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Quantum Image Processing and Its Application to Edge Detection: Theory and Experiment
Xi-Wei Yao, Hengyan Wang, Zeyang Liao, Ming-Cheng Chen, Jian Pan, Jun Li, Kechao Zhang, Xingcheng Lin, Zhehui Wang, Zhihuang Luo, Wenqiang Zheng, Jianzhong Li, Meisheng Zhao, Xinhua Peng, Dieter Suter
TL;DR
The paper addresses growing demands in digital-image storage, transmission, and processing by developing quantum image processing. It encodes pixel values in probability amplitudes and positions in computational-basis states, demonstrates common transforms, and proposes edge detection using one single-qubit operation independent of image size. The results demonstrate reduced qubit requirements, exponential speed-up for stated transform conditions, and experimental feasibility.
Problem
Growing image-data volume and increasingly challenging computational tasks motivate more efficient image processing and analysis.
Method
The paper encodes images as pure quantum states, with pixel values in probability amplitudes and positions in computational-basis states, then applies quantum transformations and edge detection.
Results
The quantum image representation reduces required qubit resources, while demonstrated transforms can achieve exponential speed-up under polynomial preparation and extraction costs; edge detection uses one single-qubit operation independent of image size.
Takeaways & Limitations
The experiments establish a first practical study toward quantum-computer applications for digital image processing and motivate further quantum image-processing research.
Abstract
from arXiv · showhide
Processing of digital images is continuously gaining in volume and relevance, with concomitant demands on data storage, transmission and processing power. Encoding the image information in quantum-mechanical systems instead of classical ones and replacing classical with quantum information processing may alleviate some of these challenges. By encoding and processing the image information in quantum-mechanical systems, we here demonstrate the framework of quantum image processing, where a pure quantum state encodes the image information: we encode the pixel values in the probability amplitudes and the pixel positions in the computational basis states. Our quantum image representation reduces the required number of qubits compared to existing implementations, and we present image processing algorithms that provide exponential speed-up over their classical counterparts. For the commonly used task of detecting the edge of an image, we propose and implement a quantum algorithm that completes the task with only one single-qubit operation, independent of the size of the image. This demonstrates the potential of quantum image processing for highly efficient image and video processing in the big data era.
I. INTRODUCTION
The growing volume and difficulty of image processing motivate quantum approaches that exploit quantum information processing. This work introduces a resource-reduced quantum image representation, demonstrates quantum image transforms, and proposes an edge-detection algorithm using one single-qubit gate independent of image size.
- Increasing image-data volume and computational difficulty create important demands for more efficient image processing and analysis.
- Quantum information processing exploits superposition, entanglement, and quantum parallelism to achieve higher computational speed for certain problems.
- The paper develops a quantum image representation that reduces qubit requirements and experimentally implements common two-dimensional image transforms.
- The proposed quantum edge-detection algorithm uses one single-qubit gate in its processing stage, independent of picture size.
- Numerical and experimental demonstrations are used to validate the quantum edge-detection algorithm.
II. FRAMEWORK OF QUANTUM IMAGE PROCESSING
Quantum image processing begins by encoding a two-dimensional image as a quantum state, with amplitudes carrying pixel values and computational-basis states carrying pixel positions.
- Quantum image processing first encodes two-dimensional image data into a quantum-mechanical system.
- The representation stores pixel values in probability amplitudes and pixel positions in computational-basis states.
- The framework includes experimental implementations of two-dimensional Fourier, Hadamard, and Haar wavelet transforms.
A. Quantum image representation
The quantum image representation flattens a two-dimensional image into a vector and encodes it in a pure state using logarithmically many qubits, with basis states marking pixel positions and amplitudes carrying values.
- A 2D image F=(F_i,j) is flattened column by column into a vector f containing ML pixel values.
- The image vector is mapped to a pure quantum state of n=⌈log2(ML)⌉ qubits.
- Computational-basis states encode pixel positions, while coefficients encode normalized pixel values.
- For an M×L image, the quantum encoding uses n qubits, whereas the classical representation requires at least 2n bits.
- Preparing the image state can take O[poly(n)] steps when image data are supplied in quantum form or prepared accordingly.
B. Quantum image transforms
Quantum image transforms apply unitary operations to encoded image states and can, under polynomial preparation and extraction costs, achieve exponential speed-ups over classical transforms.
