On the complete weight distributions of quantum error-correcting codes

In a recent paper, Hu et al. defined the complete weight distributions of quantum codes and proved the Mac Williams identities, and as applications they showed how such weight distributions may be used to obtain the singleton-type and hamming-type bounds for asymmetric quantum codes. In this paper we extend their study much further and obtain several new results concerning the complete weight distributions of quantum codes and applications. In particular, we provide a new proof of the Mac Williams identities of the complete weight distributions of quantum codes. We obtain new information about the weight distributions of quantum MDS codes and the double weight distribution of asymmetric quantum MDS codes. We get new identities involving the complete weight distributions of two different quantum codes. We estimate the complete weight distributions of quantum codes under special conditions and show that quantum BCH codes by the Hermitian construction from primitive, narrow-sense BCH codes satisfy these conditions and hence these estimate applies.

Genuine Einstein-Podolsky-Rosen steering of generalized three-qubit states via unsharp measurements

We aim to explore all possible scenarios of(1→2)(where one wing is untrusted and the others two wings are trusted)and(2→1)(where two wings are untrusted,and one wing is trusted)genuine tripartite Einstein-Podolsky-Rosen(EPR)steering.The generalized Greenberger-Horne-Zeilinger(GHZ)state is shared between three spatially separated parties,Alice,Bob and Charlie.In both(1→2)and(2→1),we discuss the untrusted party and trusted party performing a sequence of unsharp measurements,respectively.For each scenario,we deduce an upper bound on the number of sequential observers who can demonstrate genuine EPR steering through the quantum violation of tripartite steering inequality.The results show that the maximum number of observers for the generalized GHZ states can be the same with that of the maximally GHZ state in a certain range of state parameters.Moreover,both the sharpness parameters range and the state parameters range in the scenario of(1→2)steering are larger than those in the scenario of(2→1)steering.

Designing a Secure Round Function Based on Chaos

A round function based on chaos is designed combining Feistel structure’s pseudo-randomness, chaotic system’s parameter sensitivity and image data characteristics. The round function composes of two parts--data transformation based on Feistel(abbreviated as FST) and sampling output based on chaos(abbreviated as SMP). FST bases on Feistel structure and several efficient operations including bitwise xor, permutation and circulating shift. SMP is a chaos based pseudo-random sampling algorithm. It is from theoretical analysis that the round function is a pseudo-random function. The upper bounds of the average maximum differential probability and average maximum linear probability are p^2 and q^2 respectively. Finally, the good pseudo-randomness of the round function is examined with the NIST random test. The design of this round function provides an important cryptographic component for the design of chaotic image encryption algorithm.

Methods for solving equations with errors based on the HHL algorithm

To solve polynomial systems,Harrow,Hassidim,and Lloyd(HHL)proposed a quantum algorithm called HHL algorithm.Based on the HHL algorithm,Chen et al.presented an algorithm,the solving the Boolean solutions of polynomial systems(PoSSoB)algorithm.Furthermore,Ding et al.introduced the Boolean Macaulay matrix and analyzed the lower bound on the condition number.Inspired by Ding et al.’s research,several related algorithms are proposed in this paper.First,the improved PoSSoB algorithm using the Boolean Macaulay matrix is proved to have lower complexity.Second,for solving equations with errors,a quantum algorithm for the max-polynomial system solving(Max-PoSSo)problem is proposed based on the improved PoSSoB algorithm.Besides,the Max-PoSSo algorithm is extended to the learning with errors(LWE)problem and its special case,the learning parity with noise(LPN)problem,providing a quantitative criterion,the condition number,for the security of these basic problems.