Quantum Image Processing and Its Application to Edge Detection: Theory and Experiment

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.

[1]  Joshua B. Tenenbaum,et al.  Human-level concept learning through probabilistic program induction , 2015, Science.

[2]  C. Marcus,et al.  Milestones toward Majorana-based quantum computing , 2015, 1511.05153.

[3]  Masoud Mohseni,et al.  Quantum support vector machine for big feature and big data classification , 2013, Physical review letters.

[4]  Jing Li,et al.  An adaptive image Euclidean distance , 2009, Pattern Recognit..

[5]  Dong He,et al.  Satellite-based entanglement distribution over 1200 kilometers , 2017, Science.

[6]  John M. Martinis,et al.  State preservation by repetitive error detection in a superconducting quantum circuit , 2015, Nature.

[7]  Timo O. Reiss,et al.  Optimal control of coupled spin dynamics: design of NMR pulse sequences by gradient ascent algorithms. , 2005, Journal of magnetic resonance.

[8]  M. Levitt.  Spin Dynamics: Basics of Nuclear Magnetic Resonance , 2001 .

[9]  Timothy F. Havel,et al.  Nuclear magnetic resonance spectroscopy: an experimentally accessible paradigm for quantum computing , 1997, quant-ph/9709001.

[10]  C. Zu,et al.  Experimental realization of universal geometric quantum gates with solid-state spins , 2014, Nature.

[11]  E. Knill.  Quantum computing with realistically noisy devices , 2005, Nature.

[12]  D. Deutsch.  Quantum theory, the Church–Turing principle and the universal quantum computer , 1985, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[13]  S. Lloyd,et al.  Quantum algorithms for supervised and unsupervised machine learning , 2013, 1307.0411.

[14]  Sang Michael Xie,et al.  Combining satellite imagery and machine learning to predict poverty , 2016, Science.

[15]  Andrei N. Soklakov,et al.  Efficient state preparation for a register of quantum bits , 2004 .

[16]  Shi-Jie Wei,et al.  Duality quantum computer and the efficient quantum simulations , 2015, Quantum Information Processing.

[17]  Scott Aaronson,et al.  The computational complexity of linear optics , 2010, STOC '11.

[18]  Ming-Cheng Chen,et al.  Solving Systems of Linear Equations with a Superconducting Quantum Processor. , 2017, Physical review letters.

[19]  Alán Aspuru-Guzik,et al.  A two-qubit photonic quantum processor and its application to solving systems of linear equations , 2014, Scientific Reports.

[20]  H. S. Allen.  The Quantum Theory , 1928, Nature.

[21]  Jiangfeng Du,et al.  Experimental realization of a quantum support vector machine. , 2015, Physical review letters.

[22]  Seth Lloyd,et al.  Quantum random access memory. , 2007, Physical review letters.

[23]  Mile Gu,et al.  Experimental quantum computing to solve systems of linear equations. , 2013, Physical review letters.

[24]  Kaoru Hirota,et al.  A flexible representation of quantum images for polynomial preparation, image compression, and processing operations , 2011, Quantum Inf. Process..

[25]  G. Milburn,et al.  Linear optical quantum computing with photonic qubits , 2005, quant-ph/0512071.

[26]  Peter W. Shor,et al.  Algorithms for quantum computation: discrete logarithms and factoring , 1994, Proceedings 35th Annual Symposium on Foundations of Computer Science.

[27]  D. Leung,et al.  Bulk quantum computation with nuclear magnetic resonance: theory and experiment , 1998, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[28]  Christian Schneider,et al.  High-efficiency multiphoton boson sampling , 2017, Nature Photonics.

[29]  Jingfu Zhang,et al.  Experimental magic state distillation for fault-tolerant quantum computing , 2011, Nature Communications.

[30]  Dieter Suter,et al.  Colloquium : Protecting quantum information against environmental noise , 2016 .

[31]  Andrew W. Cross,et al.  Demonstration of quantum advantage in machine learning , 2015, npj Quantum Information.

[32]  H. Weinfurter,et al.  Multiphoton entanglement and interferometry , 2003, 0805.2853.

[33]  Kai Lu,et al.  NEQR: a novel enhanced quantum representation of digital images , 2013, Quantum Information Processing.

[34]  Thierry Paul,et al.  Quantum computation and quantum information , 2007, Mathematical Structures in Computer Science.

[35]  E. Knill,et al.  Liquid-state nuclear magnetic resonance as a testbed for developing quantum control methods , 2008, 0803.1982.

[36]  Isaac L. Chuang,et al.  Methodology of Resonant Equiangular Composite Quantum Gates , 2016, 1603.03996.

[37]  Jian-Wei Pan,et al.  Quantum teleportation of multiple degrees of freedom of a single photon , 2015, Nature.

[38]  D. Deng,et al.  Quantum Entanglement in Neural Network States , 2017, 1701.04844.

[39]  Chi Zhang,et al.  Single Strontium Rydberg Ion Confined in a Paul Trap , 2016, 1611.02184.

[40]  L. Teixeira,et al.  Eye , 2013, AORN journal.

[41]  S. Lloyd,et al.  Quantum principal component analysis , 2013, Nature Physics.

