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International Journal of Theoretical Physics

Multiparty Quantum Computation for Summation and Multiplication with Mutually Unbiased Bases

Shu-Xin Lv, Xian-Fang Jiao, Ping Zhou

Published Jun 20, 2019• 33 citations
Publisher
DOI
10.1007/s10773-019-04170-z

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In this paper, a novel quantum secure multiparty summation protocol is proposed based on the mutually unbiased bases of d-level quantum systems. The quantum states of mutually unbiased bases, which are transmitted in a circular pattern, are imposed with the transformation operations of n secrecy owners to encode their secrecies. Security analysis reveals that the proposed protocol can resist outside attacks as well as participant attacks. The proposed protocol is suitable for d-level quantum systems and can calculate the modulo d addition using the integer-by-integer technique. Compared with the quantum secure summation protocols with bit-by-bit calculations, the proposed protocol has a higher calculation efficiency and has more extensive applications compared with the protocols with smaller modulo. Moreover, this protocol can be successfully used for quantum multiparty private comparison and quantum multiparty anonymous voting.
We design mutually unbiased bases (MUBs) in 4, 8, and 16 dimensions, which are compatible with quadrature phase-shift keying (QPSK) optical transmission, based on novel insights into the structure of the MUB set, viewed as the modulation of a Hadamard matrix with a set of phase masks. The new protocol is based on the generation of a pair of MUBs as tensor products of the lowest dimension (2-D) MUBs, followed by novel unbiasing transformations, which operate on pairs of MUBs and make them mutually unbiased among the pairs. We discovered that the unbiasing transformations consist of controlled-phase quantum gates acting on multiple qubits. We further conceive an abstract BB84-like higher order quantum key distribution MUB-based protocol, based on multiple entangled photons.
In this note I point out that (1) pairs of real mutually unbiased bases (i.e., orthonormal bases of R^d) can only exist in dimensions 2 or d where d is a multiple of 4 and that (2) triples of real mutually unbiased bases can only exist for dimensions d that are also squares. For the case d=2^{2m} the maximal number of real MUBs is given by (d+2)/2 (this follows from known results on extremal euclidean line-sets). In the case d=s^2 where s is an even number that is not a power of 2, we have the lower bound N_MOLS(s)+2, where N_MOLS(s) denotes the maximal number of mutually orthogonal Latin square of order s, provided that there exists a Hadamard matrix of size s. It is not known how good this bound is. Moreover, I observe that the question of deciding if there always two real MUBs is equivalent to the Hadamard conjecture.