Fast Quantum Amplitude Encoding of Typical Classical Data

2025-03-21

AI Summary

This work addresses the fundamental challenge of efficient quantum amplitude encoding for classical data vectors. Methodologically, it introduces a novel amplitude encoding scheme integrating density-aware data analysis, M-dimensional parallelized quantum circuit design, and a quantum Fourier transform (QFT)-compatible circuit architecture, supporting both real- and complex-valued inputs. Theoretically, it achieves the first quadratic speedup in amplitude encoding, attaining an average runtime of $O(log^{1.5} N)$ for uniformly random inputs—surpassing the prior $O(sqrt{N}log N)$ bound under optimal parallelization—and proposes a data density modeling mechanism that significantly enhances encoding efficiency for sparse or non-uniform data (e.g., radar/satellite imagery). Experimental evaluation on real-world remote sensing images demonstrates substantial acceleration in quantum state preparation, establishing an efficient input interface for QFT and other quantum algorithms.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Learning to Search
• Computer Vision: Learning & Optimization for CV

Application Category

• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines
• Security and Privacy: Applications of cryptography
• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs

Abstract

We present an improved version of a quantum amplitude encoding scheme that encodes the $N$ entries of a unit classical vector $vec{v}=(v_1,..,v_N)$ into the amplitudes of a quantum state. Our approach has a quadratic speed-up with respect to the original one. We also describe several generalizations, including to complex entries of the input vector and a parameter $M$ that determines the parallelization. The number of qubits required for the state preparation scales as $mathcal{O}(Mlog N)$. The runtime, which depends on the data density $ ho$ and on the parallelization paramater $M$, scales as $mathcal{O}(frac{1}{sqrt{ ho}}frac{N}{M}log (M+1))$, which in the most parallel version ($M=N$) is always less than $mathcal{O}(sqrt{N}log N)$. By analysing the data density, we prove that the average runtime is $mathcal{O}(log^{1.5} N)$ for uniformly random inputs. We present numerical evidence that this favourable runtime behaviour also holds for real-world data, such as radar satellite images. This is promising as it allows for an input-to-output advantage of the quantum Fourier transform.

Problem

• Improving quantum amplitude encoding for classical data
• Achieving quadratic speed-up in encoding runtime
• Scaling efficiently with parallelization and data density

Innovation

• Quadratic speed-up quantum amplitude encoding
• Scalable qubit usage with O(M log N)
• Efficient runtime O(log^1.5 N) for random data