[42]  Yongmei Huang,et al.  Satellite-to-ground quantum key distribution , 2017, Nature.

[43]  B. J. Metcalf,et al.  Boson Sampling on a Photonic Chip , 2012, Science.

[44]  E. Knill,et al.  A scheme for efficient quantum computation with linear optics , 2001, Nature.

[45]  Demis Hassabis,et al.  Mastering the game of Go with deep neural networks and tree search , 2016, Nature.

[46]  D Marr,et al.  Theory of edge detection , 1979, Proceedings of the Royal Society of London. Series B. Biological Sciences.

[47]  A. Harrow,et al.  Quantum algorithm for linear systems of equations. , 2008, Physical review letters.

[48]  Seth Lloyd,et al.  Universal Quantum Simulators , 1996, Science.

[49]  Dieter Suter,et al.  Quantum simulation of a system with competing two- and three-body interactions. , 2008, Physical review letters.

[50]  D. Suter,et al.  NMR quantum simulation of localization effects induced by decoherence. , 2010, Physical review letters.

[51]  K. R. Brown,et al.  Microwave quantum logic gates for trapped ions , 2011, Nature.

[52]  J. O'Brien,et al.  Universal linear optics , 2015, Science.

[53]  Dieter Suter,et al.  Experimental implementation of adiabatic passage between different topological orders. , 2014, Physical review letters.

[54]  Amir Fijany,et al.  Quantum Wavelet Transforms: Fast Algorithms and Complete Circuits , 1998, QCQC.

[55]  Xinhua Peng,et al.  Experimental observation of Lee-Yang zeros. , 2015, Physical review letters.

[56]  Xinhua Peng,et al.  Preparation of pseudo-pure states by line-selective pulses in nuclear magnetic resonance , 2001 .

[57]  A. Crespi,et al.  Integrated multimode interferometers with arbitrary designs for photonic boson sampling , 2013, Nature Photonics.

[58]  T. Rudolph,et al.  Resource-efficient linear optical quantum computation. , 2004, Physical review letters.

[59]  Sougato Bose,et al.  Storing, processing, and retrieving an image using quantum mechanics , 2003, SPIE Defense + Commercial Sensing.

[60]  Ming-Cheng Chen,et al.  Efficient Measurement of Multiparticle Entanglement with Embedding Quantum Simulator. , 2015, Physical review letters.

[61]  E. Knill,et al.  Optimal quantum measurements of expectation values of observables , 2006, quant-ph/0607019.

[62]  D. Suter,et al.  Spin qubits for quantum simulations , 2010 .

[63]  Dieter Suter,et al.  Quantum adiabatic algorithm for factorization and its experimental implementation. , 2008, Physical review letters.

[64]  David Marr,et al.  VISION A Computational Investigation into the Human Representation and Processing of Visual Information , 2009 .

[65]  Philip Walther,et al.  Experimental boson sampling , 2012, Nature Photonics.

[66]  Scott Aaronson,et al.  BQP and the polynomial hierarchy , 2009, STOC '10.

[67]  Kai Lu,et al.  QSobel: A novel quantum image edge extraction algorithm , 2014, Science China Information Sciences.

[68]  Andrew G. White,et al.  Photonic Boson Sampling in a Tunable Circuit , 2012, Science.

[69]  Fei Yan,et al.  A survey of quantum image representations , 2015, Quantum Information Processing.

[70]  Seth Lloyd,et al.  Quantum algorithm for data fitting. , 2012, Physical review letters.

[71]  E M Fortunato,et al.  Implementation of the quantum Fourier transform. , 2001, Physical review letters.

[72]  Dieter Suter,et al.  Experimental protection of two-qubit quantum gates against environmental noise by dynamical decoupling. , 2015, Physical review letters.

[73]  B. Zeng,et al.  Measuring out-of-time-order correlators on a nuclear magnetic resonance quantum simulator , 2016, 1609.01246.

[74]  A. M. Turing,et al.  Computing Machinery and Intelligence , 1950, The Philosophy of Artificial Intelligence.

[75]  A. Haar.  Zur Theorie der orthogonalen Funktionensysteme , 1910 .

[76]  Lov K. Grover,et al.  Creating superpositions that correspond to efficiently integrable probability distributions , 2002, quant-ph/0208112.

[77]  Dieter Suter,et al.  Localization-delocalization transition in the dynamics of dipolar-coupled nuclear spins , 2015, Science.

[78]  C-Y Lu,et al.  Entanglement-based machine learning on a quantum computer. , 2015, Physical review letters.

[79]  Costantino S. Yannoni,et al.  Liquid-State NMR Quantum Computing , 2000, quant-ph/0012108.

[80]  Jian-Wei Pan,et al.  Ground-to-satellite quantum teleportation , 2017, Nature.

[81]  Raymond Laflamme,et al.  Selective-pulse-network compilation on a liquid-state nuclear-magnetic-resonance system , 2016 .

[82]  Lov K. Grover.  Quantum Mechanics Helps in Searching for a Needle in a Haystack , 1997, quant-ph/9706033.

[83]  Jian-Wei Pan,et al.  Experimental realization of quantum algorithm for solving linear systems of equations , 2013, 1302.1946.