### Abstract

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.

Original language | English |
---|---|

Title of host publication | Theory of Computing 2009 - Proceedings of the Fifteenth Computing |

Subtitle of host publication | The Australasian Theory Symposium, CATS 2009 |

Publication status | Published - Dec 1 2009 |

Event | Theory of Computing 2009 - 15th Computing: The Australasian Theory Symposium, CATS 2009 - Wellington, New Zealand Duration: Jan 20 2009 → Jan 23 2009 |

### Publication series

Name | Conferences in Research and Practice in Information Technology Series |
---|---|

Volume | 94 |

ISSN (Print) | 1445-1336 |

### Other

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

Country | New Zealand |

City | Wellington |

Period | 1/20/09 → 1/23/09 |

### All Science Journal Classification (ASJC) codes

- Computer Networks and Communications
- Computer Science Applications
- Hardware and Architecture
- Information Systems
- Software

## Fingerprint Dive into the research topics of 'Lower bounds on quantum query complexity for read-once decision trees with parity nodes'. Together they form a unique fingerprint.

## Cite this

*Theory of Computing 2009 - Proceedings of the Fifteenth Computing: The Australasian Theory Symposium, CATS 2009*(Conferences in Research and Practice in Information Technology Series; Vol. 94).