- Quantum image transforms: Quantum image processing evolves an input image state into an output state through a suitable Hamiltonian and unitary operator.
- Quantum image transforms: Fourier, Hadamard, and Haar wavelet transforms can be represented as two-dimensional separable transformations using unitary row and column operators.
- Quantum image transforms: Explicitly reading all components of the processed image state requires O(2^n) operations, although global features may require fewer operations.
- Quantum image transforms: Basic image transforms support applications including compression, spectral methods, and video coding, where classical running time increases sharply with data volume.
- Quantum image transforms: O[poly(m)] quantum time contrasts with O(m2^m) classical time for the corresponding one-dimensional transforms.
- Quantum image transforms: Exponential speed-up is possible when input preparation and output extraction each require no more than O[poly(n)] steps.
C. Experimental demonstrations
The experiments implement quantum image transforms on a four-qubit NMR register, prepare encoded input states, and evaluate transformed images against theory. The measured states reproduce the expected transforms with high fidelities and modest image-distance errors.
- Experimental setup: A 4 × 4 chessboard image is encoded and processed using four qubits in an iodotrifluoroethylene quantum register.The register consists of three 19F spins and one 13C spin.
- Input image state preparation: The input state is prepared by creating a pseudopure state and applying Uencode to produce the image state.UPPS1, UPPS2, and Uencode are implemented with GRAPE pulse engineering, with theoretical fidelities of about 99.9% for these operators.
- Image transforms: Haar wavelet, Fourier, and Hadamard transforms are implemented through corresponding quantum circuits and radio-frequency pulse sequences.Each unitary rotation uses Gaussian selective soft pulses, with compilation used to compensate first-order phase errors and unwanted evolutions.
- Measurement: The experiments read out fluorine-spin information indirectly by transferring it to the 13C spin with a SWAP gate.This indirect readout distinguishes the usable 13C signal from the larger background of molecules containing 12C nuclei.
- Experimental results: 0.961–0.975 is the measured fidelity range for the input and transformed image states relative to theoretical density matrices.The decoded experimental images agree well with theory; the image Euclidean distance is approximately 0.08 for input data and 0.09–0.12 after processing.
III. QUANTUM EDGE DETECTION ALGORITHM
The paper proposes quantum edge detection based on a Hadamard operation that extracts intensity differences between neighboring pixels, with additional processing for the remaining pairs. Numerical and experimental demonstrations show boundary information can be encoded and recovered from quantum image states.
- Algorithm motivation: Classical edge detection processes each pixel, whereas the proposed quantum algorithm finds region boundaries in O(1) time independent of image size.The method addresses boundaries as intensity changes between adjacent regions.
- Hadamard-based detection: A Hadamard transform on the last qubit converts neighboring pixel coefficients into sums and differences, with differences indicating intensity changes.For neighboring coefficients c_b1...bn−10 and c_b1...bn−11, the operation produces c_b1...bn−10 ± c_b1...bn−11.
- Hadamard-based detection: Measuring the last qubit extracts horizontal boundaries between pixel pairs at positions 0/1, 2/3, and subsequent even-position pairs.When the paired pixels have identical intensities, their difference vanishes; otherwise, it is nonzero.
- Complete boundary extraction: An amplitude permutation rearranges the image coefficients so a second single-qubit Hadamard rotation obtains boundaries between the remaining pairs 1/2, 3/4, and so on.The permutation can be performed in O[poly(n)] time; an ancilla-based encoding is offered as an alternative for obtaining all boundary values.
- Numerical demonstration: For a 256 × 256 binary image, the image is encoded with 16 qubits instead of 2^16 = 65536 classical bits, and the decoded output successfully detects its boundaries.The simulation uses an input image Fcat and applies the unitary operator I215 ⊗H.
- Experimental demonstration: The experimental procedure applies a single-qubit Hadamard gate to a 4-qubit image register and reads edge information for half of the pixel positions from the resulting state.Separate experiments obtain odd and even boundaries with and without amplitude permutation.
IV. CONCLUSION
The paper presents a quantum image representation with reduced qubit requirements, experimentally demonstrates fundamental transforms, and applies quantum processing to edge detection. The edge-detection algorithm uses one Hadamard gate and runs in O(1) time rather than O(2^n) classically.
- QPIE encodes pixel values in probability amplitudes and pixel positions in computational basis states, reducing required qubit resources.
