Qrtlib Enables Fast Quantum Real Transforms for Signal Processing and Data Compression

Real-valued transforms underpin numerous classical computing applications, from image processing to data compression, yet quantum versions have lagged behind, lacking a unified and efficient implementation. Armin Ahmadkhaniha, Lu Chen, and Jake Doliskani from McMaster University, along with Zhifu Sun, now address this critical gap with QRTlib, a comprehensive library for fast quantum real transforms. This work introduces practical implementations of quantum Hartley, cosine, and sine transforms, alongside novel algorithms and circuit optimizations designed for near-term quantum devices. Notably, the team achieves a fourfold reduction in circuit size for the quantum Hartley transform and simplifies the quantum sine transform by eliminating complex multi-controlled operations, representing a significant advance in quantum signal processing capabilities.

Fast Quantum Transforms for Signal Processing

This research focuses on performing signal and image processing tasks using quantum computers, specifically exploring fast quantum transforms like the Discrete Cosine Transform, Hartley Transform, and Fourier Transform. Quantum algorithms offer potential speedups over classical methods for certain signal processing operations, and this work investigates efficient implementations using quantum circuits. The study emphasizes the importance of optimizing quantum circuits to minimize complexity and make them practical for implementation on real quantum hardware. The research reviews classical discrete transforms, explaining their mathematical foundations and applications.

It then details the Quantum Fourier Transform, a fundamental building block for many quantum algorithms, highlighting its differences from the classical DFT and its advantages. Scientists also explore how to implement the Quantum DCT and DHT using quantum circuits, discussing different designs and optimization techniques. The linear combination of unitaries technique is explained as a method for implementing complex quantum algorithms by combining simpler unitary operations. Various techniques for optimizing quantum circuits, such as gate cancellation and circuit simplification, are described.

Efficient Quantum Real Transforms via Linear Combinations

Scientists developed QRTlib, a library of quantum real transforms including Hartley, cosine, and sine transforms, to address a gap in quantum computing where efficient implementations of these transforms have lagged behind the quantum Fourier transform. The study pioneered a new quantum Hartley transform algorithm based on the linear combination of unitaries technique, leveraging the quantum Fourier transform alongside oblivious amplitude amplification. This construction achieves a fourfold reduction in circuit size compared to previously established algorithms, representing a significant improvement in efficiency. Researchers also devised an improved quantum sine transform of Type I, eliminating the need for large multi-controlled operations that hinder practical implementation on near-term devices.

To further enhance practicality, scientists implemented circuit-level optimizations, including techniques like two’s-complement and or-tree constructions, to streamline the quantum circuits and minimize resource requirements. The study delivers complete implementations of these quantum real transforms within the Qiskit quantum programming framework, providing a readily accessible toolkit for researchers and developers. Experiments employed rigorous testing and benchmarking to demonstrate the performance gains achieved through these new algorithms and optimizations. This work establishes a foundation for expanding the algorithmic toolbox of quantum computing and exploring applications where quantum real transforms offer structural or efficiency advantages over existing methods, such as in quantum signal and image processing.

Quantum Transforms and Circuit Size Reduction

Scientists have achieved a breakthrough in quantum computing by developing a comprehensive library of quantum real transforms, including the Hartley, cosine, and sine transforms. This work addresses a long-standing gap in the field, as quantum counterparts to these essential classical computing tools have lagged in development. The team successfully implemented these transforms as concrete quantum circuits, paving the way for practical applications in areas like image and signal processing. A key achievement is a new quantum Hartley transform algorithm based on the linear combination of unitaries technique, delivering a fourfold reduction in circuit size compared to the best previously known algorithm.

Furthermore, the researchers developed an improved quantum sine transform of Type I, eliminating the need for large, complex multi-controlled operations that hinder performance and increase error rates. To enhance the practicality of these algorithms for near-term quantum hardware, the team introduced several circuit optimizations, including an efficient implementation of two’s complement and the strategic use of or-tree structures. Measurements confirm that these optimizations significantly improve the feasibility of running these transforms on current and near-future quantum devices. The resulting library provides the first complete implementations of these quantum real transforms, realized using Qiskit and thoroughly tested in practice.

Efficient Quantum Transforms Implemented and Advance

Researchers have developed a new library, QRTlib, that provides efficient implementations of several real-valued quantum transforms, including the Hartley, cosine, and sine transforms. This work addresses a significant gap in quantum computing, as quantum versions of these essential transforms have lagged behind their classical counterparts, hindering progress in areas like signal processing and data compression. The team achieved a fourfold reduction in circuit size for the quantum Hartley transform by employing a linear combination of unitaries technique, and they also created an improved quantum sine transform that avoids complex multi-controlled operations. Furthermore, the library’s implementations of both Type-I quantum cosine and sine transforms offer further advancements, with one method enabling simultaneous computation of both transforms using a single circuit.

This is achieved through a carefully constructed unitary transform that leverages the relationship between cosine and sine functions, effectively splitting the transform output into orthogonal subspaces. The researchers demonstrate that these new algorithms and circuit optimizations are well-suited for implementation on near-term quantum devices. The authors acknowledge that the complexity of these transforms still presents challenges for scaling to larger problem sizes, and future work will likely focus on further optimizing circuit designs and exploring the potential for hybrid quantum-classical algorithms to address these limitations. The development of QRTlib represents a substantial step towards realizing the full potential of quantum computing for real-world applications requiring efficient real-valued transforms.