Finding Stable Models via Quantum Computation
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高三科技创新英语阅读理解20题1<背景文章>Artificial intelligence (AI) is making significant inroads into the field of healthcare. AI has the potential to revolutionize medical diagnosis and treatment. One of the most prominent applications of AI in healthcare is in medical imaging. AI algorithms can analyze medical images such as X-rays, CT scans, and MRIs with a high degree of accuracy. This helps radiologists detect diseases and abnormalities more quickly and accurately.Another area where AI is making an impact is in drug discovery. By analyzing large amounts of data, AI can help identify potential drug candidates and predict their efficacy. This can significantly reduce the time and cost of drug development.AI also has the potential to improve patient care by providing personalized treatment plans. By analyzing a patient's medical history, genetic information, and other factors, AI can recommend personalized treatment options that are tailored to the individual patient's needs.However, the use of AI in healthcare also presents some challenges. One of the main challenges is the need for large amounts of high-quality data. AI algorithms require large amounts of data to train and improve their accuracy. Another challenge is the need for regulatory approval. As withany new technology, there is a need for regulatory frameworks to ensure the safety and effectiveness of AI in healthcare.Despite these challenges, the potential benefits of AI in healthcare are significant. As the technology continues to develop and improve, it is likely that we will see even more applications of AI in healthcare in the future.1. What is one of the most prominent applications of AI in healthcare?A. Surgical procedures.B. Medical imaging.C. Patient transportation.D. Hospital administration.答案:B。
最新物理实验报告(英文)Abstract:This report presents the findings of a recent physics experiment conducted to investigate the effects of quantum entanglement on particle behavior at the subatomic level. Utilizing a sophisticated setup involving photon detectors and a vacuum chamber, the experiment aimed to quantify the degree of correlation between entangled particles and to test the limits of nonlocal communication.Introduction:Quantum entanglement is a phenomenon that lies at the heart of quantum physics, where the quantum states of two or more particles become interlinked such that the state of one particle instantaneously influences the state of the other, regardless of the distance separating them. This experiment was designed to further our understanding of this phenomenon and its implications for the fundamental principles of physics.Methods:The experiment was carried out in a controlled environment to minimize external interference. A pair of photons was generated and entangled using a nonlinear crystal. The photons were then separated and sent to two distinct detection stations. The detection process was synchronized, and the data collected included the time, position, and polarization state of each photon.Results:The results indicated a high degree of correlation between the entangled photons. Despite being separated by a significant distance, the photons exhibited a consistent pattern in their polarization states, suggesting a strong entanglement effect. The data also showed that the collapse of the quantum state upon measurement occurred simultaneously for both particles, supporting the theory of nonlocality.Discussion:The findings of this experiment contribute to the ongoing debate about the nature of quantum entanglement and its potential applications. The consistent correlations observed between the entangled particles provide strong evidence for the nonlocal properties of quantum mechanics. This has implications for the development of quantum computing and secure communication technologies.Conclusion:The experiment has successfully demonstrated the robustness of quantum entanglement and its potential for practical applications. Further research is needed to explore the broader implications of these findings and to refine the experimental techniques for probing the quantum realm.References:[1] Einstein, A., Podolsky, B., & Rosen, N. (1935). Can Quantum-Mechanical Description of Physical Reality Be Considered Complete? Physical Review, 47(8), 777-780.[2] Bell, J. S. (1964). On the Einstein Podolsky RosenParadox. Physics, 1(3), 195-200.[3] Aspect, A., Grangier, P., & Roger, G. (1982). Experimental Tests of Realistic Local Theories via Bell's Theorem. Physical Review Letters, 49(2), 91-94.。
量子骗局作文高中英语Quantum deception is a fascinating concept that challenges our understanding of reality. It involves the manipulation of quantum particles to create the illusion of a certain outcome, which may not necessarily reflect the true nature of the particles.The idea of quantum deception raises questions about the nature of reality and the limits of human perception.It forces us to reconsider our assumptions about the world around us and the way we interact with it.Quantum deception also has practical implications, particularly in the field of quantum computing. By harnessing the principles of quantum deception, scientists are able to create more powerful and efficient quantum computers that can solve complex problems at a much faster rate than traditional computers.However, quantum deception also poses ethical concerns,as it challenges our ability to discern between what isreal and what is fabricated. This has implications for issues such as privacy, security, and the potential for misuse of quantum deception technology.In conclusion, quantum deception is a thought-provoking concept that challenges our understanding of reality and has both practical and ethical implications. It forces us to reconsider our assumptions about the world and raises important questions about the nature of truth and perception.。
未来科技对生活的影响英语作文高中全文共6篇示例,供读者参考篇1The Future of Tech and How It Will Change Our LivesHey guys! Today I want to talk about something that's been on my mind a lot lately - the crazy advancements happening in technology and how they're going to impact our lives in the years ahead. It's some pretty mind-blowing stuff when you really think about it!Let's start with artificial intelligence (AI). You've probably heard about AI assistants like Siri, Alexa, and ChatGPT that can understand our voices and help with all sorts of tasks. Well, get ready because AI is going to get WAY smarter and more capable. Imagine having an AI buddy that can not just answer questions but actually learn and grow smarter over time. It could help you with your homework, hobbies, or any project you're working on. Some people think advanced AI could even start creating new inventions and making scientific breakthroughs on its own! Crazy, right?Then there's virtual reality (VR) and augmented reality (AR). With VR, you can put on a headset and be transported tomind-blowing digital worlds that feel real. You could go on epic adventures, attend killer concert experiences, or hang out with friends in amazing virtual hangout spaces. AR takes things a step further by blending digital components with the real world through glasses or your phone's camera. Imagine being able to see data, graphics and digital creations hovering in the air all around you as you walk down the street! Both VR and AR are going to transform how we experience entertainment, gaming, education, you name it.But some of the most revolutionary changes could come from breakthroughs in healthcare and medicine. Get this - scientists are working on ways to edit our genes to prevent inherited diseases before we're even born! They're developing amazing bionic body parts to replace damaged organs or missing limbs. Microscopic robots could one day cruise