Quantum contextuality
Recently, quantum contextuality has been explored as a potential tool for quantum computing. In 2013, Robert Raussendorf showed that in general, any Mermin-like, inequality-free proof for quantum contextuality (such as the Kochen-Specker theorem) can be turned into a measurement-based quantum computation. More formally, "under quite natural assumptions for multi-qubit systems, MBQCs (measurement based quantum computations) which compute a non-linear Boolean function with sufficiently high success probability are contextual." Then in 2014, Mark Howard, et. al. showed that applying contextuality to magic state distillation (MSD) provides a means for arriving at universal, fault-tolerant quantum computing. MSD is a process by which a single qubit's polarization is increased along one of several "magic" directions and its state is considered "magic" once its polarization reaches a certain level. In particular, the group proved that states are non-contextual if and only if they cannot be used as inputs for magic state distillation. This provides a selection criteria for proper input states needed in MSD, and by extension aids in the efficient creation of magic states that, when coupled with fault-tolerant operators, allows for universal quantum computing. In both cases, contextuality has a potential advantage over other quantum computational techniques since contextuality itself can be thought of as a theory of information that is built directly into quantum mechanics. Comparatively, more traditional quantum phenomena such as entanglement and interference use delicate physical states which are incredibly susceptible to noise and very difficult to manipulate experimentally.
Continuous-variable quantum information
Occasionally, and somewhat confusingly, the term "continuous quantum computation" is used to refer to a different area of quantum computing: the study of how to use quantum systems having "finite"-dimensional Hilbert spaces to calculate or approximate the answers to mathematical questions involving continuous functions. A major motivation for investigating the quantum computation of continuous functions is that many scientific problems have mathematical formulations in terms of continuous quantities. A second motivation is to explore and understand the ways in which quantum computers can be more capable or powerful than classical ones. The computational complexity of a problem can be quantified in terms of the minimal computational resources necessary to solve it. In quantum computing, resources include the number of qubits available to a computer and the number of queries that can be made to that computer. The classical complexity of many continuous problems is known. Therefore, when the quantum complexity of these problems is obtained, the question as to whether quantum computers are more powerful than classical can be answered. Furthermore, the degree of the improvement can be quantified. In contrast, the complexity of discrete problems is typically unknown. For example, the classical complexity of integer factorization is unknown.