Abstract
Graph states [5] are an elegant and powerful quantum resource for measurement based quantum computation (MBQC). They are also used for many quantum protocols (error correction, secret sharing, etc.). The main focus of this paper is to provide a structural characterisation of the graph states that can be used for quantum information processing. The existence of a gflow (generalized flow) [8] is known to be a requirement for open graphs (graph, input set and output set) to perform uniformly and strongly deterministic computations. We weaken the gflow conditions to define two new more general kinds of MBQC: uniform equiprobability and constant probability. These classes can be useful from a cryptographic and information point of view because even though we cannot do a deterministic computation in general we can preserve the information and transfer it perfectly from the inputs to the outputs. We derive simple graph characterisations for these classes and prove that the deterministic and uniform equiprobability classes collapse when the cardinalities of inputs and outputs are the same. We also prove the reversibility of gflow in that case. The new graphical characterisations allow us to go from open graphs to graphs in general and to consider this question: given a graph with no inputs or outputs fixed, which vertices can be chosen as input and output for quantum information processing? We present a characterisation of the sets of possible inputs and ouputs for the equiprobability class, which is also valid for deterministic computations with inputs and ouputs of the same cardinality.
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Notes
- 1.
The other branches are taken into account by considering a different set of measurement angles e.g. the branch where all outcomes are \(1\) corresponds to the \(0\)-branch when the set of measurements is \(\{\alpha _v +\pi \}_{v\in O^C}\).
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Acknowledgements
The authors want to thank E. Kashefi for discussions. This work is supported by CNRS-JST Strategic French-Japanese Cooperative Program, and Special Coordination Funds for Promoting Science and Technology in Japan.
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Mhalla, M., Murao, M., Perdrix, S., Someya, M., Turner, P.S. (2014). Which Graph States are Useful for Quantum Information Processing?. In: Bacon, D., Martin-Delgado, M., Roetteler, M. (eds) Theory of Quantum Computation, Communication, and Cryptography. TQC 2011. Lecture Notes in Computer Science(), vol 6745. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-54429-3_12
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DOI: https://doi.org/10.1007/978-3-642-54429-3_12
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