through our bodies detecting diseases and delivering treatments. We may even find ways to dramatically slow aging so people can live considerably longer, healthier lives. Just let that sink in for a minute!The way we get around is going to transform too withself-driving cars and possibly even flying vehicles for personal travel. No more traffic jams or stressful commutes! Autonomous vehicles that are smarter and safer than human drivers could seriously reduce accidents and pollution. Getting from A to B would be a total breeze.Our homes and cities are going to evolve in wild ways as well thanks to smart home tech, renewable energy like solar and wind, and sustainable urban design. Imagine living in an ultra energy-efficient smart home that grew its own food, collected rainwater, and recycled pretty much everything. Cities of the future may be covered in vertical gardens and farms with drone transportation networks instead of roads. Mind blown yet?Of course with all these incredible innovations, there are potential downsides we'll have to watch out for too. Will advanced AI be safe and under meaningful human control? How do we stop deepfakes and other manipulated media from spreading misinformation? What happens if we grow overly reliant on tech and stop using our own skills and capabilities? There are a lot of open questions and ethical considerations.At the end of the day though, I'm super excited about the future! While change can be scary, I think emerging technologieshave amazing potential to solve worldwide problems, cure diseases, protect the environment, and make our lives more fun, productive and enriched in countless ways we can hardly imagine today.Who knows, by the time I'm an adult, humanity may have cracked the secret to eternal life, colonized Mars, or made first contact with alien civilizations! While that might sound like science fiction now, the way tech is accelerating, I wouldn't rule anything out. The 21st century is going to be one heck of a ride. I can't wait to see what world-changing breakthroughs and mind-bending technologies will arrive next!篇2Future Tech and How It Will Change Our LivesHi there! My name is Emma and I'm a high school student. Today I want to talk about something that's super exciting to me - future technology and how it's going to change our lives in crazy ways!First up, let's talk about something that's already starting to happen - self-driving cars. Can you imagine being able to just get in a car, tell it where you want to go, and then kick back and relax? No more stressful driving for your parents! The car will usesensors and cameras to see everything around it and drive itself safely to your destination. Some self-driving cars are already being tested on roads. In the future, everyone might have one!But that's just the start. What about having a robot helper at home? Companies are working on making humanoid robots that can do chores like vacuuming, cleaning, doing laundry, and even cooking! Wouldn't it be awesome to come home from school to a sparkling clean house with dinner ready? With robot helpers, your busy parents wouldn't have to worry about housework anymore.Speaking of robots, someday they might even become our friends. Scientists are trying to make robots that are great at understanding human emotions and having natural conversations. Picture having an robot buddy that can play games with you, chat with you about your day, and keep you company. So cool!Wearable technology like smartwatches is going to get much more advanced too. Imagine having a watch that can monitor your health, giving your doctor updates and alerting you to any problems. It could also amplify your physical abilities by giving you information and even controlling other devices with just hand gestures. You'd feel like a super hero!Another mind-blowing possibility is brain-computer interfaces. These could allow you to control computers and machines just by thinking about what you want them to do. Need to look something up? You could download information straight into your brain. Or play ultra-realistic virtual reality games that feel completely real by having the experience streamed right into your mind. It's straight out of a sci-fi movie!On the medical front, technologies like gene editing, artificial organs, and nanobots could help deal with injuries and diseases way better than we can now. Doctors could edit your genes while you're still an embryo to prevent genetic disorders. If you get really sick or hurt, you could receive an artificial organ to keep you going until you heal. And tiny nanobots could go inside your body to detect and fix problems.Even how we learn and go to school could be transformed by technology like virtual reality classrooms, educational AI tutors, and immersive learning games. You might attend a virtual school without even leaving your house! An AI tutor could help you with any subject you're struggling with. And instead of reading about history or science, you could experience itfirst-hand through VR.The way we shop and buy things is changing a lot too. Sooner than you think, we might have personal 3D printers that can create practically any product on-demand at home. Or personalized hologram models that let you digitally "try on" clothes and customize purchases. No more waiting for shipping! Smart homes with connected devices could automatically keep track of what supplies you need and order them.Just imagine how cool it would be to have all these technologies! Self-driving car, robot buddy, supercharged smartwatch, mind-reading computer - the future is going to be amazing. I can't wait to see what sort of other crazy inventions people come up with next. Honestly, I'm a little jealous of the kids growing up in 20 or 30 years who'll get to experience it all. But for now, I'll settle for racing home to watch my favorite sci-fi shows and dreaming about篇3The Future of Tech and How It Will Change Our LivesWhat's up guys? Today I want to talk about something that's been on my mind a lot lately - technology and how crazy advanced it's gonna get in the future. I know, I know, you're probably thinking "Ugh, another lecture about robots takingover the world." But just hear me out, because this stuff is actually kind of mind-blowing.I was watching some videos the other day about all the new tech that's in development right now. Scientists and engineers are working on some truly insane things that would have seemed like pure science fiction just a couple decades ago. I'm talking self-driving cars, artificial intelligence assistants that can learn and think like humans, advanced robotics that could automate tons of manual jobs, augmented reality that overlays digital information on the real world, brain-computer interfaces that let you control devices with your mind...the list goes on and on.At first, a lot of this future tech might seem scary or threatening. You might think "Oh no, the robots are going to take all our jobs and make humans obsolete!" And sure, I can understand that fear to some extent. Having intelligent machines that can do tons of tasks better and more efficiently than people is definitely a big shift that will disrupt a lot of industries and the workforce as we know it.But you know what? I actually think these bleeding-edge technologies could massively improve our quality of life if we harness them in the right way. Think about how much time and effort we spend on really basic, mundane tasks every single day -cooking, cleaning, driving, shopping, scheduling appointments and managing our calendars. What if we could offload a ton of that monotonous stuff to AI assistants and robots? We'd have SO much more free time to pursue our real passions, learn new skills, start businesses, spend time with loved ones, you name it.And the healthcare applications of