tests. By leveraging Quantum Random Access Memory (QRAM) for state preparation, we demonstrate that the complexity of computing the inner product of two vectors can be reduced to . We introduce an "Adaptive Stacking" framework that allows the algorithm to dynamically reconfigure its execution pattern from sequential horizontal stacking to massive vertical parallelism based on available qubit resources. This flexibility enables a tunable time-complexity range, theoretically reaching on fault-tolerant systems while maintaining compatibility with near-term hardware. We validate the numerical stability of our approach through a Quantum Machine Learning (QML) simulation, achieving 96% accuracy on the MNIST handwritten digit dataset. Our results suggest that adaptive quantum MatMul provides a viable path toward super-classical efficiency in high-dimensional linear algebra operations.

Paper Structure

This paper contains 18 sections, 4 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Methodology
  3. Mathematical Mapping: Classical to Quantum
  4. The Hadamard Test Primitive
  5. Algorithmic Stages
  6. Stage 2: Quantum Execution
  7. Discussion
  8. Comparison with Classical State-of-the-Art
  9. Comparison with Existing Quantum Approaches
  10. Precision and Sampling Overhead
  11. The QRAM Integration
  12. Information-Theoretic Noise Analysis: The Entropy Dividend
  13. Sampling Stability and Statistical Significance
  14. Simulation Results
  15. Performance Benchmarking
  16. ...and 3 more sections

Figures (2)

  • Figure 1: AQ-Stacker Adaptive Execution Modes. The framework reconfigures its layout based on qubit availability: (Top) Sequential horizontal stacking for NISQ devices, (Middle) Full vertical parallelism for $O(N^2)$ time-complexity on large-scale hardware, and (Bottom) Adaptive hybrid batching to optimize performance for intermediate resource levels.
  • Figure 2: Persistence of the Entropy Dividend across Sampling Scales. In both the low-shot (a) and high-shot (b) regimes, the Expected Variance (shot noise) for Normal States (red) shows no statistically significant correlation with entropy ($p=0.218$). Conversely, Uniform States (green) exhibit a powerful and statistically significant negative correlation ($r = -0.9365, p < 0.0001$), demonstrating that high-entropy quantum states inherently suppress measurement noise in the AQ-Stacker framework.