The Poisson equation has wide applications in many areas of science and engineering. Although there are some quantum algorithms that can efficiently solve the Poisson equation, they generally require a fault-tolerant quantum computer, which is beyond the current technology. We propose a variational quantum algorithm (VQA) to solve the Poisson equation, which can be executed on noisy intermediate-scale quantum devices. In detail, we first adopt the finite-difference method to transform the Poisson equation into a linear system. Then, according to the special structure of the linear system, we find an explicit tensor product decomposition, with only $(2{log}_{2}n+1)$ items, of its coefficient matrix under a specific set of simple operators, where $n$ is the dimension of the coefficient matrix. This implies that the proposed VQA needs fewer quantum measurements, which dramatically reduces the required quantum resources. Additionally, we design observables to efficiently evaluate the expectation values of the simple operators on a quantum computer. Numerical experiments demonstrate that our algorithm can solve the Poisson equation.

Local distinguishability of orthogonal quantum states is an area of active research in quantum information theory. However, most of the relevant results are about local distinguishability in bipartite Hilbert space and very little is known about the multipartite case. In this paper we present a generic method to construct a completable $n$-partite $(n\ge 3)$ product basis with only $2n$ members, which exhibits nonlocality without entanglement with $n$ parties, each holding a system of any finite dimension. We give an effective proof of the nonlocality of the completable multipartite product basis. In addition, we construct another incomplete multipartite product basis with a smaller number of members that cannot be distinguished by local operations and classical communication in a $d_1\otimes d_2\otimes \cdots \otimes d_n$ quantum system, where $n\ge 3$ and $d_i\ge 2$ for $i=1,2,...,n$. The results can lead to a better understanding of the phenomenon of nonlocality without entanglement in any multipartite quantum system.

By adding a parameter θ in M. Jakobi et al’s protocol [Phys. Rev. A 83, 022301 (2011)], we present a flexible quantum-key-distribution-based protocol for quantum private queries. We show that, by adjusting the value of θ, the average number of the key bits Alice obtains can be located on any fixed value the users wanted for any database size. And the parameter k is generally smaller (even k = 1 can be achieved) when θ < π/4, which implies lower complexity of both quantum and classical communications. Furthermore, the users can choose a smaller θ to get better database security, or a larger θ to obtain a lower probability with which Bob can correctly guess the address of Alice’s query.

As a new model for signing quantum messages, arbitrated quantum signature (AQS) has recently received a lot of attention. In this paper we study the cryptanalysis of previous AQS protocols from the aspects of forgery and disavowal. We show that in these protocols the receiver, Bob, can realize existential forgery of the sender's signature under known message attack. Bob can even achieve universal forgery when the protocols are used to sign a classical message. Furthermore, the sender, Alice, can successfully disavow any of her signatures by simple attack. The attack strategies are described in detail and some discussions about the potential improvements of the protocols are given. Finally we also present several interesting topics on AQS protocols that can be studied in future.

In this paper, we mainly study the local indistinguishability of mutually orthogonal product basis quantum states in the high-dimensional quantum systems. In the Hilbert space of $3\otimes 3$, Walgate and Hardy [ Phys. Rev. Lett. 89, 147901 (2002)] presented a very simple proof for nonlocality of nine orthogonal product basis quantum states which are given by Bennett et al. [ Phys. Rev. A 59, 1070 (1999)]. In the quantum system of $d\otimes d$, where $d$ is odd, we construct $d^2$ orthogonal product basis quantum states and prove these states are locally indistinguishable. Then we are able to construct some locally indistinguishable product basis quantum states in the multipartite systems. All these results reveal the phenomenon of “nonlocality without entanglement.”