Vol.21 No.1&2, February 1, 2021

Quantum Alice and silent Bob: Qubit-based Quantum Key Recycling with almost no classical communication (pp0001-0018)
         Daan Leermakers and Boris Skoric
We answer an open question about Quantum Key Recycling (QKR): Is it possible to put the message entirely in the qubits without increasing the number of qubits compared to existing QKR schemes? We show that this is indeed possible. We introduce a prepare-and-measure QKR protocol where the communication from Alice to Bob consists entirely of qubits. As usual, Bob responds with an authenticated one-bit accept/reject classical message. Compared to Quantum Key Distribution (QKD), QKR has reduced round complexity. Compared to previous qubit-based QKR protocols, our scheme has far less classical communication. We provide a security proof in the universal composability framework and find that the communication rate is asymptotically the same as for QKD with one-way postprocessing.

A limit distribution for a quantum walk driven by a five-diagonal unitary matrix (pp0019-0036)
         Takuya Machida
In this paper, we work on a quantum walk whose system is manipulated by a five-diagonal unitary matrix, and present long-time limit distributions. The quantum walk launches off a location and delocalizes in distribution as its system is getting updated. The five-diagonal matrix contains a phase term and the quantum walk becomes a standard coined walk when the phase term is fixed at special values. Or, the phase term gives an effect on the quantum walk. As a result, we will see an explicit form of a long-time limit distribution for a quantum walk driven by the matrix, and thanks to the exact form, we understand how the quantum walker approximately distributes in space after the long-time evolution has been executed on the walk.

Homogeneous open quantum walks on the line: criteria for site recurrence and absorption (pp0037-0058)
         Thomas S. Jacq and Carlos F. Lardizabal
In this work, we study open quantum random walks,  as described by S. Attal et al.\cite{attal}. These objects are given in terms of completely positive maps acting on trace-class operators, leading to one of the simplest open quantum versions of the recurrence problem for classical, discrete-time random walks. This work focuses on obtaining criteria for site recurrence of nearest-neighbor, homogeneous walks on the integer line, with the description presented here making use of recent results of the theory of open walks, most particularly regarding reducibility properties  of the operators involved. This allows us to obtain a complete criterion for site recurrence in the case for which the internal degree of freedom of each site (coin space) is of dimension 2. We also present the analogous result for irreducible walks with an internal degree of arbitrary finite dimension and the absorption problem for walks on the semi-infinite line.

Algorithms for finding the maximum clique based on continuous time quantum walks (pp0059-0079)
         Xi Li, Mingyou Wu, Hanwu Chen, and Zhibao Liu
In this work, the application of continuous time quantum walks (CTQW) to the Maximum Clique (MC) problem was studied. Performing CTQW on graphs can generate distinct periodic probability amplitudes for different vertices. We found that the intensities of the probability amplitudes at some frequencies imply the clique structure of special kinds of graphs. Recursive algorithms with time complexity $O(N^6)$ in classical computers were proposed to determine the maximum clique. We have experimented on random graphs where each edge exists with different probabilities. Although counter examples were not found for random graphs, whether these algorithms are universal is beyond the scope of this work.

Quantum algorithmic differentiation (pp0080-0094)
         Giuseppe Colucci and Francesco Giacosa
In this work we present an algorithm to perform algorithmic differentiation in the context of quantum computing. We present two versions of the algorithm, one which is fully quantum and one which employees a classical step (hybrid approach). Since the implementation of elementary functions is already possible on quantum computers, the scheme that we propose can be easily applied. Moreover, since some steps (such as the CNOT operator) can (or will be) faster on a quantum computer than on a classical one, our procedure may ultimately emonstrate that quantum algorithmic differentiation has an advantage relative to its classical counterpart.

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