future tech could be absolutely game-changing too. Imagine nanobots that could detect diseases at the cellular level and treat them before you even have symptoms. Or advanced prosthetics and brain implants that could cure paralysis, blindness, and other disabilities. Heck, some scientists are even talking about radically extending the human lifespan with genetic engineering and anti-aging therapies. How crazy is that?There are a lot of ethical questions we'll need to figure out too. If we can create hyper-realistic virtual worlds and experiences, does that make actual reality less meaningful or important? Should we use gene editing to enhance babies and create designer children? Where do we draw the line between healing disabilities and enhancing abilities beyond normal human limits?Those are some deep philosophical questions that don't have any easy answers. But I think it's exciting that ourgeneration will get to grapple with and solve those big challenges as these world-changing technologies come to fruition.In the meantime, I can't wait to see what the future holds. Maybe by the time I'm in my 30s, I'll be commuting to work in a flying car while my AI assistant schedules all my appointments and handles mundane tasks for me. Or taking vacations in photorealistic virtual worlds that let me experience anything I can imagine. Or just kicking back at home while housekeeping robots clean and cook for me. A guy can dream, right?The point is, buckle up guys, because the next few decades are going to be one heck of a revolutionary ride as science fiction becomes reality. Will it be a utopia or a dystopia? That's up to us to decide how we develop and use these world-transforming technologies. But I'm optimistic that if we're smart about it, the future of tech can massively elevate the human experience and reshape society in awesome ways that we can barely even imagine today. Let's go invent the future!篇4The Impact of Future Technology on LifeHi there! My name is Alex, and I'm a high school student who loves learning about science and technology. Today, I want to talk to you about the exciting world of future tech and how it's going to change our lives in incredible ways.First, let's talk about something that's already starting to become a reality – artificial intelligence (AI). AI is kind of like a really smart computer program that can learn and make decisions on its own. It's already being used in things like virtual assistants, self-driving cars, and even medical diagnosis. But in the future, AI is going to get way smarter and more advanced.Imagine having a personal AI assistant that can help you with homework, chores, and even making important decisions. It could be like having a super-smart friend who knows everything and is always there to help you out. Or what about AI robots that can do dangerous jobs like firefighting or construction work, so humans don't have to risk their lives? That would be pretty awesome, right?Another area where we're going to see some crazy advancements is in biotechnology. Scientists are working on ways to edit genes and create new kinds of living things, like super-crops that can grow in harsh environments or animals that are resistant to diseases. They're even trying to figure out how togrow organs and body parts in labs, which could save millions of lives!But that's not all – what about nanotechnology? That's the science of working with really, really tiny particles called nanoparticles. With nanotechnology, we could createsuper-strong and lightweight materials, tiny robots that can repair our bodies from the inside, and even devices that can clean up pollution or generate clean energy. It's like something straight out of a sci-fi movie!But I believe that if we approach these challenges with wisdom, caution, and a commitment to ethics, we can harness the power of future technology to make the world a better place. Imagine a future where everyone has access to clean water, nutritious food, and quality healthcare, thanks to advancements in biotechnology and nanotechnology. Or a future where we've solved the problem of climate change and pollution, thanks to clean energy technologies and AI-powered environmental solutions.And that's just the tip of the iceberg! There are so many other exciting possibilities, like space exploration, virtual reality, and even technologies that we haven't even dreamed of yet.So, what do you think? Are you as excited about the future of technology as I am? I can't wait to see what kinds of amazing innovations and discoveries are in store for us in the years to come. And who knows, maybe some of you reading this will be the ones to make those discoveries and shape the future of our world!篇5The Future is Here: How Technology Will Change Our LivesHi there! My name is Emma, and I'm a high school student who's super excited about the future of technology. I know, I know, you're probably thinking, "Ugh, not another boring essay about robots and computers." But trust me, this is going to be way more exciting than that!Imagine a world where you can talk to your computer or phone like it's a real person. No more typing or swiping – you just say what you want, and boom! It's done. That's the power of artificial intelligence (AI) and voice recognition technology. Pretty cool, right?But that's just the tip of the iceberg. What about self-driving cars? No more fighting traffic or worrying about parking! You just hop in, tell the car where you want to go, and it takes youthere safely and efficiently. And with electric and solar-powered vehicles, we'll be doing our part to save the planet too.Speaking of saving the planet, have you heard about vertical farming? It's this amazing way of growing crops in tall, indoor buildings using special lights and hydroponics (growing plants without soil). This could help us produce way more food with way less land and water. No more worrying about droughts or famines!And let's not forget about virtual reality (VR) and augmented reality (AR). With VR, you can transport yourself to different worlds and experiences without ever leaving your room. Want to explore ancient Rome? Done. Want to go on a space adventure? No problem. And with AR, you can overlay digital information and graphics onto the real world around you. Imagine being able to see directions or reviews for restaurants and shops just by looking at them!But perhaps the most exciting (and maybe a little scary) development is the rise of genetic engineering and biotechnology. Scientists are already working on ways to edit our genes to cure diseases, enhance our abilities, and even prolong our lives. Can you imagine living to be 150 or 200 years old? Orhaving super-strength or super-intelligence? It sounds like something straight out of a sci-fi movie!But overall, I'm super optimistic about the future. Technology has the potential to solve so many of the world's problems – from hunger and disease to climate change and energy shortages. And it can open up new frontiers of exploration, creativity, and human potential that we can't even imagine yet.So, there you have it – my thoughts on how technology is going to shape our lives in the years and decades to come. It's going to be an amazing ride, full of mind-blowing innovations and game-changing advancements. But it's also going to require us to think carefully about the ethical and societal implications of these new technologies.At the end of the day, technology is just a tool – it's up to us to use it wisely and responsibly. But if we can do that, the future is going to be pretty awesome. So buckle up, because the future is coming, and it's going to be one wild ride!