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An ordered binary decision diagram (OBDD) is a directed acyclic graph that represents a Boolean function. OBDDs are also known as special cases of oblivious read-once branching programs in the field of complexity theory. Since OBDDs have many nice properties as data structures, they have been extensively studied for decades in both theoretical and practical fields, such as VLSI design, formal verification, machine learning, and combinatorial problems. Arguably, the most crucial problem in using OBDDs is that they may vary exponentially in size depending on their variable ordering (i.e., the order in which the variable are to read) when they represent the same function. Indeed, it is NP hard to find an optimal variable ordering that minimizes an OBDD for a given function. Hence, numerous studies have sought heuristics to find an optimal variable ordering. From practical as well as theoretical points of view, it is also important to seek algorithms that output optimal solutions with lower (exponential) time complexity than trivial brute-force algorithms do. Friedman and Supowit provided a clever deterministic algorithm with time/space complexity O * (3 n ), where n is the number of variables of the function, which is much better than the trivial brute-force bound O * (n!2 n ). This paper shows that a further speedup is possible with quantum computers by demonstrating the existence of a quantum algorithm that produces a minimum OBDD together with the corresponding variable ordering in O * (2.77286 n ) time and space with an exponentially small error. Moreover, this algorithm can be adapted to constructing other minimum decision diagrams such as zero-suppressed BDDs, which provide compact representations of sparse sets and are often used in the field of discrete optimization and enumeration.

One important question for quantum computing is whether a computational gap exists between models that are allowed to use quantum effects and models that are not. Several types of quantum computation models have been proposed, including quantum finite automata and quantum pushdown automata (with a quantum pushdown stack). It has been shown that some quantum computation models are more powerful than their classical counterparts and others are not since quantum computation models are required to obey such restrictions as reversible state transitions. In this paper, we investigate the power of quantum pushdown automata whose stacks are assumed to be implemented as classical devices, and show that they are strictly more powerful than their classical counterparts under the perfect-soundness condition, where perfect-soundness means that an automaton never accepts a word that is not in the language. That is, we show that our model can simulate any probabilistic pushdown automata and also show that there is a non-context-free language which quantum pushdown automata with classical stack operations can recognize with perfect soundness.