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Bounds on Oblivious Multiparty Quantum Communication Complexity

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LATIN 2022: Theoretical Informatics (LATIN 2022)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13568))

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Abstract

The main conceptual contribution of this paper is investigating quantum multiparty communication complexity in the setting where communication is oblivious. This requirement, which to our knowledge is satisfied by all quantum multiparty protocols in the literature, means that the communication pattern, and in particular the amount of communication exchanged between each pair of players at each round is fixed independently of the input before the execution of the protocol. We show, for a wide class of functions, how to prove strong lower bounds on their oblivious quantum k-party communication complexity using lower bounds on their two-party communication complexity. We apply this technique to prove tight lower bounds for all symmetric functions with AND gadget, and in particular obtain an optimal \(\varOmega (k\sqrt{n})\) lower bound on the oblivious quantum k-party communication complexity of the n-bit Set-Disjointness function. We also show the tightness of these lower bounds by giving (nearly) matching upper bounds.

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Notes

  1. 1.

    This way of distributing inputs is called the number-in-hand model. There exists another model, called the number-on-the-forehead model, which we do not consider in this paper.

  2. 2.

    We will show later (in Theorem 3 in Sect. 5) how to obtain an improved \(O(k\sqrt{n})\) upper bound based on the protocol from [1].

  3. 3.

    Note that in the two-party setting, the notions of oblivious communication complexity and non-oblivious communication complexity essentially coincide, since any non-oblivious communication protocol can be converted into an oblivious communication protocol by increasing the complexity by a factor at most two. To see this, without loss of generality assume that each player sends only one qubit at each round.

  4. 4.

    Trivially, players can send classical messages using quantum communication in this communication model.

  5. 5.

    To show \({\text {QCC}}^{2M}(f, \varepsilon ) \le {\text {QCC}}^M_\textrm{Co}(f, \varepsilon )\), assign \(P_1\) the role of the coordinator. To show \({\text {QCC}}^M_\textrm{Co}(f, \varepsilon ) \le 2{\text {QCC}}^M(f, \varepsilon )\), consider the coordinator only passes messages without performing any operation.

  6. 6.

    Although a function \(f: \{0, 1\}^n \rightarrow \{0, 1\}\) is generally said to be symmetric when any permutation on the input does not change the value of f, in this paper we focus on functions of the form \(f : \{0, 1\}^n \times \{0, 1\}^n \rightarrow \{0, 1\}\), and use the same definition for symmetric functions (predicates) as in [24].

  7. 7.

    In Fig. 1, \(U_i\) and \(U_i^\textrm{out}\) denote classical or quantum operations and \(\otimes \) denotes the operation of attaching registers. \(U_i^\textrm{out}\) usually includes measurement operations to output \(f_k(x_1, x_2, x_3)\).

  8. 8.

    If \(n^{1/3}\) is not an integer, inputs are embedded to a larger cube of size \(\lceil n^{1/3}\rceil ^3\). In this case, for any coordinate \(i \in \lceil n^{1/3}\rceil ^3 \setminus [n]\), the i-th inputs \(x_i\) and \(y_i\) are set to 0.

  9. 9.

    Here we use the fact that for any \(n_0 \le \frac{n}{2}\), \(\log \left( \varSigma _{m = n - n_0 + 1}^n \left( {\begin{array}{c}n\\ m\end{array}}\right) \right) = O(n_0 \log n)\).

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Acknowledgement

FLG was partially supported by JSPS KAKENHI grants Nos. JP16H01705, JP19H04066, JP20H00579, JP20H04139 and by MEXT Q-LEAP grants Nos. JPMXS0118067394 and JPMXS0120319794. DS would like to take this opportunity to thank the “Nagoya University Interdisciplinary Frontier Fellowship” supported by JST and Nagoya University.

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  1. Department of Mathematics, Nagoya University, Furocho, Chikusa-ku, Nagoya, 464-860, Japan

    François Le Gall & Daiki Suruga

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Correspondence to Daiki Suruga .

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  1. Universidad Nacional Autónoma de México, Mexico City, Mexico

    Armando Castañeda

  2. Centro de investigación y de Estudios Avanzados, Mexico City, Mexico

    Francisco Rodríguez-Henríquez

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Le Gall, F., Suruga, D. (2022). Bounds on Oblivious Multiparty Quantum Communication Complexity. In: Castañeda, A., Rodríguez-Henríquez, F. (eds) LATIN 2022: Theoretical Informatics. LATIN 2022. Lecture Notes in Computer Science, vol 13568. Springer, Cham. https://doi.org/10.1007/978-3-031-20624-5_39

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  • DOI: https://doi.org/10.1007/978-3-031-20624-5_39

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