篇6The Future of Tech and How It'll Change Our LivesHey guys! Today I wanted to talk about something that's been on my mind a lot lately - the crazy ways that technology is developing and how it's going to transform pretty much every aspect of our lives in the coming years and decades. I find this topic really fascinating and also a little scary if I'm being honest. The pace of technological change just seems to be accelerating so quickly!Let's start by looking at some of the major areas where we're likely to see huge leaps forward driven by new technologies emerging from labs and research centers around the world. One big area is artificial intelligence and machine learning. You've probably heard a lot about AI assistants like Siri, Alexa and ChatGPT that can understand our speech, answer questions, and even create content like articles and essays. Well, that's just the very beginning!Scientists are working on developing artificial general intelligence (AGI) which would be AI systems with human-level intelligence that can think, learn, reason and plan just like us. If they succeed with AGI, it could lead to an "intelligence explosion" where AIs rapidly become way smarter than humans at pretty much everything. It's mind-blowing to think about, but superintelligent AIs could help solve problems like curing alldiseases, reversing aging, colonizing other planets, and figuring out solutions to big challenges facing humanity.Another area that's going to transform our lives is biotechnology and genetics. Pretty soon, it will become routine to have your full genome sequenced and analyzed. Based on your genes, doctors and AIs will be able to assess your risks for various diseases and customize treatments, diets and lifestyle recommendations to optimize your health and lifespan. Gene editing tools like CRISPR will allow us to edit genes before birth to prevent inheriting certain conditions. Further in the future, we may develop the capability for advanced genetic engineering of humans based on a desired "design."The biotechnology revolution will also produce lab-grown organs, tissue engineering to regrow limbs, and technologies to dramatically extend human lifespan and prevent aging. It's pretty crazy to think about, but some scientists believe the first person who will live to 150 or even 200 years old has already been born. Just imagine how different life would be if our lifespans were doubled, tripled or even longer!Nanotechnology is another area that I think is going to have mind-blowing impacts. Nanotech involves engineering and manufacturing at the molecular scale - we're talking aboutmaking machines that are literally the size of molecules! By the 2030s, we may develop the ability to build molecular machines that can go inside our bodies, repair damaged cells, augment our immune systems, and even back up our memories and consciousness.Nanobots could also be used to manufacture virtually anything from the molecular level up in nanofactories - making physical objects the way a 3D printer makes things, but on a unimaginably small scale. This could lead to a world of radical abundance where any object can be replicated or "printed" for pennies. But nanotech also brings potential risks like weaponized nanobots or molecular assemblers that make lethal pollutants or viruses.In the coming decades, we'll also likely see major advances in areas like quantum computing, nuclear fusion energy, space exploration and off-world colonies, brain-computer interfaces that connect our minds to machines, and even potentially the development of useful thermonuclear weapons that aren't as destructive as nukes. The scope of technological progress on the horizon is just staggering.So how will all these mind-bending technologies impact our actual day-to-day lives and the societies we live in? Well, for onething, the nature of work and the job market will be massively disrupted as AI and automation start handling more and more types of jobs that humans currently do. Jobs focused on manual labor, data entry, driving, bookkeeping, customer service, and knowledge work involving analysis, writing, and research could all become obsolete or change dramatically.There will likely need to be big economic and political reforms like a universal basic income (UBI) so that people can have their basic needs met even if robots have taken all the jobs. Otherwise, you could end up with mass unemployment and social unrest. Governments may also have to introduce policies to slow down the automation of certain job categories to prevent too much disruption happening too quickly.Education and training programs will have to adapt rapidly to focus less on imparting static knowledge that AIs can just look up instantly, and more on developing uniquely human skills like creativity, emotional intelligence, designing & instructing AIs, and continuously learning new in-demand skillsets throughout our lives.Our relationships and dating will evolve too as AI companions become more sophisticated and widespread. A lot of people may prefer partnering with AI spouses that arecustomized to their ideal preferences and needs rather than dealing with the challenges of human relationships. Of course, procreating with an AI isn't possible, so human reproduction methods like IVF and surrogacy may see a resurgence.Things like global communication, media, art and entertainment will be radically transformed by AI's ability to flawlessly dub movies and shows into any language, create photorealistic video imagery indistinguishable from reality, and compose incredible music, literature, films and other creative content. Every home could have an AI composer, novelist and entertainment studio built-in that learns your preferences. But on the flip side, we'll have to grapple with new challenges like not being able to trust any visual or audio evidence because everything could be a fake AI-generated deepfake.Overall though, the incredible power of advanced AI, nanotech, biotech and future disruptive technologies means we may eventually be able to eliminate virtually all social and economic challenges plaguing humanity - aging, disease, poverty, hunger, pollution, lack of energy, conflict, and even death itself. Of course, we have to navigate the potential risks of these technologies first and govern their development wisely.We may finally be entering an era when the only limitations we face are the boundaries we choose to impose on ourselves.Anyway, those are just some of my thoughts on where technology seems to be heading and how it could massively alter life as we know it in the generations ahead. In some ways, I'm really excited for all the amazing benefits - curing disease, reversing aging, space travel, abundance for all, you name it. But I'm also a bit unnerved by the sheer power these new technologies will unleash and the challenge of trying to control such fundamentally disruptive forces. Only time will tell how it all unfolds, but one thing seems clear - this technological revolution really is going to change everything! I'd love to hear your thoughts too. Catch you later!。
