Efficient loading of classical data into quantum states remains a central bottleneck of quantum computing. We propose Quantum-Gray Phase Injection for Pixel Encoding (Q-PIPE), which writes continuous data into the eigenphases of a diagonal oracle by phase kickback and converts them coherently into computational-basis states by quantum phase estimation. Sums and differences between data sets are obtained by sequential oracle application, without quantum arithmetic circuits. We apply Q-PIPE to quantum image processing, using edge detection by directional finite differences as a proof of concept. Synthesized with uniformly controlled rotations, the oracle requires O(qN) controlled-NOT gates without ancillas for N pixels and q estimation qubits. This is the same scaling as optimized preparation of the novel enhanced quantum representation, whose output form Q-PIPE reproduces. A half-spectrum normalization removes phase aliasing, and a probability threshold derived from the phase-estimation kernel and scaled with image size controls spectral leakage. In ideal simulation, edge detection is exact for inputs commensurate with the estimation register and has low mean absolute error for continuous data. Under device noise models, multi-controlled-phase synthesis drives the output close to the maximally mixed state. The uniformly controlled synthesis instead reduces the compiled depth by a factor of 53--59 and preserves the edge structure (Pearson correlation r = 0.99 with the classical gradient). The limiting resource on current hardware is therefore the compiled oracle depth, not the encoding principle.
Efficient loading of classical data into quantum states remains a central bottleneck of quantum computing. We propose Quantum-Gray Phase Injection for Pixel Encoding (Q-PIPE), which writes continuous data into the eigenphases of a diagonal oracle by phase kickback and converts them coherently into computational-basis states by quantum phase estimation. Sums and differences between data sets are obtained by sequential oracle application, without quantum arithmetic circuits. We apply Q-PIPE to quantum image processing, using edge detection by directional finite differences as a proof of concept. Synthesized with uniformly controlled rotations, the oracle requires O(qN) controlled-NOT gates without ancillas for N pixels and q estimation qubits. This is the same scaling as optimized preparation of the novel enhanced quantum representation, whose output form Q-PIPE reproduces. A half-spectrum normalization removes phase aliasing, and a probability threshold derived from the phase-estimation kernel and scaled with image size controls spectral leakage. In ideal simulation, edge detection is exact for inputs commensurate with the estimation register and has low mean absolute error for continuous data. Under device noise models, multi-controlled-phase synthesis drives the output close to the maximally mixed state. The uniformly controlled synthesis instead reduces the compiled depth by a factor of 53--59 and preserves the edge structure (Pearson correlation r = 0.99 with the classical gradient). The limiting resource on current hardware is therefore the compiled oracle depth, not the encoding principle.
This work introduces a novel quantum algorithm for gradient-based edge detection that operates entirely within the quantum circuit model. Grayscale images are encoded using the Novel Enhanced Quantum Representation (NEQR), allowing exact arithmetic on pixel intensities. Directional gradients are computed by generating superpositions of neighboring pixels via cyclic shift operations and performing subtraction with an exact quantum arithmetic circuit. To refine accuracy, we introduce a direction-aware shifting mechanism that aligns edges with the darker side of intensity transitions. Our novel Quantum Partitioning Algorithm enables efficient in-place thresholding of edge candidates. This work exhibits polynomial-time improvements and optimizes the ancilla count compared to previous NEQR-based quantum edge detection algorithms. These results demonstrate a resource-efficient and fully quantum approach to edge detection, highlighting a practical quantum advantage in image processing.
Quantum phase estimation (QPE) is fundamental to quantum algorithms for eigenvalue problems, chemistry simulation, and number-theoretic applications. However, canonical QPE implementations employ controlled unitary operations and quantum Fourier transform, incurring substantial circuit depth and control overhead that severely limit applicability on nearterm quantum hardware. This work develops a novel class of control-free phase estimation algorithms that eliminate both controlled unitaries and QFT by leveraging signal processing principles. Our approach applies uncontrolled powers of the unitary U = e −iHτ to an eigenstate or eigenstate-rich state, measures in multiple bases to capture interference patterns, and reconstructs eigenphases via classical post-processing techniques including Prony's method, compressed sensing, and harmonic inversion. We establish that for eigenstates or states with sparse eigendecomposition, our algorithms achieve phase estimation precision ϵ using Õ(1/ϵ) applications of U , matching standard QPE query complexity up to logarithmic factors while dramatically reducing circuit depth and eliminating control overhead. We provide detailed error analysis characterizing robustness to measurement noise, finite sampling, and state preparation errors. Applications to electronic structure estimation in small molecules and variational quantum eigensolvers demonstrate the practical viability of control-free QPE on both NISQ and early fault-tolerant quantum computers. Our results suggest that signal-processing-based phase estimation may offer more efficient implementations of quantum algorithms on realistic hardware.