- The experiments demonstrate 2D Fourier, Hadamard, and Haar wavelet transforms as fundamental quantum image-processing operations.
- Exponential speed-ups are reported for the demonstrated quantum image transforms over their classical counterparts.
- The edge-detection algorithm processes global image-edge information with one single-qubit Hadamard gate in O(1) time, compared with O(2^n) for classical algorithms.
- The experiment is presented as a first experimental study toward practical digital image-processing applications of quantum computers.
APPENDIX A: COMPARISON OF QIMRS
The paper compares quantum image representations and introduces QPIE, which encodes image information using amplitudes and computational basis states. QPIE is reported to require fewer resources than FRQI and NEQR.
- The qubit lattice model represents each pixel with a qubit, requiring 2^n qubits for an image of 2^n pixels.
- FRQI integrates pixel value and position information into an (n + 1)-qubit quantum representation using an angle for each pixel value.
- NEQR stores each pixel value in the computational basis state of d qubits rather than using FRQI’s angle encoding.
- QPIE encodes pixel values in probability amplitudes and pixel positions in computational basis states, requiring fewer resources than FRQI and NEQR.
APPENDIX B: QUANTUM WAVELET TRANSFORM
The appendices describe quantum circuits for Haar wavelet transforms and linear spatial filtering. The Haar transform has O(m^3) elementary-gate complexity, while general spatial filtering is generally nonunitary.
- Quantum Haar wavelet transform: The M × M Haar wavelet transform uses A_M with M = 2^m and has A_2 equal to a Hadamard transform.
- Quantum Haar wavelet transform: The Haar-transform circuit recursively combines controlled gates with qubit cyclic right-shift permutations implemented by SWAP gates.
- Quantum Haar wavelet transform: O(m^3) elementary gates implement the quantum Haar wavelet transform when controlled Hadamard and controlled NOT gates have linear complexity.
- Spatial filtering: Spatial filtering maps the vectorized input image f to the vectorized output image g through the linear transformation g = Uf.
- Spatial filtering: General spatial filtering is nonunitary; U is unitary only when w22 = ±1 and all other elements in W are zero.
APPENDIX D: DETECTING SYMMETRY BY QIMP
The paper proposes a quantum algorithm for recognizing inversion symmetry by rotating an image with NOT gates and measuring overlap with the original state. The overlap provides information about the image’s inversion symmetry.
- The algorithm rotates an encoded image by 180° using a NOT operation on every position qubit.
- A SWAP test measures overlap between the original and rotated image states to assess inversion symmetry.
- A larger measured overlap indicates better inversion symmetry of the original image.
- Estimating inner products between image-state vectors is reported to take O(log ML) time on a quantum computer.
- For a 2 × 2 example, applying NOT to both qubits transforms the input amplitudes into those of the image rotated 180° about its center.
APPENDIX E: VARIANT OF QHED ALGORITHM
The variant of QHED adds an auxiliary qubit to encode the image and produce full boundary values in a single step. Its circuit combines Hadamard gates with an amplitude-permutation operation before conditional measurement.
- APPENDIX E: VARIANT OF QHED ALGORITHM: An auxiliary qubit enables the variant to produce full boundary values in a single step.The image is redundantly encoded across the original n-qubit state and the auxiliary qubit.
- APPENDIX E: VARIANT OF QHED ALGORITHM: A Hadamard gate on the auxiliary qubit creates duplicated amplitudes for each input coefficient.The resulting n+1-qubit state contains each amplitude c_i twice, scaled by 2^-1/2.
- APPENDIX E: VARIANT OF QHED ALGORITHM: The amplitude permutation rearranges the redundant state so neighboring image amplitudes become paired, including the wraparound pair c_N−1 and c_0.This arrangement prepares adjacent-amplitude sums and differences after the second Hadamard gate.
- APPENDIX E: VARIANT OF QHED ALGORITHM: A Hadamard gate on the last qubit transforms each neighboring pair into sum and difference amplitudes.The state contains terms such as c_i + c_i+1 and c_i − c_i+1, including the boundary pair.
- APPENDIX E: VARIANT OF QHED ALGORITHM: The circuit consists of Hadamard gates and the n+1-qubit amplitude-permutation operation D2n+1.Figure 9 depicts the circuit and identifies H as the Hadamard gate and D2n+1 as the permutation operation.