Given two poly-sized quantum circuits $C_1$ and $C_2$ on $n$ qubits with a universal gate set generated by some finite set of one and two qubit gates. I'm thinking of the gates $\langle H, T, CNOT\rangle$, but other universal gate sets should work as well. Notice that $C_1$ and $C_2$ each correspond to a $2^n \times 2^n$ unitary matrix, $U_1$ and $U_2$, respectively. How hard is it to determine if $U_1 == U_2$, given the circuits $C_1$ and $C_2$?
Clearly the problem is in coQMA since if $U_1$ and $U_2$ are different there exists an input state $r$ such that $C_1(r)$ not equal to $C_2(r)$ which can be checked with a quantum computer. Has this problem been studied? Is it complete for this class? Is this class known as something else in the literature, since I cannot find much about it?