  The standard circuit model for quantum computation presumes the ability to
directly perform gates between arbitrary pairs of qubits, which is unlikely to
be practical for large-scale experiments. Power-law interactions with strength
decaying as $1/r^\alpha$ in the distance $r$ provide an experimentally
realizable resource for information processing, whilst still retaining
long-range connectivity. We leverage the power of these interactions to
implement a fast quantum fanout gate with an arbitrary number of targets. Our
implementation allows the quantum Fourier transform (QFT) and Shor's algorithm
to be performed on a $D$-dimensional lattice in time logarithmic in the number
of qubits for interactions with $\alpha \le D$. As a corollary, we show that
power-law systems with $\alpha \le D$ are difficult to simulate classically
even for short times, under a standard assumption that factoring is classically
intractable. Complementarily, we develop a new technique to give a general
lower bound, linear in the size of the system, on the time required to
implement the QFT and the fanout gate in systems that are constrained by a
linear light cone. This allows us to prove an asymptotically tighter lower
bound for long-range systems than is possible with previously available
techniques.
