Synthesis of quantum circuits based on incompletely specified functions and if-decision diagrams

The problem of synthesis and optimisation of logical reversible and quantum circuits from functional descriptions represented as decision diagrams is considered. It is one of the key problems being solved with the aim of creating quantum computing technology and quantum computers. A new method of stepwise transformation of the initial functional specification to a quantum circuit is proposed, which provides for the following project states: reduced ordered binary decision diagram, if-decision diagram, functional if-decision diagram, reversible circuit and quantum circuit. The novelty of the method consists in extending the Shannon and Davio expansions of a Boolean function on a single variable to the expansions of the same Boolean function on another function with obtaining decomposition products that are represented by incompletely defined Boolean functions. Uncertainty in the decomposition products gives remarkable opportunities for minimising the graph representation of the specified function. Instead of two outgoing branches of the binary diagram vertex, three outgoing branches of the if-diagram vertex are generated, which increase the level of parallelism in reversible and quantum circuits. For each transformation step, appropriate mapping rules are proposed that reduce the number of lines, gates and the depth of the reversible and quantum circuit. The comparison of new results with the results given by the known method of mapping the vertices of binary decision diagram into cascades of reversible and quantum gates shows a significant improvement in the quality of quantum circuits that are synthesised by the proposed method.

Logic-Based Benders Decomposition and Binary Decision Diagram Based Approaches for Stochastic Distributed Operating Room Scheduling

The distributed operating room (OR) scheduling problem aims to find an assignment of surgeries to ORs across collaborating hospitals that share their waiting lists and ORs. We propose a stochastic extension of this problem where surgery durations are considered to be uncertain. In order to obtain solutions for the challenging stochastic model, we use sample average approximation and develop two enhanced decomposition frameworks that use logic-based Benders (LBBD) optimality cuts and binary decision diagram based Benders cuts. Specifically, to the best of our knowledge, deriving LBBD optimality cuts in a stochastic programming context is new to the literature. Our computational experiments on a hospital data set illustrate that the stochastic formulation generates robust schedules and that our algorithms improve the computational efficiency. Summary of Contribution: We propose a new model for an important problem in healthcare scheduling, namely, stochastic distributed operating room scheduling, which is inspired by a current practice in Toronto, Ontario, Canada. We develop two decomposition methods that are computationally faster than solving the model directly via a state-of-the-art solver. We present both some theoretical results for our algorithms and numerical results for the evaluation of the model and algorithms. Compared with its deterministic counterpart in the literature, our model shows improvement in relevant evaluation metrics for the underlying scheduling problem. In addition, our algorithms exploit the structure of the model and improve its solvability. Those algorithms also have the potential to be used to tackle other planning and scheduling problems with a similar structure.

A method for transformation from dynamic fault tree to binary decision diagram

Dynamic fault tree (DFT) is a powerful modeling approach for reliability analysis of complex system with dynamic failure behaviors. In reality, the tree structure may be highly coupled either by shared basic events or by the high-level dynamic gates. Currently, the application of sequential binary decision diagram (SBDD)-based method for quantitative analysis of such highly coupled DFTs is mainly limited to DFTs whose dynamic gates locate in the bottom of the tree. Moreover, there is no efficient way dealing with the dependencies among different nodes of a SBDD 1-path. This paper makes an improvement to the SBDD-based approach. A generation procedure is proposed to directly construct the binary decision diagram (BDD) model for a DFT with arbitrary tree structure. During the construction, the sequential-dependent information of the tree is derived as several BDD nodes, each indicates a binary-sequential event representing the sequence of two occurred basic events. A topological sorting is applied on each 1-path of the resultant BDD to obtain its contained disjoint cut sequences. Based on this, both qualitative and quantitative analysis can be performed on the DFT with no limitations on tree structure, and its minimal cut sequence set (CSS) is obtained as disjoint. Examples are provided for verification and comparison, and the results illustrate the merits of the proposed approach.