a r X i v :q u a n t -p h /9702050v 3 5 M a r 1998Quantum Isomorphic SimulationHaiqing Wei and Xin XueDepartment of ECE,McGill University,Montreal,Quebec,Canada H3A 2A7ABSTRACTA new type of quantum simulator is proposed which can simulate any quantum many-body system in an isomorphic manner.It can actually synthesize a du-plicate of the system to be simulated.The isomor-phic simulation has the great advantage that the in-evitable coupling of the simulator to the environment can be fully exploited in simulating thermodynamic processes.PACS numbers:89.80.+h,71.45.-dSimulating quantum many-body systems on a classical computer is hard in the sense that the simulation takes exponentially long time and large memory as the size of the system increases [1,2].The difficulty stems from the fact that the Hilbert space of the system consists of ex-ponentially many states as the function of the number of physical variables involved.The ef-ficiency of simulating a system by another well controlled quantum system (quantum computer)was conjectured by Feynman [1]and has been justified by Lloyd [2].Because of the decoher-ence problem [3],despite the recent advancement of the quantum computation theory especially on quantum error correction [4,5],it remains to see whether one can construct a large size quantum computer and maintain the delicate quantum co-herence in order to accomplish meaningful com-putations or simulations,although many efforts have been made to implement quantum gates or small size quantum computers [6].Here we propose that a universal quantum iso-morphic simulator (IS)can be constructed by tai-loring the many-body interactions among a suit-able physical system.Just for convenience in this paper,the system being simulated is called the simulatee.One may choose physical systems con-sisting of,for example,many isolated quantum wires each with a freely moving electron inside,or isolated one-dimensional arrays of Josephson junctions [7]each having a magnetic fluxon freely moving back and forth.The position of the elec-tron in the wire or the fluxon in the Josephson junction array provides one continuous degree of freedom which can be assigned to a continuous variable of the simulatee.For simplicity,we will use the terms wire and particle in the following,whether they are a quantum wire and an elec-tron or a Josephson junction array and a fluxon.Given the HamiltonianH (t )=H 0+H ′(t ),where H 0describes the unperturbed system of simulatee and H ′(t )is the external perturba-tion,one can tailor the many-body interactions among the system of wires which is called a quan-tum isomorphic simulator such that the Hamil-tonian describing the simulator is identical (in some sense)to H (t ),the Hamiltonian of the sim-ulatee.This is why it is called an isomorphic simulator.First note that H 0−K can be de-composed into a sum of two-body interactions in general,H 0−K =ij h ij (x i ,x j ),where x i ,x j are continuous variables which can be simulated by two wires (of course,discrete variables can also be simulated using wires with double poten-tial wells or something else),and the externalperturbation H ′(t )is a sum of one-body poten-tials,H ′(t )= k g k (x k ),K is the total kinetic energy of the simulatee which is naturally mim-icked by the inherent inertial motion of the parti-cles (electrons or fluxons)of the simulator.One may divide each wire into N parts (discretiza-tion with precision 1/N ),the position-dependent interaction h ij (x i ,x j )between two wires can be achieved by drawing N 2connections between the two wires so that the interaction potential of the i th particle at position x i and the j th particle at position x j is proportional to h ij (x i ,x j ).For example,if the wires are Josephson junction ar-rays and the particles are magnetic fluxons,one may enclose the magnetic fluxes at each pair of discrete positions (x i ,x j )by a superconducting 1ring so that the subsystem of the ring and the i th and j thfluxons has a total interaction energy h ij(x i,x j)when the i thfluxon is at position x i and the j thfluxon is at x j[8-10].The one-body potentials g k(x k)can be easily realized by ap-plying position-dependent externalfields to the particles.A simulatee with M continuous vari-ables can be simulated by an IS consisting of M wires,at most1vironment can be fully and naturally exploited when simulating real thermodynamic processes. At present the well-established technology for su-perconductive devices is very promising to imple-ment an IS using Josephson junction arrays[8-10].The continuing advancement in fabricating small size devices may eventually make it possi-ble to implement IS’in terms of semiconductor nanostructures[13].It is exciting to expect that in the near future quantum isomorphic simula-tors are widely used to synthesize and test vari-ous complex systems.。
量子计算中的量子搜索算法量子计算是一种基于量子力学原理的计算模型,它利用量子比特(qubit)的叠加和纠缠特性,能够在某些特定情况下实现比传统计算机更高效的计算。
量子搜索算法是量子计算中的重要算法之一,它能够在大量数据中高效地搜索到目标项。
本文将介绍量子搜索算法的原理、应用以及相关的研究进展。
一、量子搜索算法的原理量子搜索算法的核心思想是利用量子叠加和纠缠的特性,在一次计算中同时处理多个搜索项,从而实现高效的搜索。
与经典计算中的线性搜索算法相比,量子搜索算法能够在O(√N)的时间复杂度内找到目标项,其中N是搜索空间的大小。
量子搜索算法的基本原理是利用量子叠加和纠缠的特性构建一个量子态,使得目标项的概率振幅被放大,从而能够更容易地找到目标项。
其中最著名的量子搜索算法是Grover算法,它是由美国计算机科学家Lov Grover于1996年提出的。
Grover算法的基本步骤如下:1. 初始化:将搜索空间中的所有项都置于均匀叠加态。
2. 量子搜索:通过一系列的量子门操作,将目标项的概率振幅放大。
3. 相位反转:将目标项的相位反转。
4. 量子搜索:再次进行量子搜索,直到找到目标项为止。
通过这样的迭代过程,Grover算法能够在O(√N)的时间复杂度内找到目标项。
相比于经典的线性搜索算法的时间复杂度为O(N),Grover算法具有更高的搜索效率。
二、量子搜索算法的应用量子搜索算法在实际应用中具有广泛的潜力和应用价值。
以下是一些典型的应用场景:1. 数据库搜索:量子搜索算法能够在大规模数据库中高效地搜索到目标项,从而提高数据库的查询效率。
2. 图像识别:利用量子搜索算法可以在大规模图像库中快速地搜索到与目标图像相似的图像,从而实现高效的图像识别。
3. 密码破解:量子搜索算法可以用于破解一些基于传统计算机的加密算法,从而对密码安全提出新的挑战。
4. 优化问题求解:量子搜索算法可以应用于一些优化问题的求解,例如旅行商问题、物流路径优化等。
人工智能辅助蛋白质结构预测的研究进展随着现代科技的不断发展,人工智能技术正越来越广泛地应用于各个领域,其中蛋白质结构预测便是其中一个重要的领域。
蛋白质结构预测是生物学和生物化学领域中的一个重要问题,对于解决许多生物学问题有着重要的作用。
然而,在过去的几年中,世界各地的科学家们一直在致力于开发新的算法和技术,以改善蛋白质结构预测的效率和准确性。
近年来,人工智能技术在蛋白质结构预测领域的应用成为了一种热门趋势,并且已经取得了一定的研究进展。
“蛋白质是生命的基础”这一经典的说法,准确地传达着蛋白质生物学的重要性。
蛋白质是由氨基酸组成的复杂生物大分子,具有广泛的功能。
这些功能包括:酶反应催化、免疫系统的维持、信号传递、运输、细胞内骨架的形成、细胞外基质的结构支撑以及核酸的包装等多种生物学功能。
通常,这些功能是和蛋白质的特定结构密切关联的。
因此,蛋白质结构的研究对于科学家们更好地理解生命的机理和进化历史有非常重要的意义。
蛋白质结构预测是一个既重要而且挑战性的问题。
基于实验成果的结构解析,是通常被采用的方法,但是这种方法代价昂贵、耗时长,并且时常难以获得比较准确的结果。
相对于实验方法,生物信息学的技术在蛋白质结构预测领域具有很高的效率、时间、成本优势,因而广泛应用于这个领域。
绝大多数将人工智能技术应用于蛋白质结构预测的研究方法仍然是依据先前具有基因库和蛋白库的分析。
其中,结构预测分为三个步骤:(1)在数据库中寻找相似的蛋白结构;(2)确定候选蛋白结构的序列和氨基酸序列特征;(3)使用算法对蛋白质的结构进行预测。
虽然这种机器学习方法已经有了很多改善,但一直依赖于生物学家的背景知识。
然而,可靠使用机器学习技术的方法之一是使用人工神经网络(ANN)来学习蛋白质结构所涉及的规则和相互作用。
而这项技术在当前的研究中取得了成功。
另一个重要的技术是基于机器学习的静态和动态模拟的组合。
基于机器学习的模型和模拟学习的模型不同。
物理经典书籍教材推介个人以为国内好的物理教材不太多,很多教材内容严谨可是易读性不强。
因此那个书单特意精选了一些适宜自学且架构严谨的国外大学热点教材分享~~最重要的是,这套教材组成一个完整的物理教材体系,都是教得专门深切浅出的专著,专门适合自学提高。
学物理是一件宝贵的乐事,什么缘故不学得乐在其中呢?这些教材在保留了趣味的情形下不失学术水平,因此专门推荐。
以下是依照学习推荐进度排序的《费曼物理学讲义》卷一、卷二、卷3及《习题解答》诺奖大师费曼乐趣横生的经典之作,读起来津津有趣,没有大段令人畏惧的公式和推导,几乎全文都在说明物理思想和有趣的现象,关于明白得物理思想的本质有极大帮忙。
《Mathematical methods for Physics and engineering》by Riley涵盖了几乎所有物理研究需要的数学知识,没有过度苛求证明严格性且说明形象有趣读起来超级有成绩感,不像国内的数理方式写得过于太抽象,用户界面不太友好呵呵。
《All the Mathematics you missed but need to know》by Garrity轻松的读物,高屋建瓴地整合了之前学过的数学知识,使读者很容易看透其中的数学本质。
举重假设轻地谈了很多深刻的数学领域,例如拓扑和“形式(form)”。
能够给大三看,也能够给研一看,必然会有专门大收成。
《Classical Mechanics- Systems of Particles and Hamiltonian Dynamics》by Greiner清楚地讲述了理论力学的内容,尽管书厚,但说明清楚没有冗余,超级适合自学。
《Introduction to Electrodynamics》及《习题解答》by Griffiths深切浅出的一本书,把人轻松地从电磁学带进电动力学的世界。
《Classical Electrodynamics》及《习题解答》by JacksonJackson那个词几乎已经专指这本神作了,难度高但循序渐进不失自学性。
Finding Stable Models via Quantum ComputationD.A.Meyer and J.B.RemmelDepartment of Mathematics University of California/San DiegoLa Jolla,CA 92903-0112J.PommersheimDepartment of Mathematics University of California/San DiegoLa Jolla,CA 92903-0112andDepartment of Mathematics and Computer SciencePomona College Claremont,CA 91711AbstractQuantum computers have the potential to out-perform classi-cal computers—certain quantum algorithms run much faster than any known alternative classical algorithm.For example,Grover showed that a quantum computer can search an un-ordered list of N items in time O (√N ),providing a quadratic speed-up over the classical algorithm.In this paper,we show that we can modify Grover’s search algorithm to give an al-gorithm that finds stable models of an Answer Set Program with a similar quadratic improvement over the classical al-gorithm.Marek and Remmel showed that Answer Set Pro-gramming (ASP)programs can uniformly solve all NP-search problems,so our quantum algorithm to find stable models of ASP programs also solves all NP-search problems.It follows that Answer Set Programming could provide a programming language for quantum computation.IntroductionThe history of electronic computing has been marked by the fact that the computers of each new generation are dramat-ically smaller and faster than