Noisy Intermediate-Scale Quantum (NISQ) technology will be available in the near future. Quantum computers with 50-100 qubits may be able to perform tasks which surpass the capabilities of today's classical digital computers, but noise in quantum gates will limit the size of quantum circuits that can be executed reliably. NISQ devices will be useful tools for exploring many-body quantum physics, and may have other us…

Linear optics with photon counting is a prominent candidate for practical quantum computing. The protocol by Knill, Laflamme, and Milburn [2001, Nature (London) 409, 46] explicitly demonstrates that efficient scalable quantum computing with single photons, linear optical elements, and projective measurements is possible. Subsequently, several improvements on this protocol have started to bridge the gap between theore…

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The subject of quantum computing brings together ideas from classical\ninformation theory, computer science, and quantum physics. This review aims to\nsummarise not just quantum computing, but the whole subject of quantum\ninformation theory. It turns out that information theory and quantum mechanics\nfit together very well. In order to explain their relationship, the review\nbegins with an introduction to classical…

Practical challenges in simulating quantum systems on classical computers have been widely recognized in the quantum physics and quantum chemistry communities over the past century. Although many approximation methods have been introduced, the complexity of quantum mechanics remains hard to appease. The advent of quantum computation brings new pathways to navigate this challenging and complex landscape. By manipulati…

In 2001, all-optical quantum computing became feasible with the discovery that scalable quantum computing is possible using only single-photon sources, linear optical elements, and single-photon detectors. Although it was in principle scalable, the massive resource overhead made the scheme practically daunting. However, several simplifications were followed by proof-of-principle demonstrations, and recent approaches…

Recently a great deal of attention has focused on quantum computation following a sequence of results suggesting that quantum computers are more powerful than classical probabilistic computers. Following Shor's result that factoring and the extraction of discrete logarithms are both solvable in quantum polynomial time, it is natural to ask whether all of NP can be efficiently solved in quantum polynomial time. In thi…

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Trapped ions are among the most promising systems for practical quantum computing (QC). The basic requirements for universal QC have all been demonstrated with ions, and quantum algorithms using few-ion-qubit systems have been implemented. We review the state of the field, covering the basics of how trapped ions are used for QC and their strengths and limitations as qubits. In addition, we discuss what is being done,…