Wavelength Conversion in All-Optical Networks with Shortest-Path Routing

We consider all-optical networks with shortest-path routing that use wavelength-division multiplexing and employ wavelength conversion at specific nodes in order to maximize their capacity usage. We present efficient algorithms for deciding whether a placement of wavelength converters allows the network to run at maximum capacity, and for finding an optimal wavelength assignment when such a placement of converters is known. Our algorithms apply to both undirected and directed networks. Furthermore, we show that the problem of designing such networks, i.e., finding an optimal placement of converters, is MAX SNP-hard in both the undirected and the directed case. Finally, we give a linear-time algorithm for finding an optimal placement of converters in undirected triangle-free networks, and show that the problem remains NP-hard in bidirected triangle-free planar network

Optimal Space-Depth Trade-Off of CNOT Circuits in Quantum Logic Synthesis

Due to the decoherence of the state-of-the-art physical implementations of quantum computers, it is essential to parallelize the quantum circuits to reduce their depth. Two decades ago, Moore et al. demonstrated that additional qubits (or ancillae) could be used to design "shallow" parallel circuits for quantum operators. They proved that any $n$-qubit CNOT circuit could be parallelized to $O(\log n)$ depth, with $O(n^2)$ ancillae. However, the near-term quantum technologies can only support limited amount of qubits, making space-depth trade-off a fundamental research subject for quantum-circuit synthesis. In this work, we establish an asymptotically optimal space-depth trade-off for the design of CNOT circuits. We prove that for any $m\geq0$, any $n$-qubit CNOT circuit can be parallelized to $O\left(\max \left\{\log n, \frac{n^{2}}{(n+m)\log (n+m)}\right\} \right)$ depth, with $O(m)$ ancillae. We show that this bound is tight by a counting argument, and further show that even with arbitrary two-qubit quantum gates to approximate CNOT circuits, the depth lower bound still meets our construction, illustrating the robustness of our result. Our work improves upon two previous results, one by Moore et al. for $O(\log n)$-depth quantum synthesis, and one by Patel et al. for $m = 0$: for the former, we reduce the need of ancillae by a factor of $\log^2 n$ by showing that $m=O(n^2/\log^2 n)$ additional qubits suffice to build $O(\log n)$-depth, $O(n^2/\log n)$ size --- which is asymptotically optimal --- CNOT circuits; for the later, we reduce the depth by a factor of $n$ to the asymptotically optimal bound $O(n/\log n)$. Our results can be directly extended to stabilizer circuits using an earlier result by Aaronson et al. In addition, we provide relevant hardness evidences for synthesis optimization of CNOT circuits in term of both size and depth.Comment: 25 pages, 5 figures. Fixed several minor typos and a mistake about CNOT+Rz circui

Approximating fluid schedules in packet-switched networks

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2004.This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.Includes bibliographical references (p. 145-151).We consider a problem motivated by the desire to provide exible, rate-based, quality of service guarantees for packets sent over switches and switch networks. Our focus is solving a type of on-line, traffic scheduling problem, whose input at each time step is a set of desired traffic rates through the switch network. These traffic rates in general cannot be exactly achieved since they treat the incoming data as fluid, that is, they assume arbitrarily small fractions of packets can be transmitted at each time step. The goal of the traffic scheduling problem is to closely approximate the given sequence of traffic rates by a sequence of switch uses throughout the network in which only whole packets are sent. We prove worst-case bounds on the additional delay and buffer use that result from using such an approximation. These bounds depend on the network topology, the resources available to the scheduler, and the types of fluid policy allowed.by Michael Aaron Rosenblum.Ph.D