their predecessors.Neverthe-less,from a computational point of view,each generation of computers is essentially the same:the machines are built out of simple logic gates.The Church-Turing thesis asserts that this is inevitable since any computer can be simulated with at most a polynomial factor slowdown by a probabilis-tic Turing machine.The promise of quantum computation is that we may be able to use the laws of quantum mechan-ics to build a quantum computer that can out-perform clas-sical computers,violating the classical Church-Turing the-sis.Quantum computation originated with a suggestion by Feynman that although there appears to be no efficient way of simulating a multiparticle quantum system on a classical computer,there would be a way to run a simulation effi-ciently on a computer that took advantage of the properties of the quantum world (Feynman 1982).Two formal mod-els for quantum computers—the quantum Turing machine (Deutsch 1985)and quantum computational networks,i.e.,quantum gate arrays (Deutsch 1989)—were defined subse-quently by Deutsch.Since these initial results,evidence that quantum com-puters can out-perform classical computers has been accu-mulating.In the work that stimulated the greatest surge of interest in the subject,Shor showed that there exist polyno-mial time quantum algorithms for two of the most famous problems in computer science:factoring and discrete log (Shor 1994).Bernstein and Vazirani gave the first formal evidence that quantum computers violate the modern form of the Church-Turing thesis.They proved that the recursive Fourier sampling problem can be solved in polynomial time on a quantum Turing machine,but relative to an oracle,re-quires superpolynomial time on a classical probabilistic Tur-ing machine (Bernstein and Vazirani 1997).Nevertheless,there are limits to the power of quantum computers.For example,Bennett,Bernstein,Brassard and Vazirani proved that,relative to a random oracle,with prob-ability 1,the class NP cannot be solved on a quantum Turing machine in time o (2n/2)(Bennett et.al.1997).This bound is tight,since Grover’s search algorithm (Grover 1996)shows that one can accept any language in NP in time O (2n/2)on a quantum Turing machine.These results on the power of quantum computation leave a wide range of possible applications of quantum algo-rithms to be explored.Our explorations in this paper are guided by recent developments in Knowledge Represen-tation,especially the appearance of a new generation of systems (Cholewi´n ski et.al.1996;Niemel¨a and Simons 1996;Eiter et.al.1998)based on the so-called An-swer Set Programming (ASP)paradigm (Niemel¨a 1998;Cadoli and Palipoli 1988;Marek and Truszczy´n ski 1999;Lifschitz 1999).We shall focus on one particular ASP for-malism,namely,the Stable Semantics for Logic Programs (SLP)(Gelfond and Lifschitz 1988).The underlying methods of ASP are similar to those used in Logic Programming (Apt 1990)and Constraint Program-ming (Jaffar and Maher 1994;Marriott and Stuckey 1998).That is,like Logic Programming,ASP is a declarative for-malism and the semantics of all ASP systems are based on logic.Like Constraint Programming,certain clauses of an ASP program act as constraints .There is,however,a fun-damental difference between ASP programs and Constraint Logic programs:in Constraint Programming,the constraints act on individual elements of the Herbrand base of the pro-gram,while the constraint clauses in ASP programs act more globally,placing restrictions on which subsets of the Her-brand base can be acceptable answers for the program.For example,suppose that we have a problem Πwhose solutions are subsets of some Herbrand base H .In order tosolve the problem,an ASP programmer essentially writes a logic program P that describes the constraints on the subsets of H that can be answers toΠ.The basic idea is that the program P should have the property that there is an easy decoding of solutions ofΠfrom stable models of P,and that all solutions ofΠcan be obtained from stable models of P through this decoding.The program P is then submitted to an ASP engine such as smodels(Niemel¨a and Simons1996), dlv(Eiter et.al.1998)or DeReS(Cholewi´n ski et.al. 1996),which computes the stable models of the program P. Thus the ASP enginefinds the stable models of the program (if any exists)and one reads off the solutions toΠfrom these stable models.Notice that all solutions are equally good in the sense that any solution found in the process described above is acceptable.In the Aswer Set Programming paradigm,the semantics of logic program P can be defined in two stages.First,we assume,as in standard Logic Programming,that we interpret P over the Herbrand universe of P determined by the pred-icates and constants that occur in P.Since the set of con-stants occurring in the program isfinite,we can ground the program in these constants to obtain afinite propositional logic program P g.The stable models of P are by defini-tion the stable models of P g.The process of grounding is performed by a separate grounding engine such as lparse (Niemel¨a and Simons1996).Second,the grounded program P g is passed to the engine computing the stable models. The basic complexity result for SLP propositional pro-grams is due to Marek and Truszczy´n ski,who showed that the problem of deciding whether afinite propositional logic program has a stable model is NP-complete(Marek and Truszczy´n ski1991).For DATALOG(with negation), an analogous result has been obtained by Schlipf(Schlipf 1995).Marek and Remmel strengthened both of these re-sults by showing that ASP based on SLP is complete for NP-search problems(Marek and Remmel2003).This means that the search engines developed so far are appropriate to solve a vast array of practical problems.Currently,systems based on the ASP paradigm are being tested on problems re-lated to planning,product configuration,combinatorial op-timization problems and other domains.For example,ASP systems have been applied to circuit verification problems (Heljanko and Niemel¨a2001),product configuration prob-lems(Soininen et.al.2001),information extraction engines for the web(?),and updating database specifications(Eiter et.al.2001).The main goal of this paper is show that if we had a quan-tum computer,then we could develop a quantum search en-gine for Answer Set Programming using a suitable extension of Grover’s basic quantum search algorithm.Our extension of Grover’s algorithm gives a quadratic speed-up over the na¨ıve search algorithm for stable models.The importance of this marriage of Answer Set Programming with quantum computation is that ASP provides an effective language for computation and thus could the basis of a practical“quan-tum programming language”for quantum computation.Quantum algorithmsAs we noted in the Introduction,in practice any classi-cal computation is implemented by an array of logic gates, each acting on afinite set of bits.A somewhat surpris-ing result,discovered in the investigation of thermodynamic limits to computation,is that in principle these gates can each be reversible(Bennett1973).For example,Toffoli (Toffoli1981)showed that afinite set of reversible gates is universal for classical computation,since any reversible boolean operation can be decomposed as an array of three gates:N OT:b→b+1,C-N OT:(a,b)→(a,b+a), C-C-N OT:(a,b,c)→(a,b,c+ab),where a,b and c are bits and+is addition modulo2.These gates permute the set of possible states of1,2,and3bits,respectively.Thus any classical computation can be implemented as a sequence of permutations acting on the possible states of the classical computer—n-bit strings,b1...b n—each of which acts non-trivially only on a subset of1,2,or3bits.In an analogous formulation of quantum computation,the possible states are complex linear combinations(superposi-tions)of n-bit strings, a