While signal processing is a mature area, its connections with quantum computing have received less attention. In this work, we propose approaches that perform classical discrete-time signal processing using quantum systems. Our approaches encode the classical discrete-time input signal into quantum states, and design unitaries to realize classical concepts of finite impulse response (FIR) filters. We also develop strategies to cascade lower-order filters to realize higher-order filters through designing appropriate unitary operators. Finally, a few directions for processing quantum states on classical systems after converting them to classical signals are suggested for future work.
While signal processing is a mature area, its connections with quantum computing have received less attention. In this work, we propose approaches that perform classical discrete-time signal processing using quantum systems. Our approaches encode the classical discrete-time input signal into quantum states, and design unitaries to realize classical concepts of finite impulse response (FIR) filters. We also develop strategies to cascade lower-order filters to realize higher-order filters through designing appropriate unitary operators. Finally, a few directions for processing quantum states on classical systems after converting them to classical signals are suggested for future work.
Many signal processing tasks require intensive computations. Quantum computing promises to accelerate certain tasks, but algorithms must be designed around the limitations of quantum mechanics. This paper provides a quantum implementation of finite impulse response (FIR) filters, which are a widely used tool in classical signal processing. The filter can be parallelized and composed as part of larger quantum algorithms, with potential for speedups using quantum amplitude estimation and future fault-tolerant quantum hardware. To accomplish these properties, we introduce a state-space framework wherein sample-based signal processing is performed with unitary quantum circuits. We present the quantum delay gate as the quantum analog of the delay line in discrete-time signal processing. Unitary circuits intrinsically describe lossless systems, but we also implement lowpass and other bounded filters by emulating a projection operator. The FIR filter implementation is tested and demonstrated with a quantum circuit simulator.
In digital signal processing, the finite impulse response (FIR) filter is a fundamental tool for processing discrete-time signals. This paper explores the implementation of FIR filters using quantum computation methods. In this study, a quantum circuit for the FIR filter is designed using a normalized filter coefficient vector, orthogonal triangular decomposition commonly known as QR decomposition, and the transpilation tools provided by IBM’s software Qiskit SDK V2.3. Then, each block of the input signal is normalized to a unit-norm vector, loaded into a quantum register, and processed by the FIR filter quantum circuit to produce an output state. Quantum measurement is then performed on the output state to obtain a histogram, from which the first-bin data are scaled to compute the output sample of the filter. Finally, signal filtering experiments using FIR mean filters are conducted to demonstrate the effectiveness of the proposed quantum computation approach.
A quantum computer directly manipulates information stored in the state of quantum mechanical systems. The available operations have many attractive features but also underly severe restrictions, which complicate the design of quantum algorithms. We present a divide-and-conquer approach to the design of various quantum algorithms. The class of algorithm includes many transforms which are well-known in classical signal processing applications. We show how fast quantum algorithms can be derived for the discrete Fourier transform, the Walsh-Hadamard transform, the Slant transform, and the Hartley transform. All these algorithms use at most O(log^2 N) operations to transform a state vector of a quantum computer of length N.
Signal processing stands as a pillar of classical computation and modern information technology, applicable to both analog and digital signals. Recently, advancements in quantum information science have suggested that quantum signal processing (QSP) can enable more powerful signal processing capabilities. However, the developments in QSP have primarily leverageddigitalquantum resources, such as discrete-variable (DV) systems like qubits, rather thananalogquantum resources, such as continuous-variable (CV) systems like quantum oscillators. Consequently, there remains a gap in understanding how signal processing can be performed on hybrid CV-DV quantum computers. Here we address this gap by developing a new paradigm of mixed analog-digital QSP. We demonstrate the utility of this paradigm by showcasing how it naturally enables analog-digital conversion of quantum signals— specifically, the transfer of states between DV and CV quantum systems. We then show that such quantum analog-digital conversion enables new implementations of quantum algorithms on CV-DV hardware. This is exemplified by realizing the quantum Fourier transform of a state encoded on qubits via the free-evolution of a quantum oscillator, albeit with a runtime exponential in the number of qubits due to information theoretic arguments. Collectively, this work marks a significant step forward in hybrid CV-DV quantum computation, providing a foundation for scalable analog-digital signal processing on quantum processors.
This work focuses into the emerging field of quantum image representation, which is a key for processing grayscale and color images on quantum computing platforms. We explore foundational methods like the qubit lattice model, the flexible representation for quantum images, the novel enhanced quantum representation, and the generalized quantum image representation, and categorize them into state-based, amplitude-based, and quaternion-based representations - the latter offering holistic color processing. A novel discrete signal representation mapping data on the unit circle will be introduced to simplify the Fourier transform concepts in quantum systems. We will examine crucial quantum operations, including quantum gradients, multiplication, square roots, Fourier transform, and filtering. Challenges in implementing the circular convolution on quantum hardware will be addressed with practical examples such as quantum low-pass and high-pass filters and 3-qubit convolution circuits. Additionally, we will discuss ongoing research into quantum algorithms inspired by classical transform processing, including the Fourier transforms, that leverage quantum states’ unique properties to achieve efficient encoding, transformation, and decoding of images.
This paper presents a theoretical classification framework for phase-only digital signal processing filters interpreted as Quantum Fourier Transform diagonal quantum operators. It builds on an earlier DSP-to-QFT filtering framework and focuses on identifying which DSP-inspired filter responses can be directly implemented as valid quantum operations.The main result is that only phase-only sampled filter responses are directly compatible with unitary quantum gates. This means that ordinary magnitude-changing filters, such as classical low-pass or high-pass filters, cannot be implemented directly as quantum gates without additional methods such as ancilla systems, block encoding, post-selection, or unitary dilation.The paper connects the framework to circulant operator theory, showing that direct QFT-diagonal quantum filters correspond to circulant unitary operators. This gives the framework a clearer mathematical foundation and separates valid unitary quantum filters from non-unitary classical filtering operations.Several structured examples are analysed, including delay filters, linear phase filters, polynomial phase filters, and chirp-like filters. Delay filters are shown to produce cyclic shifts in the computational basis, making them useful as simple building blocks for modular shifts and quantum-walk-style operations. The paper also discusses how structured phase filters can have compact descriptions, while arbitrary phase filters may require exponentially many parameters.The central design principle is that direct QFT-diagonal quantum filters must be phase-only, while practical or efficient implementations require compact phase structure. The paper does not claim a new quantum speedup, but instead provides a DSP-inspired classification and design perspective for valid unitary QFT-diagonal quantum filters.