# Given a non-Clifford quantum circuit $U$, is it possible to construct a commuting Clifford circuit $C$?

Given a non-Clifford circuit $$U$$, say $$U = \prod_{i=1}^k e^{i \theta_i P_i}$$ for $$P_i \in \{ I,X,Y,Z\}^{\otimes n}$$ and $$\theta_i \in \mathbb{R}$$.

Is it possible to construct a non-trivial Clifford circuit $$C$$ such that it commutes with $$U$$? .

• I would guess not since it is possible to decompose U as {Clifford+T}. If there is C that commutes with U while being Clifford shouldn't we have Cliffords commuting with T, which is not possible I guess? Interesting question (+1)
– R.W
Apr 17 at 6:42
• @R.W Any diagonal Clifford commutes with $T$ ... Apr 18 at 11:42