b1...b n|b1...b n .Here|· is the standard physics notation for a vector in(C2)⊗n,in whichwe have chosen a basis |b1···b n |b i∈{0,1} .That is, C2has a basis|0 = 10and|1 = 01.Thus ifH=1√2111−1,thenH|0 =1√ 111−11=1√211=1√2|0 +1√2|1 .When we write|b1...b n where b i∈{0,1},we mean the column vector of size2n whose rows are indexed by bits strings c1...c n∈{0,1}n such that there is a1in the row indexed by the bit string b1...b n and all other entries are0. The states a b1...b n|b1...b n should be thought of as be-ing analogous to the states of a classical probabilistic com-puter,which are probability distributions over n-bit strings, i.e.,linear combinations p b1...b n b1...b n,where the co-efficients are probabilities,so p b i...b n=1.In quantum mechanics,it is the norm-squares of the components that are probabilities,so states satisfy |a b1...b n|2=1.This con-dition allows measurement of the state to be described as a choice of basis(which in this paper will always be this com-putational basis),and the outcome of a measurement to be one of the basis vectors,with probability given by the norm-squared of that component.We will also need the notation ·|to indicate a vector in the dual space to(C2)⊗n.Any quantum computation can be implemented as a se-quence of unitary transformations acting on(C2)⊗n,each of which acts non-trivially only on1or2of the factors(qubits) in this tensor product.That only such1and2qubit quantumgate operations are necessary follows from the result thatarbitrary unitary transformations of a single qubit,togetherwith the linear extension of C-N OT to a unitary transfor-mation of C2⊗C2,are universal for quantum computation (Adelman et.al.1997).Notice that since permutations areunitary transformations,if we also allow a C-C-N OT quan-tum gate,i.e.,the linear extension of C-C-N OT to a unitarytransformation of C2⊗C2⊗C2,every(reversible)classi-cal computation is trivially,exactly,a quantum computation. Without this3qubit gate we must rely on the universality theorem to ensure that every(reversible)classical computa-tion can be approximated efficiently(i.e.,with poly(n)over-head)by a quantum computation.But the remarkable dis-coveries of Deutsch(Deutsch1989),Simon(Simon1997), Bernstein and Vazirani(Bernstein and Vazirani1997),Shor (Shor1994),and Grover(Grover1996)show that in some cases fewer quantum gates are required to transform the quantum computer into a state that upon measurement re-turns the correct bit string with probability greater than2/3 (any probability bounded above1/2is polynomially equiv-alent,by repeating the quantum computation multiple times and choosing the most frequent outcome).We will explain exactly how this works in the case of Grover’s quantum search algorithm,butfirst we describe in detail the classi-cal problems to which we will apply it.ASP search problemsA search problem is a set P offinite instances such that, given any instanceΠ∈P,there is a set SΠof solutions to P for instanceΠ,where SΠcan be empty.For example, the search problem might be tofind Hamiltonian paths in a graph.In this case,the set of instances of the problem is the set of allfinite graphs.Given any instance,i.e.,a graphΓ, SΓis the set of all Hamiltonian paths ofΓ.We say that an algorithm solves the search problem P if it returns a solution S∈SΠwhenever SΠis non-empty and it returns the string “empty”otherwise.We say that a search problem P is in NP if there is such an algorithm which can be computed by a non-deterministic polynomial time Turing machine.We say that search problem P is solved by a uniform logic program if there exists a single logic program P P,a polynomial time extensional data base transformation function edb P(·),and a polynomial time solution decoding function sol P(·,·),such that for every instanceΠ∈P,1.edb P(Π)is afinite set of facts,i.e.,clauses with emptybodies and no variables,and2.whenever SΠis not empty,sol P(Π,·)maps the set ofstable models of edb P(Π)∪P P onto the set of solutions SΠofΠ,and3.whenever SΠis empty,edb P(Π)∪P P has no stable mod-els.We note that decision problems can be viewed as special cases of search problems.Schlipf’s result(Schlipf1995) shows,in fact,that the class of decision problems in NP is captured precisely by uniform logic programs.Specifically he proved that a decision problem is solved by a uniformlogic program if and only if it is in NP.Marek and Remmelshowed that Schlipf’s result can be extended to all NP searchproblems.That is,Marek and Remmel showed that there isa single logic program P T uring that is capable of simulat-ing polynomial time nondeterministic Turing machines inthe sense that given any polynomial time nondeterministicTuring machine M,any inputσ,and any run-time poly-nomial p(x),there is a set of facts edb T uring,p,σ(M)suchthat a stable model of P T uring∪edb T uring,p,σ(M)codes an accepting computation of M started with inputσthatterminates in p(|σ|)or fewer steps and any such accept-ing computation of M is coded by some stable model ofP T uring∪edb T uring,p,σ(M).This result shows that logic programs without function symbols under the stable logic semantics capture all NP-search problems.The converse, that a search problem computed by a uniform logic program P is an NP-search problem,is obvious since one can com-pute a stable model s of a program by guessing s and then doing a polynomial time check to verify that s is a stable model of the program.In this paper we will consider only so-calledDATALOG¬programs.These consist of clauses ofthe formp(X)←q1(X),...,q m(X),¬r1(X),...,¬r n(X),(1) where p,q1,...,q m,r1,...,r n are atoms,possibly with variables and/or constants.Here we abuse notation by writ-ing p(X)to mean that the variables that occur in the pred-icate p are contained in X.A program is afinite set P of clauses of the form(1).We assume that the underlying lan-guage L P of any given program P is determined by the con-stants and predicate symbols that occur in the program.Thus the Herbrand universe U P of P is just the set of all constant terms occurring in P,and the Herbrand base H P of P is the set of all ground atoms of the language L P.Since there are no function symbols in our programs,both the Herbrand universe and the Herbrand base of the program arefinite.A ground instance of a clause C of the form(1)is the re-sult of a simultaneous substitution of constants for variablesoccurring in C.Given a program P,P g is the propositionalprogram consisting of all ground substitutions of clauses ofP.Given a propositional program P and a set S included in its Herbrand base,H P,the Gelfond-Lifschitz transforma-tion of P relative to S,is the program GL(P,S)arising from P as follows.First,eliminate all clauses C in P such that for some j,1≤j≤n,r j∈M.Second,in any remain-ing clause,eliminate all negated atoms.The resulting set of clauses forms a program,GL(P,S),which is a Horn pro-gram and hence possesses a least model M S.We say that S is a stable model of the propositional program P if S=M S. Finally,given any program P with variables,we say that S is a stable model of a program P if S is a stable model of the propositional program P g.Finding a unique stable modelFinding a stable model for an ASP program P is a search problem among the2|H P|possible models for an S⊂H Psuch that S =M S .The simplest classical algorithm over which there is no known worst-case improvement is 1.choose S ⊂H P ;2.construct the Gelfond-Lifschitz transform GL (P,S );3.find the least model M S of GL (P,S )by iterating the one-step provability operator T GL(P,S )on ∅;(This can be done in linear time,see (Dowling and Gallier 1984).)4.if S =M S ,return S ,else repeat.Each step here is polynomial in the problem size,but in the worst case the steps must be repeated O (2n )times,where n =|H P |.Notice that we could formulate each iteration of this algo-rithm