Abstract In signal processing, resampling algorithms can modify the number of resources encoding a collection of data points. Downsampling reduces the cost of storage and communication, while upsampling interpolates new data from limited one, e.g., when resizing a digital image. We present a toolset of quantum algorithms to resample data encoded in the probabilities of a quantum register, using the quantum Fourier transform to adjust the number of high-frequency encoding qubits. We discuss advantage over classical resampling algorithms.
Abstract In signal processing, resampling algorithms can modify the number of resources encoding a collection of data points. Downsampling reduces the cost of storage and communication, while upsampling interpolates new data from limited one, e.g., when resizing a digital image. We present a toolset of quantum algorithms to resample data encoded in the probabilities of a quantum register, using the quantum Fourier transform to adjust the number of high-frequency encoding qubits. We discuss advantage over classical resampling algorithms.
We present efficient methods to interpolate data with a quantum computer to complement uploading techniques and quantum postprocessing. The quantum algorithms make use of the efficient quantum Fourier transform (QFT) as well as classical signal and imaging processing techniques, and may facilitate quantum advantages for some relevant families of data. We showcase a QFT interpolation method, a quantum cosine transform (QCT) interpolation geared towards natural data, and we improve upon them by utilizing a quantum circuit's capabilities of processing data in superposition. A circuit for the QCT is presented. We illustrate the methods on probability distributions and quantum encoded images, and discuss the precision of the resulting interpolations.
The joint photographic expert group algorithm compresses a digital image by filtering its high spatial-frequency components. Similarly, we introduce a quantum algorithm that uses the quantum Fourier transform to discard the high spatial-frequency qubits of an image, downsampling it to a lower resolution. This allows one to capture, compress, and send images even with limited quantum resources for storage and communication. We show under which conditions this protocol is advantageous with respect to its classical counterpart.
Algorithms for processing data in short-time batches are critical for both online and offline processing of streamed and large data respectively due to the quadratic relation between signal length and computational cost of convolution-based processing schemes. Whilst quantum analogs to some digital signal processing algorithms have been discovered, including the quantum Fourier transform (QFT), there has been no development of short-time processing techniques in the quantum domain. In this manuscript, we introduce the short-time QFT (STQFT) processing technique to bridge this gap in research. We develop a novel overlap-add reconstruction technique in the quantum domain using a permutation gate to combine subsequent windows. With this in mind, we discuss convolution under our novel STQFT processing scheme. We demonstrate filtering in the quantum Fourier domain with a filter stored in a quantum register as well as in a block encoded unitary gate. Throughout the paper, we elaborate upon implementation details such as applying DC offsets to input signals, skipping input data frames whenever necessary, the use of overlap-save as a reconstruction technique and mitigating time-varying scaling due to normalization of the windowed input data and filters.
Image processing is a fascinating field for exploring quantum algorithms. However, achieving quantum speedups turns out to be a significant challenge. In this work, we focus on image filtering to identify a class of images that can achieve a substantial speedup. We show that for images with efficient encoding and a lower bound on the signal-to-noise ratio, a quantum filtering algorithm can be constructed with a polynomial complexity in terms of the qubit number. Our algorithm combines the quantum Fourier transform with the amplitude amplification technique. To demonstrate the advantages of our approach, we apply it to three typical filtering problems. We highlight the importance of efficient encoding by illustrating that for images that cannot be efficiently encoded, the quantum advantage will diminish. Our work provides insights into the types of images that can achieve a substantial quantum speedup.
Amplitude encoding of real-world data on quantum computers is often the workflow bottleneck: direct amplitude encoding scales poorly with input size and can offset any speedups in subsequent processing. Fourier-based sparse amplitude encoding lowers cost by retaining only a small subset of dominant coefficients, but its fixed, non-adaptive basis leads to significant information loss. In this work, we replace the Fourier transform with the adaptive interpolating quantum transform (AIQT) in the sparse amplitude encoding workflow. The AIQT learns a data-adapted basis that concentrates information into a small number of coefficients. Consequently, at matched sparsity, the AIQT retains more information and achieves lower reconstruction error compared to the Fourier baseline. On financial time-series data, the AIQT reduces reconstruction error by 40% relative to the Fourier baseline, and on image datasets the reduction is up to 50% at the same sparsity level, with nearly identical encoding gate cost. Crucially, the approach preserves the efficiency of Fourier-based methods: the AIQT is built on the structure of the quantum Fourier transform circuit. Its gate count scales quadratically with the number of qubits, while classical evaluation can be carried out in quasilinear time. In addition, the AIQT is trained without labels and does not require sampling from quantum hardware or a simulator, removing a major bottleneck in data-driven amplitude-encoding methods.
Synthetic Aperture Radar (SAR) has played an important role in Earth observation, with applications in disaster monitoring, agriculture, forestry, ice surveillance, defense, and planetary exploration. The reconstruction of SAR images is a costly operation that uses Fourier Transform, while quantum computing has some key advantages that potentially enhance the algorithm to make it faster. Therefore, in the first part of the thesis, a major image processing method used in SAR systems is studied, known as the Range Doppler Algorithm (RDA). RDA is implemented on a simulated data set and a real data set from ICEYE Oy. In the second half of the thesis, quantum computing basic is studied and reviewed, including qubits, 1-qubit gate and 2-qubits gates, as well as the Quantum Fourier Transform (QFT) that are specific to replace the classical Fourier Transform. A framework for implementing the Range Doppler Algorithm (RDA) on a quantum computer, referred to as the Quantum Range Doppler Algorithm (QRDA), is proposed. In addition, the required resources under ideal quantum-computing assumptions are analyzed, including time complexity (algorithmic scaling) and space complexity (the number of qubits required). Lastly, the potential applicability of the proposed. QRDA in the context of current quantum device, Noisy Intermediate-Scale Quantum (NISQ) machine, is discussed.