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Determinism and Computational Power of Real Measurement-Based Quantum Computation

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Fundamentals of Computation Theory (FCT 2017)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10472))

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Abstract

Measurement-based quantum computing (MBQC) is a universal model for quantum computation. The combinatorial characterisation of determinism in this model, powered by measurements, and hence, fundamentally probabilistic, is the cornerstone of most of the breakthrough results in this field. The most general known sufficient condition for a deterministic MBQC to be driven is that the underlying graph of the computation has a particular kind of flow called Pauli flow. The necessity of the Pauli flow was an open question. We show that Pauli flow is not necessary, providing several counter examples. We prove however that Pauli flow is necessary for determinism in the real MBQC model, an interesting and useful fragment of MBQC.

We explore the consequences of this result for real MBQC and its applications. Real MBQC and more generally real quantum computing is known to be universal for quantum computing. Real MBQC has been used for interactive proofs by McKague. The two-prover case corresponds to real-MBQC on bipartite graphs. While (complex) MBQC on bipartite graphs are universal, the universality of real MBQC on bipartite graphs was an open question. We show that real bipartite MBQC is not universal proving that all measurements of real bipartite MBQC can be parallelised leading to constant depth computations. As a consequence, McKague’s techniques cannot lead to two-prover interactive proofs.

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Notes

  1. 1.

    A completely positive trace-preserving map describes the evolution of a quantum system which state is represented by a density matrix. See for instance [15] for details.

  2. 2.

    In both cases the unique measurement consists of measuring a qubit in state \(\left| + \right\rangle \) according to the observable \(-X\) which produces the signal \(s_1=1\) with probability 1.

  3. 3.

    In [4], an example of deterministic MBQC with no Pauli flow is given. This is however not a counter example to the necessity of the Pauli flow as the example is not robustly deterministic. More precisely not all the branches of computation occur with the same probability: with the notation of Fig. 8 in [4] if measurements of qubits 4, 6, 8 produce the outcome 0, then the measurement of qubit 10 produces the outcome 0 with probability 1.

  4. 4.

    The proof of Theorem 3 is available in the pre-print version of the present article [16].

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Authors and Affiliations

  1. CNRS, LORIA, Université de Lorraine, Inria-Carte, Nancy, France

    Simon Perdrix

  2. LORIA, CNRS, Université de Lorraine, Inria-Caramba, Lycée Poincaré, Nancy, France

    Luc Sanselme

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  1. Simon Perdrix
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  2. Luc Sanselme
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Correspondence to Simon Perdrix .

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Editors and Affiliations

  1. CNRS, University of Bordeaux, Talence cedex, France

    Ralf Klasing

  2. LaBRI, University of Bordeaux, Talence cedex, France

    Marc Zeitoun

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Perdrix, S., Sanselme, L. (2017). Determinism and Computational Power of Real Measurement-Based Quantum Computation. In: Klasing, R., Zeitoun, M. (eds) Fundamentals of Computation Theory. FCT 2017. Lecture Notes in Computer Science(), vol 10472. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-55751-8_31

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