In quantum image processing, a fundamental step is encoding classical image data into quantum states. This can be achieved using methods such as Flexible Representation of Quantum Images (FRQI), Quantum Probability Image Encoding (QPIE), and Novel Enhanced Quantum Representation (NEQR). However, on real quantum hardware, these encodings can quickly lead to circuits with many gates, large circuit depth, and high qubit usage, which is a problem for Noisy Intermediate-Scale Quantum (NISQ) devices. In this work, we investigate whether low-rank state approximation, formulated via Schmidt decomposition, can help reduce this complexity. The method keeps only the most significant parts of a quantum state's entanglement structure, making state preparation more efficient while preserving most of the image information. We compare the three encoding techniques in their original form and with low-rank approximation, evaluating metrics such as circuit depth, CNOT count, MSE, and visual quality of reconstructed images. The results reveal meaningful trade-offs between accuracy and resource efficiency, with the FRQI model achieving a 97 percent reduction in circuit depth while maintaining a near-perfect reconstruction (MSE of about 0.27). This demonstrates the potential of low-rank techniques for advancing practical quantum image processing on near-term hardware.
Abstract Quantum Image Representation (QIR) is a fundamental concept in quantum image processing. It describes how classical images are converted into quantum states for processing on quantum computers. Researchers proposed several QIR models, with FRQI and NEQR being the most popular ones. These models show the possibilities of the quantum image encoding. Nevertheless, they are also limited in some way. Mostly, pixel intensity accuracy, the required number of qubits, and circuit complexity have trade-offs. Their application to Noisy Intermediate-Scale Quantum (NISQ) devices becomes challenging, the quantum resources being small in such devices, due to this. In the following paper, we introduce our own approach, which is the Intensity-Preserving Quantum image Representation (IP-QIR). The key aim of IP-QIR is to maintain the information about the grayscale intensities and decrease the amount of quantum resources. IP-QIR quantifies pixel intensities by a controlled rotation scheme. The measurement statistics of a single qubit hold the intensity information and the position qubits represent the spatial information. The strategy uses small patches of images rather than complete images to make it easy to implement. This will decrease the depth of circuit design and makes the approach less advanced to the near-term quantum hardware. IBM Qiskit simulations are used in assessing the performance of IP-QIR. The three types of grayscale images of which experiments are run are the synthetic image patches, SAR images and medical TB chest X-ray images. It is revealed that IP-QIR maintains the intensity information compared to the FRQI and NEQR. SAR and medical data can reach values of fidelity of 84.12\%. The other notable point is that the size of a 4 x 4 image patch only needs five qubits which are much lower than in NEQR. Meanwhile, good reconstruction accuracy is achieved. These findings demonstrate the fact that IP-QIR is an effective and realistic approach to quantum image representation on the NISQ-era quantum devices.
We construct quantum circuits that exactly encode the spectra of correlated electron models up to errors from rotation synthesis.By invoking these circuits as oracles within the recently introduced "qubitization" framework, one can use quantum phase estimation to sample states in the Hamiltonian eigenbasis with optimal query complexity Oðλ=ϵÞ, where λ is an absolute sum of Hamiltonian coefficients and ϵ is the target precision.For both the Hubbard model and electronic structure Hamiltonian in a second quantized basis diagonalizing the Coulomb operator, our circuits have T-gate complexity O(N þ logð1=ϵÞ), where N is the number of orbitals in the basis.This scenario enables sampling in the eigenbasis of electronic structure Hamiltonians with T complexity O(N 3 =ϵ þ N 2 logð1=ϵÞ=ϵ).Compared to prior approaches, our algorithms are asymptotically more efficient in gate complexity and require fewer T gates near the classically intractable regime.Compiling to surface code fault-tolerant gates and assuming per-gate error rates of one part in a thousand reveals that one can error correct phase estimation on interesting instances of these problems beyond the current capabilities of classical methods using only about a million superconducting qubits in a matter of hours.
Quantum computing offers the potential to enhance computational efficiency beyond classical methods, but practical implementation remains challenging due to the limitations of Noisy Intermediate-Scale Quantum (NISQ) hardware, namely, restricted qubit counts, limited connectivity, and the presence of noise and decoherence. This study presents a novel approach to edge detection by leveraging a recently developed Quantum Fuzzy Inference Engine, implemented on a NISQ device. We introduce an optimized quantum circuit for its implementation, reducing qubit requirements and gate depth to improve execution on NISQ hardware. To overcome constraints related to large-scale image processing, a hybrid quantum–classical lookup table approach is employed. Edge detection performance is evaluated on the Berkeley Segmentation Data Set and Benchmarks 500 dataset under different conditions, including classical execution, ideal quantum simulation, noisy quantum simulation, and NISQ hardware calculation. Results demonstrate that the quantum fuzzy logic-based edge detection achieves outcomes comparable to classical methods by using fewer operations, marking a step toward practical quantum-enhanced image processing.