An Introduction to Quantum Complexity Theory

Abstract

1985年にD. Deutschは,いわゆる量子並列計算を行うことができるTuring機械として,量子Turing機械(QTMと略す)を導入した.その後,1994年にP. Shorが,QTMは多項式時間内に任意に小さな誤り確率で,整数を因数分解できることを示した.QTMに基づく計算量理論を量子計算量理論という.本論では,最初にQTMと,EQP,BQP,ZQP等の主要な量子計算量クラスの定義を述べる.次に,この分野ですでに知られている結果と,主要な未解決問題を紹介する.

In 1985, D. Deutsch introduced quantum Turing machines (QTMs for short) as Turing machines which can perform so called quantum parallel computations. Then, in 1994, P. Shor showed that QTM can factor integers with arbitrary small error probability in polynomial time. The quantum complexity theory is the computational complexity theory based on QTMs. In this paper, we first review the definitions of the QTM and major quantum complexity classes EQP, BQP, ZQP, etc. Then, we present the known results and major open questions in this field.