In the Name of the Pasta, and of the Sauce, and of the Holy Meatballs … – Bobby Henderson, The Gospel of the Flying Spaghetti Monster, 2006 After the conceptual comes the practical. While the study of quantum foundations is as old as quantum theory itself, the field of quantum computing is relatively new. So new in fact that large-scale, practical quantum computing is still not a reality. A typical ‘quantum computer’ takes many months to set up before performing such astounding tasks as factoring 6 into 3 × 2. Nonetheless, if those machines would exist, we know that we would gain amazing speed-ups in solving some hard (classical) computational problems, such as those involved in breaking a huge portion of cryptographic systems in use today. Before we get into ‘quantum computing’, we should say a couple of things about ‘computing’. What is computing? Our answer by now probably won't come as such a shock: it's a process theory! Computation is indeed all about wiring the inputs and outputs of small processes together to make bigger processes. More specifically, a computation consists of a (finite) set of basic processes, which are wired together according to some (also finite) instructions, which we refer to as an algorithm or simply a program . The only essential difference between classical and quantum computation is the contents of the basic processes. For classical computation, these operations consist of things like logical operations (e.g. XOR) or reading/writing locations in memory. For quantum computation, we can extend this with quantum processes and classical-quantum interactions such as measurements. So quantum computing is all about figuring out how to write new kinds of programs that exploit these new building blocks to build faster algorithms or accomplish new kinds of tasks that aren't possible classically. The first quantum algorithms were ‘proofs of concept’, in the sense that they solved some problem much faster than a classical computer, but the kinds of problems they solved were not particularly interesting in their own right. However, this changed drastically with the advent of Grover's quantum search and Shor's factoring algorithms.
In the Name of the Pasta, and of the Sauce, and of the Holy Meatballs … – Bobby Henderson, The Gospel of the Flying Spaghetti Monster, 2006 After the conceptual comes the practical. While the study of quantum foundations is as old as quantum theory itself, the field of quantum computing is relatively new. So new in fact that large-scale, practical quantum computing is still not a reality. A typical ‘quantum computer’ takes many months to set up before performing such astounding tasks as factoring 6 into 3 × 2. Nonetheless, if those machines would exist, we know that we would gain amazing speed-ups in solving some hard (classical) computational problems, such as those involved in breaking a huge portion of cryptographic systems in use today. Before we get into ‘quantum computing’, we should say a couple of things about ‘computing’. What is computing? Our answer by now probably won't come as such a shock: it's a process theory! Computation is indeed all about wiring the inputs and outputs of small processes together to make bigger processes. More specifically, a computation consists of a (finite) set of basic processes, which are wired together according to some (also finite) instructions, which we refer to as an algorithm or simply a program . The only essential difference between classical and quantum computation is the contents of the basic processes. For classical computation, these operations consist of things like logical operations (e.g. XOR) or reading/writing locations in memory. For quantum computation, we can extend this with quantum processes and classical-quantum interactions such as measurements. So quantum computing is all about figuring out how to write new kinds of programs that exploit these new building blocks to build faster algorithms or accomplish new kinds of tasks that aren't possible classically. The first quantum algorithms were ‘proofs of concept’, in the sense that they solved some problem much faster than a classical computer, but the kinds of problems they solved were not particularly interesting in their own right. However, this changed drastically with the advent of Grover's quantum search and Shor's factoring algorithms.
After some remarks on the fundamental physical nature of information, Bennett and Fredkin’s ideas reversible computation are introduced. This leads to the suggestions of Benioff and Feynman as to the possibility of a new type of essentially ‘quantum computer’. If we happen to build such devices, Deutsch scientists showed that‘quantum parallelism’ leads to new algorithms and new complexity classes. This is dramatically illustrated by Shor's quantum algorithm for factorization which is polynomial in time in contrast to algorithms for factorization on a classical Turing computer. This discovery has potentially important implications for the security of many modern cryptographic systems. The fundamentals of quantum computing are then introduced - reversible logic gates, qubits and quantum registers. The key quantum property of ‘entanglement’ is described, with due homage to Einstein and Bell. As an illustration of a quantum program, Grover's database search algorithm is described in some detail. After all this theory, the status of experimental attempts to build a quantum computer is reviewed: it will become evident that we have a long way to go before we can factorize even small numbers. Finally, we end with some thoughts about the process of ‘quantum compilation’ - translating a quantum algorithm into actual physical operations on a quantum system - and some comments on prospects for future progress.
