Abstract Quantum evolution is time-reversible and yet little advantage is gained from this in the circuit model of quantum computing. Indeed, most quantum algorithms expressed in the circuit model compute strictly from the present to the future, preparing initial states and proceeding forward with unitary transformations and measurements. We may call this predictive computation. In contrast, retrodictive quantum theory, retrocausality, and the time-symmetry of physical laws all suggest that quantum computation embodies richer --untapped-- modes of computation, which could exploit knowledge about the future for a computational advantage. We demonstrate that by using symbolic partial evaluation, retrodictive reasoning can indeed be used as a computational resource that exhibits richer modes of computation at the boundary of the classical/quantum divide. Specifically, instead of fully specifying the initial conditions of a quantum circuit and computing forward, it is possible to compute, classically, in both the forward and backward directions starting from partially-specified initial and final conditions. Furthermore, this mixed mode of computation (i) can solve problems with fewer resources than the conventional forward mode of execution, sometimes even purely classically (de-quantization), (ii) can be expressed in a symbolic representation that immediately exposes global relational properties of the wavefunction that are needed for quantum algorithms and (iii) reveals that the entanglement patterns inherent in genuine quantum algorithms with no known classical counterparts are artifacts of the chosen symbolic representation.
Abstract Quantum evolution is time-reversible and yet little advantage is gained from this in the circuit model of quantum computing. Indeed, most quantum algorithms expressed in the circuit model compute strictly from the present to the future, preparing initial states and proceeding forward with unitary transformations and measurements. We may call this predictive computation. In contrast, retrodictive quantum theory, retrocausality, and the time-symmetry of physical laws all suggest that quantum computation embodies richer --untapped-- modes of computation, which could exploit knowledge about the future for a computational advantage. We demonstrate that by using symbolic partial evaluation, retrodictive reasoning can indeed be used as a computational resource that exhibits richer modes of computation at the boundary of the classical/quantum divide. Specifically, instead of fully specifying the initial conditions of a quantum circuit and computing forward, it is possible to compute, classically, in both the forward and backward directions starting from partially-specified initial and final conditions. Furthermore, this mixed mode of computation (i) can solve problems with fewer resources than the conventional forward mode of execution, sometimes even purely classically (de-quantization), (ii) can be expressed in a symbolic representation that immediately exposes global relational properties of the wavefunction that are needed for quantum algorithms and (iii) reveals that the entanglement patterns inherent in genuine quantum algorithms with no known classical counterparts are artifacts of the chosen symbolic representation.
Quantum models of computation are widely believed to be more powerful than classical ones. Efforts center on proving that, for a given problem, quantum algorithms are more resource efficient than any classical one. All this, however, assumes a standard predictive paradigm of reasoning where, given initial conditions, the future holds the answer. How about bringing information from the future to the present and exploit it to one's advantage? This is a radical new approach for reasoning, so-called Retrodictive Computation, that benefits from the specific form of the computed functions. We demonstrate how to use tools of symbolic computation to realize retrodictive quantum computing at scale and exploit it to efficiently, and classically, solve instances of the quantum Deutsch-Jozsa, Bernstein-Vazirani, Simon, Grover, and Shor's algorithms.
The field of Quantum Information Theory provides the theoretical foundation for the pursuit of quantum computers. The ongoing questions of how quantum computers will be realized and what they will achieve, are both very uncertain. However, worldwide efforts are beginning to converge on some answers, and the future of quantum computers is looking brighter than ever. In contribution to the grand goal that is quantum computing, this thesis serves as a demonstration to the usefulness of quantum over classical computing. The central theme of my work, and my collaborators, is the exploration of using quantum systems as a tool for path finding and search algorithms. First, we explore a specific problem of sequential measurements in Quantum Information Theory, namely quantum retrodiction. In experiments involving quantum systems with numerous intermediate interactions, often times the only thing that matters to the experimenters is the final measurement. However, quantum retrodiction is the problem of extrapolating information about the past of a quantum system, by studying the present state. The case studied in this dissertation is the retrodiction of a series of two-outcome measurements, the double and triple qubit-interferometers. In both cases, we are presented with the problem of inferring the results of the two-outcome measurements, using only the final state of the system. Specifically, all of the possible outcomes from the measurements result in a different final state, and we are tasked with distinguishing which final state we have. One version of this problem can be seen as light traversing a series of half-silver mirrors, and we would like to know the path the light took. To distinguish which final state we are given, we employ the use of POVMs (Positive-Operator Valued Measurements) that reflect the symmetry of the system. The effectiveness of these POVMs in solving the retrodiction problem are then presented, along with a few comments as to other types of information that can be extracted. Second, we examine a different type of path finding via a quantum system. For a given graph G that can be expressed by connections and nodes, the same graph can be mapped onto a quantum system with a Hilbert space H. All of the connections in the graph can be represented as states in H, and the nodes act as sites for local unitary operations. The sum of all these local operations give rise to U, which is the unitary operator that acts on H, and describes the behavior of the system. Using this formalism, we explore a particular type of U called a scattering quantum random walk. The quantum random walk is the quantum mechanical analogy to the classical random walk, and serves as a powerful tool for searching on a graph. In this dissertation, we present the case of searching for a marked final node, on a graph that consists of a series of stars. The initial state of the system has an equal probability distribution among all states, reflecting no a priori knowledge about the location of the final node. But by applying the quantum random walk, the final state of the system reaches a superposition where nearly all of the probability is concentrated along the path leading to the marked final node. Lastly, we extend this idea of using a quantum random walk to reveal a path even further. We examine a specific type of graph G, called an “N-tree” graph, and apply our quantum random walk scheme. An N-tree graph can be described as a maze where the solver is presented with a junction consisting of N choices. Beyond each choice, is an identical junction of N choices, and so on. The depth of the maze is categorized by a parameter M, thus, to correctly solve the maze one must make M correct choices in succession. Labeling the marked final nodes as F, we present a quantum algorithm that uses a quantum random walk to locate F faster than any classical algorithm. Specifically, the quantum random walk results in a final state of the system with a high probability concentration on all the states leading to F. Two methods for searching for F are then presented. The first approach mimics the ideology of a Grover Search, relying on repetitive searches until F is found. The second approach presents a novel use of past measurements to dictates how the algorithm proceeds next. Specifically, we take advantage of the fact that states along the path leading to F hold most of the probability in the system. In the end, the second quantum algorithm approach provides an speedup over the classical algorithm of the order O().