as an evaluation of a function f P :P (H P )→{0,1}on S ∈P (H P )(the power set of H P ),wheref P (S )=1if S =M S ;0otherwise,and the problem is to identify an argument at which f P takes the value 1.If we encode P (H P )as n -bit strings,then we can write f P :{0,1}n →{0,1}.Consider the function from {0,1}n ×{0,1}to itself defined by (x,b )→ x,b +f P (x ),which is a permutation.Since the collection of all |x,b is a basis (the computational basis)of the vector space V =(C 2)⊗n ⊗C 2,this permutation induces a linear map U f P from V to V ,which satisfiesU f P |x,b =|x,b +f P (x ) .Since U f P permutes the computational basis,the same (polynomial size)reversible gate array that computes this permutation classically can be also be used to implement the unitary map U f P .We will set the last qubit to be in the state |− = |0 −|1 /√2so that this unitary transformation takes a particularly simple form:U f P |x,− =(−1)f P (x )|x,− .Thus we can abuse notation slightly and writeU f P |x =(−1)f P (x )|x .More generally,we can define U g similarly,for any function g :{0,1}n →{0,1}.We need one more definition before we can describe the application of Grover’s algorithm (Grover 1996)to this problem.Let H =1√2 111−1denote the discrete Fourier transform on C 2.In the quan-tum computing literature this is usually called the Hadamard transform .Notice thatH |0 =1√2|0 +|1 ;this is used in Step 2below.Grover’s algorithm consists of the following steps:1.prepare the state|0...0 ∈(C 2)⊗n ;2.apply H ⊗n to obtain the state1√2n x |x ;3.repeat T times:(a)apply U f P ;(b)apply −H ⊗n U δ0H ⊗n ;(δ0(x )=1if x =0,and van-ishes otherwise)4.measure the state and return the outcome x of the mea-surement.A simple analysis determines the appropriate value of T in Grover’s algorithm.Assume first that P has a unique sta-ble model S ,as,for example,is the case in stratified pro-grams (Apt et.al.1988).Then U f P acts as a reflection in the hyperplane orthogonal to |s ,where s is the n -bit string encoding S .Similarly,since U δ0is a reflection in the hy-perplane orthogonal to |0...0 ,and H ⊗n maps |0...0 to the equal superposition state obtained in Step 2,the unitary transformation −H ⊗n U δ0H ⊗n is a reflection in the hyper-plane orthogonal to the equal superposition vector.Recall that the product of two hyperplane reflections is a rotation in the plane spanned by the vectors orthogonal to the hyper-planes,through twice the angle between them.In this case that angle is θ=arcsin(1/√2n ),so each iteration of Step 3rotates the state by 2θtowards |s ,and choosingT = π/2−θ2θ ∼π4√n as n →∞ensures that when the state is measured in Step 4,it returns the encoding of the stable model with probability at least cos 2θ∼1−2−n.But it does so with only O (√2n )itera-tions,quadratically faster than the classical algorithm at the beginning of this section.Finding one of multiple stable modelsOf course,in general there are multiple stable models for an ASP program.Suppose P has k ∈N stable models,and we apply Grover’s algorithm as described in the previous sec-tion.Now U f P is reflection through the hyperplane orthog-onal to x stable |x ,so the two reflections in Step 3create a rotation by 2θwheresin θ=1√ x stable x |·1√2n x |x =k2.Thus,if we knew the number of stable models,k ,we could iterate Step 3T ∼π4 n times,after whichthe state would be within an angle θofx stable |x /√k .Then measuring the state in Step 4would,with probability 1−O (2−n ),return one of the stable models,each with equal probability.Unfortunately,we do not know how many stable models there are a priori .Brassard,Høyer,Mosca and Tapp haveshown,however,that even when the number of solutions toGrover’s search problem is unknown,one can still be found with expected number of iterations O( n(Brassard et. al.2002).Their algorithm is based on the observation thatpicking a random number of iterations of Step3would cor-respond to picking an angle0≤φ<2πat random,where φis the angle between thefinal state and x stable|x /√.A measurement returns the encoding of a stable model withprobability cos2φ,so for random choices ofφ,E(successful measurement)≈ 2π0cos2φdφ=12. Since we can check efficiently whether the outcome is a sta-ble model,we could repeat this procedure r times and re-duce the failure probability to O(2−r),without increasing the computational complexity.Of course,we cannot really pick a random positive integer number of iterations.Never-theless,this observation suggests the following algorithm:1.let M=1;2.pick T uniformly at random from the integers in[1,M];3.run Grover’s algorithm with this value of T;4.if outcome is stable model,stop,else let M=cM andrepeat from Step2.For1<c<2,Brassard,Høyer,Mosca and Tapp’s result(Brassard et.al.2002)shows that this algorithm succeeds after an expected O( n total number of Grover itera-tions.Thus there is a quadratic improvement in this case as well as in the unique stable model case.DiscussionWe have shown that Grover’s algorithm and Brassard,Høyer,Mosca and Tapp’s generalization can be applied tothe problem offinding stable models for ASP programsand that these quantum algorithms run quadratically fasterthan existing classical solvers in the worst case.Since An-swer Set Programming is a general formalism for NP-searchproblems,this result shows how to solve any such problemwith a quantum ed in this way,ASP would bea general“quantum programming language”.Of course,since Shor’s quantum algorithm for factoringprovides a superpolynomial improvement over the best clas-sical algorithm known,the hope is for more than a quadraticimprovement in other problems.Evidence from highlystructured search problems(Hunziker and Meyer2002)sug-gests that generalizing f P to return more information abouta possible model S than only whether it is M S or not couldprovide a greater improvement.A similar idea motivates quantum adiabatic algorithms(Farhi et.al.2001).These utilize a Hermitian matrix(theHamiltonian)that acts on potential solutions of the problemencoded as quantum states.Applied to3-SAT,for example,this matrix is constructed from the clauses of the probleminstance in such a way that quantum states encoding truthassignments are eigenstates with eigenvalues that are posi-tively proportional to the number of clauses violated.Thealgorithm is designed to evolve an initial state into a quan-tum state that has minimum eigenvalue,i.e.,corresponds to a truth assignment with the smallest number of violated clauses.This idea has also been applied within the discrete time model of quantum computation described in this paper. Hogg describes this idea as the application of‘quantum heuristics’(Hogg2000).Just as in the corresponding quan-tum adiabatic algorithm,his algorithm applies a multiplica-tive phase associated with each truth assignment,one that increases linearly with the number of violated clauses.This multiplicative phase generalizes the(−1)f P phase that ap-pears in Grover’s algorithm.Numerical results for simulations of these quantum algo-rithms indicate that search complexity may be reduced to O(2n/6)(Hogg2000),or possibly even to poly(n)(Farhi et. al.2001;Hogg2000)by using this extra information be-yond whether a truth assignment is satisfying or not.Thus, in our context offinding stable models,we are currently in-vestigating the use of functions(beyond the indicator func-tion for S=M S)that can be computed in polynomial time, e.g.,some function of|S|−|M S|.We anticipate numerical results hinting at similar reduction in complexity.AcknowledgementsThis work was supported in part by the National Security Agency(NSA)and Advanced Research and Development Activity(ARDA)under Army Research Office(ARO)Grant No.DAAD19-01-1-0520.ReferencesL.M.Adelman,J.Demarrais and M.-D.A.Huang.Quan-tum computability.SIAM put.26:1524–1540,1997. 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