CS 880 : Complexity of Counting Problems

The difference between EVALP(C,D) and EVAL(C,D) is that EVALP(C,D) fixes the value of a vertex w by i. We want to prove EVALP(C,D) ≡ EVAL(C,D). It is easy to see that EVAL(C,D) ≤ EVALP(C,D). Thus, we only need to prove the other direction. First, we define the notion of a discrete unitary matrix. Definition 1. Let F ∈ C be a matrix. We say F is M-discrete unitary for some positive integer M if 1. Every entry Fi,j is a root of unity, and M = lcm{ the order of Fi,j : i, j ∈ [m]} 2. F1,i = Fi,1 = 1 for all i ∈ [m] 3. For any i, j ∈ [m], i 6= j, 〈Fi,∗,Fj,∗〉 = 0 and 〈F∗,i,F∗,j〉 = 0 We can prove Lemma 1 by assuming the following pinning condition on the pair (C,D): 1. Every entry of F is a power of wN where wN = e 2πi/N for some positive integer N . 2. F is a discrete unitary matrix. 3. D is the 2m× 2m identity matrix.