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Communication in Engineered Quantum Networks

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Quantum State Transfer and Network Engineering

Part of the book series: Quantum Science and Technology ((QST))

Abstract

We review the engineering of passive quantum networks for performing fundamental quantum communications tasks, such as the transfer, routing, and splitting of signals that are associated with quantum states. After an introduction to fundamental concepts and notions, the problem of quantum state transfer is discussed for networks of various physical and logical topologies. The discussion is cast in terms of a unified theoretical formalism, which is perfectly suited to addressing the problem in the context of various physical realizations.

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Notes

  1. 1.

    For a d-dimensional system the unit of quantum information is the qudit.

  2. 2.

    The energy separation of successive orbitals in a node is assumed to be much larger than the coupling constants between adjacent nodes, as well as than the typical energy scales of various excitation mechanisms.

  3. 3.

    It should be sufficiently large with respect to decoherence mechanisms, but at the same time sufficiently small with respect to other system parameters e.g., the energy separation between successive orbitals, so that the dynamics are restricted to equivalent orbitals only.

  4. 4.

    In a classical setting, this can be achieved by means of the so-called internet-protocol (IP) addresses. As the message arrives at each node, the corresponding device checks the destination address contained in the message to see if it matches its own address. If the two addresses match the device processes the message, otherwise it does nothing. Clearly, no addressing is required in a PP topology since any data transmitted from one node is intended for the other node.

  5. 5.

    For instance, if the network is viewed as part of a larger fault-tolerant quantum-information processor, the probability of errors at the end of the transfer must be below ≈10−4 [1].

  6. 6.

    For the sake of simplicity in Eq. (2.30) we assume that the labels of the source and the destination nodes are successive. This is not important for the present formalism, but in general one can always relabel the sites in a given network so that the permutation is of the form (2.30).

  7. 7.

    For more general mappings, one has to relax the constraint of the evolution operator being a permutation (e.g., see [13]).

  8. 8.

    A closed cycle is a permutation or sub-permutation, which cannot be decomposed further.

  9. 9.

    Clearly, the labelling of the basis states in \(\mathbb{B}_{j}\) is to some extent ambiguous. One may consider any ordering for which Eq. (2.33) yields a complete set of eigenstates of \(\hat{\mathcal{P}}_{j}\). In this case, the ambiguity is unimportant, and does not affect our approach.

  10. 10.

    Thereby the notation (2, 3) means that starting from the original site ordering of the sites {1, 2, 3, 4}, the second site is replaced by the third and the third site by the second.

  11. 11.

    We have simplified the notation relative to the previous section, but the overall approach remains the same.

  12. 12.

    We focus on the case of Hamiltonians with non-degenerate spectrum. The case of degenerate spectrum can be treated similarly and leads to NN-type Hamiltonians with vanishing couplings i.e., to a broken network and thus QST from the first to the last site is impossible.

  13. 13.

    Recall here that, in view of the previous discussion, it is sufficient to consider the transformation of the basis states for our purposes.

  14. 14.

    In the same fashion (i.e., by setting U j, ξ j, ξ →∞) one can formally describe the Pauli principle, which does not allow two electrons with the same state, to occupy the same site.

  15. 15.

    As long as we are interested in networks that are insensitive to different degrees of freedom, the subspaces \(\mathbb{H}_{2}^{(<)}\) and \(\mathbb{H}_{2}^{(>)}\) are equivalent, and the case of s 1 > s 2 is covered by the present discussion on the case s 1 < s 2.

  16. 16.

    By contrast to passive networks of logical bus topology where the state is transferred successively to different destination nodes, here the transfer pertains to one of the two available output nodes, and the choice is performed in a controlled manner.

  17. 17.

    As we have seen, the main effect of the unperturbed Hamiltonians \(\hat{\mathcal{H}}_{k}^{(0)}\) is the introduction of an accumulated phase at the end of the transfer which, however, is fixed and known in the absence of disorder and other imperfections. Hence, for the time being we focus on the interaction part of the Hamiltonian.

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Acknowledgements

I. J. received funding from MSM 6840770039, RVO 68407700 and GACR 13-33906S. A. H. was supported by the Grant Agency of the Czech Technical University in Prague, grant No. SGS13/217/OHK4/3T/14.

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Authors and Affiliations

  1. Foundation for Research and Technology – Hellas, Institute of Electronic Structure and Laser, 1385, GR-711 10, Heraklion, Greece

    Georgios M. Nikolopoulos

  2. Department of Physics, University of Strathclyde, 107 Rottenrow, Glasgow, G4 0NG, Scotland, UK

    Thomas Brougham

  3. Faculty of Nuclear Sciences and Physical Engineering, Department of Physics, Czech Technical University in Prague, Břehová 7, Praha 1, Staré Město, 115 19, Czech Republic

    Antonin Hoskovec & Igor Jex

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  1. Georgios M. Nikolopoulos
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  2. Thomas Brougham
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Correspondence to Georgios M. Nikolopoulos .

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  1. Institute of Electronic Structure and Laser, Foundation for Research and Technology - Hellas, Heraklion, Greece

    Georgios M. Nikolopoulos

  2. Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University Department of Physics, Prague, Czech Republic

    Igor Jex

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Nikolopoulos, G.M., Brougham, T., Hoskovec, A., Jex, I. (2014). Communication in Engineered Quantum Networks. In: Nikolopoulos, G., Jex, I. (eds) Quantum State Transfer and Network Engineering. Quantum Science and Technology. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-39937-4_2

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  • DOI: https://doi.org/10.1007/978-3-642-39937-4_2

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