Preliminaries and Our Reult

By using a “witness erasing technique” proposed in [OT01], we prove that AM is contained in QMA. (In this report, we only give the proof of our main result. The other explanation will be given in a full paper.) 1 Preliminaries and Our Reult We assume the reader is familiar with the basic notions from quantum computation and structural complexity. See, for example, a book by Balcázar, Dı́az, and Gabarró [BDG90]. We first define two classes, AM and QMA. Note that the following definition of AM is different from the original one given in [Bab85]; we simply use the well-known characterization of AM. Definition 1 A set L is in AM if there exist a polynomial p and polynomial-time computable predicate R such that for any x of length n, x ∈ L ⇒ Pry∈{0,1}p(n){ ∃z ∈ {0, 1}p(n) R(x, y, z) } = 1, and x 6∈ L ⇒ Pry∈{0,1}p(n){ ∃z ∈ {0, 1}p(n) R(x, y, z) } < 1/3. Throughout this note, we fix any language L in AM; let p and R be a polynomial and a predicate satisfying the above condition for L. Also define N = 2. We use y and z to denote respectively a random string of {0, 1}p(n) and a witness string in {0, 1}p(n). String y and z are also referred as an Arthur’s message and a Merlin’s answer respectively. Definition 2 (Watrous [Wat00]) A set L is in QMA if there exist a polynomial q and a polynomial-time uniformly generated family of quantum circuits {Qx}x∈{0,1}∗ such that for any x of length n, x ∈ L ⇒ ∃|φ〉 : q(n) qubit [ Pr{ Qx accepts |φ〉 } > 2/3 ], and x 6∈ L ⇒ ∀|φ〉 : q(n) qubit [ Pr{ Qx accepts |φ〉 } < 1/3 ]. A quantum string |φ〉 is called a quantum certificate for x if it satisfies Pr{ Qx accepts |φ〉 } > 2/3. For these classes we show the following relation. Theorem 1 AM ⊆ QMA.