Quantum Walks on the Line with Phase Parameters

In this paper, a study on discrete-time coined quantum walks on the line is presented. Clear mathematical foundations are still lacking for this quantum walk model. As a step towards this objective, the following question is being addressed: {\it Given a graph, what is the probability that a quantum walk arrives at a given vertex after some number of steps?} This is a very natural question, and for random walks it can be answered by several different combinatorial arguments. For quantum walks this is a highly non-trivial task. Furthermore, this was only achieved before for one specific coin operator (Hadamard operator) for walks on the line. Even considering only walks on lines, generalizing these computations to a general SU(2) coin operator is a complex task. The main contribution is a closed-form formula for the amplitudes of the state of the walk (which includes the question above) for a general symmetric SU(2) operator for walks on the line. To this end, a coin operator with parameters that alters the phase of the state of the walk is defined. Then, closed-form solutions are computed by means of Fourier analysis and asymptotic approximation methods. We also present some basic properties of the walk which can be deducted using weak convergence theorems for quantum walks. In particular, the support of the induced probability distribution of the walk is calculated. Then, it is shown how changing the parameters in the coin operator affects the resulting probability distribution.Comment: In v2 a small typo was fixed. The exponent in the definition of N_j in Theorem 3 was changed from -1/2 to 1. 20 pages, 3 figures. Presented at 10th Asian Conference on Quantum Information Science (AQIS'10). Tokyo, Japan. August 27-31, 201

DDMF: An Efficient Decision Diagram Structure for Design Verification of Quantum Circuits under a Practical Restriction

Recently much attention has been paid to quantum circuit design to prepare for the future "quantum computation era." Like the conventional logic synthesis, it should be important to verify and analyze the functionalities of generated quantum circuits. For that purpose, we propose an efficient verification method for quantum circuits under a practical restriction. Thanks to the restriction, we can introduce an efficient verification scheme based on decision diagrams called Decision Diagrams for Matrix Functions (DDMFs). Then, we show analytically the advantages of our approach based on DDMFs over the previous verification techniques. In order to introduce DDMFs, we also introduce new concepts, quantum functions and matrix functions, which may also be interesting and useful on their own for designing quantum circuits.Comment: 15 pages, 14 figures, to appear IEICE Trans. Fundamentals, Vol. E91-A, No.1

p42MAPK-mediated phosphorylation of xEIAP/XLX in Xenopus cytostatic factor-arrested egg extracts

BACKGROUND: BIR family proteins are evolutionarily conserved anti-apoptotic molecules. One member of Xenopus BIR family proteins, xEIAP/XLX, is a weak apoptosis inhibitor and rapidly degraded in a cell-free apoptotic execution system derived from interphase egg extracts. However, unfertilized eggs are naturally arrested at the metaphase of meiosis II by the concerted activities of Mos-MEK-p42MAPK-p90Rsk kinase cascade (cytostatic factor pathway) and many mitotic kinases. Previous studies suggest that cytostatic factor-arrested egg extracts are more resistant to spontaneous apoptosis than interphase egg extracts in a p42MAPK-dependent manner. We tested whether xEIAP/XLX might be phosphorylated in cytostatic factor-arrested egg extracts, and also examined whether xEIAP/XLX could be functionally regulated by phosphorylation. RESULTS: We found that p42MAPK was the major kinase phosphorylating xEIAP/XLX in cytostatic factor-arrested egg extracts, and three Ser residues (Ser 235/251/254) were identified as p42MAPK-mediated phosphorylation sites. We characterized the behaviors of various xEIAP/XLX mutants that could not be phosphorylated by p42MAPK. However, neither protein stability nor anti-apoptotic ability of xEIAP/XLX was significantly altered by the substitution of Ser with either Ala or Asp at these three sites. CONCLUSION: xEIAP/XLX is physiologically phosphorylated by p42MAPK in Xenopus unfertilized eggs. However, this protein may not serve as an essential mediator of p42MAPK-dependent anti-apoptotic activity