Quantum Circuits for the Unitary Permutation Problem

Abstract

We consider the Unitary Permutation problem which consists, given $n$ unitary gates $U_1, \ldots , U_n$ and a permutation $\sigma$ of $\{1,\ldots , n\}$, in applying the unitary gates in the order specified by $\sigma$, i.e. in performing $U_{\sigma (n)}\circ \ldots \circ U_{\sigma (1)}$.

This problem has been introduced and investigated in [6] where two models of computations are considered. The first is the (standard) model of query complexity: the complexity measure is the number of calls to any of the unitary gates $U_i$ in a quantum circuit which solves the problem. The second model provides quantum switches and treats unitary transformations as inputs of second order. In that case the complexity measure is the number of quantum switches. In their paper, Colnaghi et al. [6] have shown that the problem can be solved within $n^2$ calls in the query model and $\frac{n(n-1)}{2}$ quantum switches in the new model, moreover both results was claimed to be optimal.

We refine these results and contradict their optimality, by proving that $n\log _2(n) +\Theta (n)$ quantum switches are necessary and sufficient to solve this problem, whereas $n^2-2n+4$ calls are sufficient to solve this problem in the standard quantum circuit model. We prove, with an additional assumption on the family of gates used in the circuits, that $n^2-o(n^{7/4+\epsilon })$ queries are required, for any $\epsilon >0$. The upper and lower bounds for the standard quantum circuit model are established by pointing out connections with the permutation as substring problem introduced by Karp.