A computer is a physical device that helps us process information by executing algorithms. An algorithm is a well-defined procedure, with finite description, for realizing an information-processing task. An information-processing task can always be translated into a physical task. When designing complex algorithms and protocols for various information-processing tasks, it is very helpful, perhaps essential, to work with some idealized computing model. However, when studying the true limitations of a computing device, especially for some practical reason, it is important not to forget the relationship between computing and physics. Real computing devices are embodied in a larger and often richer physical reality than is represented by the idealized computing model. Quantum information processing is the result of using the physical reality that quantum theory tells us about for the purposes of performing tasks that were previously thought impossible or infeasible. Devices that perform quantum information processing are known as quantum computers. In this book we examine how quantum computers can be used to solve certain problems more efficiently than can be done with classical computers, and also how this can be done reliably even when there is a possibility for errors to occur. In this first chapter we present some fundamental notions of computation theory and quantum physics that will form the basis for much of what follows. After this brief introduction, we will review the necessary tools from linear algebra in Chapter 2, and detail the framework of quantum mechanics, as relevant to our model of quantum computation, in Chapter 3. In the remainder of the book we examine quantum teleportation, quantum algorithms and quantum error correction in detail. We are often interested in the amount of resources used by a computer to solve a problem, and we refer to this as the complexity of the computation. An important resource for a computer is time. Another resource is space, which refers to the amount of memory used by the computer in performing the computation. We measure the amount of a resource used in a computation for solving a given problem as a function of the length of the input of an instance of that problem.
Computation is a process of performing a large number of simple operations. The main building blocks of classical computers are bits. Each of them can take either the value 0 or 1. A series of gates can change these values by acting on them selectively. This process is structured to fulfil a purpose. For example, if we are interested in adding two numbers, the numbers are encoded into bits and gates are performed such that the final state of the bits reveals the desired answer: a number which is the sum of the initial ones. While it is possible to build algorithms that can perform almost any computation, it is also desirable to find the answer within a reasonable length of time or with a reasonable amount of resources. There is an ever-growing demand in computational power for both scientific and commercial purposes. Indeed, it is surprisingly easy to find an application that can jam even the fastest computer. This fuels a vast effort in the research of information science to increase the speed and processing power of computers. Modern computational models are based on the universal Turing machine (Turing, 1937). This is a theoretical information processing model that employs the elementary gate processes described above. It can efficiently simulate any other device capable of performing an algorithmic process. Since the introduction of the Turing machine, physics has influenced computation in many ways.
Quantum computing is a big change in the way computers work, and it promises to be much faster than traditional systems. This new technology brings both huge benefits and huge problems, especially when it comes to cryptographic security measures. Classical encryption algorithms, like RSA and ECC, depend on the fact that some math problems are hard, like discrete logarithms and integer factorization. Quantum algorithms, like Shor's algorithm, can solve these problems quickly. Because of this, the development of scalable quantum computers poses a danger to the basic safety of the cryptography methods that are widely used today. This short summary looks at the big effects that quantum computing will have on the safety of cryptography. It looks at the security holes that quantum algorithms create and stresses how important it is to find answers for post-quantum cryptography (PQC). PQC wants to make programs that can't be broken by quantum attacks. This will make sure that digital interactions can still be private, secure, and real in a world powered by quantum computers. Also, switching to PQC comes with a lot of problems, such as implementing algorithms, making sure they are all the same, and getting people to use them in a lot of different technology environments. The abstract talks about current research projects and foreign partnerships that aim to standardize and implement PQC. It stresses how important it is to plan ahead to reduce the risks of the future.
