A Near-Quadratic Lower Bound for the Size of Quantum Circuits of Constant Treewidth

We show that any quantum circuit of treewidth t, built from r-qubit gates, requires at least Omega(n(2)/2(O(r.t)). log(4)n) gates to compute the element distinctness function. Our result generalizes a near-quadratic lower bound for quantum formula size obtained by Roychowdhury and Vatan [SIAM J. on Computing, 2001]. The proof of our lower bound follows by an extension of Neciporuk's method to the context of quantum circuits of constant treewidth. This extension is made via a combination of techniques from structural graph theory, tensor-network theory, and the connected-component counting method, which is a classic tool in algebraic geometry. In particular, an essential key to proving our lower bound is the development of a new algorithm for tensor network contraction which may be of independent interest.