Quantum computation is a subject born out of the combination between physics and computer science. It studies how the laws of quantum mechanics can be exploited to perform computations much more efficiently than current computers (termed classical computers as oppose to quantum computers). The thesis starts by introducing ideas from quantum physics and theoretical computer science and based on these ideas, introducing the basic concepts in quantum computing. These introductory discussions are intended for non-specialists to obtain the essential knowledge needed for understanding the new results presented in the subsequent chapters. After introducing the basics of quantum computing, we focus on the recently proposed quantum algorithm for linear systems. The new results include i) special instances of quantum circuits that can be implemented using current experimental resources; ii) detailed quantum algorithms that are suitable for a broader class of linear systems. We show that for some particular problems the quantum algorithm is able to achieve exponential speedup over their classical counterparts.
Quantum computing improves substantially on known classical algorithms for various important problems, but the nature of the relationship between quantum and classical computing is not yet fully understood. This relationship can be clarified by free models, that add to classical computing just enough physical principles to represent quantum computing and no more. Here we develop an axiomatisation of quantum computing that replaces the standard continuous postulates with a small number of discrete equations, as well as a free model that replaces the standard linear-algebraic model with a category-theoretical one. The axioms and model are based on reversible classical computing, isolate quantum advantage in the ability to take certain well-behaved square roots, and link to various quantum computing hardware platforms. This approach allows combinatorial optimisation, including brute force computer search, to optimise quantum computations. The free model may be interpreted as a programming language for quantum computers, that has the same expressivity and computational universality as the standard model, but additionally allows automated verification and reasoning.
Why quantum computing? As we all know very well, conventional (electronic) computers have enormous power which, moreover, is constantly increasing. With their help gigantic quantities of data can be handled and evaluated. This is of crucial importance, for instance, in modern high energy physics experiments. On the other hand, computers can be used to model very complex systems with the aim of predicting, for example, future developments in our environment, notably climatic changes. Thanks to their incredible speed computers are, of course, destined to treat mathematical problems for which specific algorithms exist very efficiently. To mention a simple example, the digits of π are readily evaluated up to a very high order. Last, but not least, their potential to solve differential equations numerically, even when they are nonlinear, makes computers indispensable in modern research. So, one might ask, is there any reason to look for fundamentally new ways of computing as they are seemingly offered by quantum processes? What advantages can be expected when the laws of quantum mechanics, rather than those of classical physics, are exploited for computation? Two answers were given: first, the simulation of complex quantum systems on classical computers meets serious difficulties that might be overcome with the help of quantum computers. This idea was suggested in 1982 by R. Feynman, and in the 1990s it was indeed shown by several groups of researchers that quantum mechanical systems for which no efficient simulation on a classical computer is known can be efficiently simulated on a quantum computer.
Abstract The discovery of physical systems that exhibit what we now call quantum mechanical behaviour was unquestionably one of the most astonishing events of modern physics. No less remarkable was the realisation that quantum mechanics presents a much more powerful paradigm for computation and information processing than its classical counterpart. But this power is somewhat subtle, and only by building up a rigorous model of quantum computation from the fundamental physics can it be fully understood and analysed. Developing and then using the quantum circuit model as a suitable representation of quantum mechanics, this textbook presents a canon of the most important quantum algorithms. Starting with historically important query algorithms, ‘classic’ results such as Grover’s and Shor’s algorithm are then covered, followed by more recent breakthroughs including the quantum algorithm for solving linear systems (HHL) and the quantum singular value transformation. The thread that runs through the book is the hunt for quantum advantage, in all of its manifestations, and to this end the final chapter moves from the idealised setting of closed quantum systems, to address real ‘noisy’ open quantum systems. Quantum error correction is introduced, culminating in the seminal result, the threshold theorem, which roughly says that the idealised performance predicted by the analysis of quantum mechanics as a model of computation can actually be realised on real-world quantum computers.