Do you remember your first computer? Mine was a Mac Plus that possessed a random access memory (RAM) of 1 MB! In 1987 this was enough to run the first version of MacRepertory; now not enough to save one snapshot! In 25 years the technical abilities of computers have skyrocketed, and the impact of new developments in the coming 25 years can only be imagined. For homeopathy the video and the computer have had huge influence. Video cases improved the quality of teaching tremendously. Who can better teach you a remedy than a patient telling you about his state? Earlier we could read or hear about a remedy, but now we could, as it were, taste it. A huge difference similar to reading about a red fruit the size of a chestnut, with tiny seeds on its surface and then getting the opportunity to taste the real thing – a strawberry. A complete library of homeopathy books can now be on your hard disk, and what is more, you can search in it at an amazing speed. What will be available to us in 2037? A development that may open incredible new vistas, also for homeopathy, has been set in motion by Dutch nanoscientist Leo Kouwenhoven . He caused great excitement among scientists with research published in Science in April 2012. In the 1930s, the brilliant Italian physicist Ettore Majorana deduced from quantum theory the possibility of the existence of a very special particle, a particle that is its own anti-particle. That Majorana fermion would be right on the border between matter and anti-matter. Leo Kouwenhovenʼs research group managed to create a nanoscale electronic device in which a pair of Majorana fermions ‘appear’ at either end of a nanowire. The discovery of Majorana fermions opens up a new and uncharted chapter of fundamental physics; they may also play a role in cosmology, as a proposed theory assumes that the mysterious ‘dark matter’, which forms the greatest part of the universe, is composed of Majorana fermions. Furthermore, scientists view the particles as fundamental building blocks for the quantum computer. Such a computer is far more powerful than the best supercomputer because it has the ability to be in more than one state simultaneously. Whereas digital computers require data to be encoded into binary digits (bits), quantum computation utilizes quantum properties to represent data and perform operations on these data. So far quantum computers have only existed in theory, now they may soon become a reality. In 2037 your quantum computer might record everything your patient says while simultaneously analyzing the information and comparing it with a huge memory of provings, materia medica and cured cases. By the time the interview is over and 3-D photos of the patient, DNA and all sorts of physical parameters have been entered into the equation, a differential diagnosis of homeopathic substances is proposed together with a calculation of the likelihood of a substance being the simillimum. This seems a possible scenario. Whether it would actually still involve you or whether the patient would simply run an app at home – who knows? You and homeopathy may have moved on to completely new ways of determining, creating and applying the simillimum. Practically all theoretic particles that are predicted by quantum theory have been found in the last decades, with just a few exceptions, including the enigmatic Majorana particle and the well-known Higgs boson. But Ettore Majorana the person is every bit as mysterious as the particle. In 1938 he withdrew all his money and disappeared during a boat trip from Palermo to Naples. Whether he killed himself, was murdered or lived on under a different identity is still not known. You donʼt withdraw your money and then kill yourself, do you? But then he did write what seems to be a suicide note. Was he killed by the Nazis because he refused to be involved in developing a nuclear weapon, or because they feared he might do so for the allies? Was he indeed seen in Argentina? No trace of Majorana was ever found. As if he had merged with his own anti-particle. As if he changed roles with the mysterious dark matter the particle named after him possibly explains. Could homeopathy offer an answer as to his state and possibly his fate? Itʼs tempting to at least have a look at Origanum majorana . The remedy has suicidal disposition as one of its known symptoms. The Lamiaceae are ambitious and have a great need of acknowledgement. Ettore Majorana was extremely brilliant, which would be a good way to get recognition, but theoretical physics and the sphere of action for which Origanum is known donʼt really seem to compute. Also praise may not have been an important motivation for him as he left writing a paper about another discovery he made to James Chadwick who won the Nobel Prize for it. We could analyse the books and films made about Ettore and speculate about his remedy. But we may better wait until 2037, then feed all personal writings by Ettore Majorana, his DNA and his photo into the quantum computer that he should posthumously receive a Nobel Prize for, and accept the – also posthumously – suggested remedy as his most likely simillimum. Harry van der Zee