### 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 |
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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 |
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Volume | 94 |

ISSN (Print) | 1445-1336 |

### Other

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

City | Wellington |

Period | 1/20/09 → 1/23/09 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

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

### 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).

**Lower bounds on quantum query complexity for read-once decision trees with parity nodes.** / Fukuhara, Hideaki; Takimoto, Eiji.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*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, Theory of Computing 2009 - 15th Computing: The Australasian Theory Symposium, CATS 2009, Wellington, New Zealand, 1/20/09.

}

TY - GEN

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

AU - Fukuhara, Hideaki

AU - Takimoto, Eiji

PY - 2009/12/1

Y1 - 2009/12/1

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

ER -