Improving Gate-Level Simulation of Quantum Circuits

Simulating quantum computation on a classical computer is a difficult problem. The matrices representing quantum gates, and the vectors modeling qubit states grow exponentially with an increase in the number of qubits. However, by using a novel data structure called the Quantum Information Decision Diagram (QuIDD) that exploits the structure of quantum operators, a useful subset of operator matrices and state vectors can be represented in a form that grows polynomially with the number of qubits. This subset contains, but is not limited to, any equal superposition of n qubits, any computational basis state, n-qubit Pauli matrices, and n-qubit Hadamard matrices. It does not, however, contain the discrete Fourier transform (employed in Shor's algorithm) and some oracles used in Grover's algorithm. We first introduce and motivate decision diagrams and QuIDDs. We then analyze the runtime and memory complexity of QuIDD operations. Finally, we empirically validate QuIDD-based simulation by means of a general-purpose quantum computing simulator QuIDDPro implemented in C++. We simulate various instances of Grover's algorithm with QuIDDPro, and the results demonstrate that QuIDDs asymptotically outperform all other known simulation techniques. Our simulations also show that well-known worst-case instances of classical searching can be circumvented in many specific cases by data compression techniques. PACS: 03.67.Lx, 03.65.Fd, 03.65.Vd, 07.05.Bx     

Cell shape changes and the mechanism of inversion in Volvox

Inversion is a dominant aspect of morphogenesis in Volvox. In this process, the hollow, spheroidal Volvox embryo turns inside-out through a small opening called the phialopore to bring flagella from its inner to its outer surface. Analyses of intact, sectioned, and fragmented embryos by light, scanning electron, and transmission electron microscopy, suggest that shape changes preprogrammed into the cells cause inversion. First, cells throughout the embryo change from pear to spindle shape, which causes the embryo to contract and the phialopore to open. Then cells adjacent to the phialopore become flask-shaped, with long, thin stalks at their outer ends. Simultaneously, the cytoplasmic bridges joining all adjacent cells migrate from the midpoint of the cells to the stalk tips. Together, these changes cause the lips of cells at the phialopore margin to curl outward. Now cells progressively more distal to the phialopore become flask-shaped while the more proximal cells become columnar, causing the lips to curl progressively further over the surface of the embryo until the latter has turned completely inside-out. Fine structural analysis reveals a peripheral cytoskeleton of microtubules that is apparently involved in cellular elongation. Cell clusters isolated before inversion undergo a similar program of shape changes; this suggests that the changes in cellular shape are the cause rather than an effect of the inversion process.