CS 286 . 2 Lectures 5 - 6 : Introduction to Hamiltonian Complexity , QMA - completeness of the Local Hamiltonian problem

Definition 1. A language L ∈ BQP if there exists a classical polynomial time algorithm A that maps inputs x ∈ {0, 1}∗ to quantum circuits Cx on n = poly(|x|) qubits, where the circuit is considered a sequence of unitary operators each on 2 qubits, i.e Cx = UTUT−1...U1 where each Ui ∈ L ( C2 ⊗C2 ) , such that: i. Completeness: x ∈ L⇒ Pr(Cx accepts |0n〉) ≥ 2 3 ii. Soundness: x 6∈ L⇒ Pr(Cx accepts |0n〉) ≤ 3 We say that the circuit “Cx accepts |ψ〉” if the first output qubit measured in Cx|ψ〉 is 0. More specifically, letting Π 1 = |0〉〈0|1 be the projection of the first qubit on state |0〉, Pr(Cx accepts |ψ〉) =‖ (Π 1 ⊗ In−1)Cx|ψ〉 ‖ 2 2