TY - GEN

T1 - Lower bounds on quantum query complexity for read-once decision trees with parity nodes

AU - Fukuhara, Hideaki

AU - Takimoto, Eiji

PY - 2009

Y1 - 2009

N2 - We introduce a complexity measure for decision trees called the soft rank, which measures how wellbalanced a given tree is. The soft rank is a somehow relaxed variant of the rank. Among all decision trees of depth d, the complete binary decision tree (the most balanced tree) has maximum soft rank √d, the decision list (the most unbalanced tree) has minimum soft rank √d, and any other trees have soft rank between d and d. We show that, for any decision tree T in some class G of decision trees which includes all read-once decision trees, the soft rank of T is a lower bound on the quantum query complexity of the Boolean function that T represents. This implies that for any Boolean function f that is represented by a decision tree in G, the deterministic query complexity of f is only quadratically larger than the quantum query complexity of f.

AB - We introduce a complexity measure for decision trees called the soft rank, which measures how wellbalanced a given tree is. The soft rank is a somehow relaxed variant of the rank. Among all decision trees of depth d, the complete binary decision tree (the most balanced tree) has maximum soft rank √d, the decision list (the most unbalanced tree) has minimum soft rank √d, and any other trees have soft rank between d and d. We show that, for any decision tree T in some class G of decision trees which includes all read-once decision trees, the soft rank of T is a lower bound on the quantum query complexity of the Boolean function that T represents. This implies that for any Boolean function f that is represented by a decision tree in G, the deterministic query complexity of f is only quadratically larger than the quantum query complexity of f.

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M3 - Conference contribution

AN - SCOPUS:84864018670

SN - 9781920682750

T3 - Conferences in Research and Practice in Information Technology Series

BT - Theory of Computing 2009 - Proceedings of the Fifteenth Computing

T2 - Theory of Computing 2009 - 15th Computing: The Australasian Theory Symposium, CATS 2009

Y2 - 20 January 2009 through 23 January 2009

ER -