Gate simulation and lower bounds on the simulation time

Abstract

Unitary operations are the building blocks of quantum programs. Our task is to design efficient or optimal implementations of these unitary operations by employing the intrinsic physical resources of a given $n$-qubit system. The most common versions of this task are known as Hamiltonian simulation and gate simulation, where Hamiltonian simulation can be seen as an infinitesimal version of the general task of gate simulation. We present a Lie-theoretic approach to Hamiltonian simulation and gate simulation. From this, we derive lower bounds on the time complexity in the $n$-qubit case, generalizing known results to both even and odd $n$. To achieve this we develop a generalization of the so-called magic basis for two-qubits. As a corollary, we note a connection to entanglement measures of concurrence-type.