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Register a new userSymmetry constraints on temporal order in measurement-based quantum computation
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We discuss the interdependence of resource state, measurement setting and temporal order in measurement-based quantum computation. The possible temporal orders of measurement events are constrained by the principle that the randomness inherent in quantum measurement should not affect the outcome of the computation. We provide a classification for all temporal relations among measurement events compatible with a given initial stabilizer state and measurement setting, in terms of a matroid. Conversely, we show that classical processing relations necessary for turning the local measurement outcomes into computational output determine the resource state and measurement setting up to local equivalence. Further, we find a symmetry transformation related to local complementation that leaves the temporal relations invariant.
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Quantum computation offers a promising new kind of information processing, where the non-classical features of quantum mechanics can be harnessed and exploited. A number of models of quantum computation exist, including the now well-studied quantum circuit model. Although these models have been shown to be formally equivalent, their underlying elementary concepts and the requirements for their practical realization can differ significantly. The new paradigm of measurement-based quantum computation, where the processing of quantum information takes place by rounds of simple measurements on qubits prepared in a highly entangled state, is particularly exciting in this regard. In this article we discuss a number of recent developments in measurement-based quantum computation in both fundamental and practical issues, in particular regarding the power of quantum computation, the protection against noise (fault tolerance) and steps toward experimental realization. Moreover, we highlight a number of surprising connections between this field and other branches of physics and mathematics.
We show, under natural assumptions for qubit systems, that measurement-based quantum computations (MBQCs) which compute a non-linear Boolean function with high probability are contextual. The class of contextual MBQCs includes an example which is of practical interest and has a super-polynomial speedup over the best known classical algorithm, namely the quantum algorithm that solves the Discrete Log problem.
Measurement-based quantum computation (MBQC) represents a powerful and flexible framework for quantum information processing, based on the notion of entangled quantum states as computational resources. The most prominent application is the one-way quantum computer, with the cluster state as its universal resource. Here we demonstrate the principles of MBQC using deterministically generated graph states of up to 7 qubits, in a system of trapped atomic ions. Firstly we implement a universal set of operations for quantum computing. Secondly we demonstrate a family of measurement-based quantum error correction codes, and show their improved performance as the code length is increased. We show that all our graph states violate a multipartite Bell inequality and are therefore capable of information processing tasks that cannot be described by a local hidden variable model. The methods presented can directly be scaled up to generate graph states of several tens of qubits.
Certain physical systems that one might consider for fault-tolerant quantum computing where qubits do not readily interact, for instance photons, are better suited for measurement-based quantum-computational protocols. Here we propose a measurement-based model for universal quantum computation that simulates the braiding and fusion of Majorana modes. To derive our model we develop a general framework that maps any scheme of fault-tolerant quantum computation with stabilizer codes into the measurement-based picture. As such, our framework gives an explicit way of producing fault-tolerant models of universal quantum computation with linear optics using protocols developed using the stabilizer formalism. Given the remarkable fault-tolerant properties that Majorana modes promise, the main example we present offers a robust and resource efficient proposal for photonic quantum computation.
Measurement based quantum computation (MBQC), which requires only single particle measurements on a universal resource state to achieve the full power of quantum computing, has been recognized as one of the most promising models for the physical realization of quantum computers. Despite considerable progress in the last decade, it remains a great challenge to search for new universal resource states with naturally occurring Hamiltonians, and to better understand the entanglement structure of these kinds of states. Here we show that most of the resource states currently known can be reduced to the cluster state, the first known universal resource state, via adaptive local measurements at a constant cost. This new quantum state reduction scheme provides simpler proofs of universality of resource states and opens up plenty of space to the search of new resource states, including an example based on the one-parameter deformation of the AKLT state studied in [Commun. Math. Phys. 144, 443 (1992)] by M. Fannes et al. about twenty years ago.
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