# finding generic Quantum circuits for k-local hamiltonians

Let $P_n$ denote the Pauli group on $n$ qubits (think of n as a large number).

Let $G=\left<g_1,...,g_n\right> < P_n$ be some abelian subgroup such that each $g_i$ acts on at most $k\ll n$ qubits. Assume that $g_i$ are independent and $-I\notin G$.

• Does there always exist some unitary $U$ such that $U^\dagger G U=\left<X_1,...,X_n\right>$?

• If so, what does $U$ look like? Can we ensure $U$ is "simple" enough? More precisely, can we find such a $U$ when we restrict ourselves to constant depth unitary quantum circuits?

The Clifford group is generated by CNOT, Hadamard, and the phase gate $\left(\begin{smallmatrix}1\\&i\end{smallmatrix}\right)$, so your $U$'s indeed have a special form.
However, $U$ can't always be chosen to be local. Consider, e.g., the Toric Code on a sphere: It is given by $n$ local and commuting stabilizers, and its ground state (=the joint $+1$ eigenstate of all stabilizers) has global (topological) entanglement, and thus cannot be converted to a product state with constant depth circuits.
• The thing is: you can NOT express any of the globally entangled ground states of the toric code with n independant AND local constraints: If the code is on $n^2$ qubits, then you can describe it's ground space as the stabilized space of at most $n^2-2$ local constraints. Here it's not the case as the stabilized space is unique since we have n commuting pauli elements, and therefore, there is no global entanglement involved. – user3001348 Aug 24 '15 at 12:27