Representation of a matrix product state for system S that contains N qudits. Each qudit is initialized at ∣ 0 〉 , the trace on which leads to a channel E n on the ancilla A, which is dilated to a unitary Ue _n . A single qudit gate affects a local U _n , while a two-qudit gate on qudits n and n + 1 affects Ue _n and U n + 1 .