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Register a new userQuantum information and quantum computing: an overview and some mathematical aspects
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Added by
Maurice Robert Kibler
Publication date
2018
fields
Physics
and research's language is
English
Authors
Maurice Robert Kibler
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The aim of the present paper is twofold. First, to give the main ideas behind quantum computingand quantum information, a field based on quantum-mechanical phenomena. Therefore, a shortreview is devoted to (i) quantum bits or qubits (and more generally qudits), the analogues of theusual bits 0 and 1 of the classical information theory, and to (ii) two characteristics of quantummechanics, namely, linearity (which manifests itself through the superposition of qubits and theaction of unitary operators on qubits) and entanglement of certain multi-qubit states (a resourcethat is specific to quantum mechanics). Second, to focus on some mathematical problems relatedto the so-called mutually unbiased bases used in quantum computing and quantum informationprocessing. In this direction, the construction of mutually unbiased bases is presented via twodistinct approaches: one based on the group SU(2) and the other on Galois fields and Galois rings.
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We discuss how quantum computation can be applied to financial problems, providing an overview of current approaches and potential prospects. We review quantum optimization algorithms, and expose how quantum annealers can be used to optimize portfolios, find arbitrage opportunities, and perform credit scoring. We also discuss deep-learning in finance, and suggestions to improve these methods through quantum machine learning. Finally, we consider quantum amplitude estimation, and how it can result in a quantum speed-up for Monte Carlo sampling. This has direct applications to many current financial methods, including pricing of derivatives and risk analysis. Perspectives are also discussed.
Quantum computers, if fully realized, promise to be a revolutionary technology. As a result, quantum computing has become one of the hottest areas of research in the last few years. Much effort is being applied at all levels of the system stack, from the creation of quantum algorithms to the development of hardware devices. The quantum age appears to be arriving sooner rather than later as commercially useful small-to-medium sized machines have already been built. However, full-scale quantum computers, and the full-scale algorithms they would perform, remain out of reach for now. It is currently uncertain how the first such computer will be built. Many different technologies are competing to be the first scalable quantum computer.
I consider some promising future directions for quantum information theory that could influence the development of 21st century physics. Advances in the theory of the distinguishability of superoperators may lead to new strategies for improving the precision of quantum-limited measurements. A better grasp of the properties of multi-partite quantum entanglement may lead to deeper understanding of strongly-coupled dynamics in quantum many-body systems, quantum field theory, and quantum gravity.
We describe spacelike and timelike (causally connected) events on an equal footing by utilizing detectors coupled to timers that store information about a given system and the moment of measurement. By tracing out the system and focusing on the detectors and timers states, events are represented by a tensor product structure. Furthermore, including a time register gives rise to a temporal superposition analogous to the familiar spatial superposition in quantum mechanics. We verify that the presence of coherence can ensure a causal connection between events. We also propose a causal correlation function involving the detection times to characterize the type of events. Finally, we verify that our formalism allows us to simultaneously apply quantum information concepts to spacelike and timelike events. In this context we observe, in the limit of instantaneous measurements, a deterministic relationship between causally connected events similar to that of spatially entangled physical systems; i.e. observing the state of one of the systems (in our case, knowing a previous event), enables us to learn precisely the state of the other system (we delineate a later event).
We perform a quantum information analysis for multi-mode Gaussian approximate position measurements, underlying noisy homodyning in quantum optics. The Gaussian maximizer property is established for the entropy reduction of these measurements which provides explicit formulas for computations including their entanglement-assisted capacity. The case of one mode is discussed in detail.
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