  The dynamics of a quantum system can be simulated using a quantum computer by
breaking down the unitary into a quantum circuit of one and two qubit gates.
The most established methods are the Trotter-Suzuki decompositions, for which
rigorous bounds on the circuit size depend on the number of terms $L$ in the
system Hamiltonian and the size of the largest term in the Hamiltonian
$\Lambda$. Consequently, Trotter-Suzuki is only practical for sparse
Hamiltonians. Trotter-Suzuki is a deterministic compiler but it was recently
shown that randomised compiling offers lower overheads. Here we present and
analyse a randomised compiler for Hamiltonian simulation where gate
probabilities are proportional to the strength of a corresponding term in the
Hamiltonian. This approach requires a circuit size independent of $L$ and
$\Lambda$, but instead depending on $\lambda$ the absolute sum of Hamiltonian
strengths (the $\ell_1$ norm). Therefore, it is especially suited to electronic
structure Hamiltonians relevant to quantum chemistry. Considering propane,
carbon dioxide and ethane, we observe speed-ups compared to standard
Trotter-Suzuki of between $306\times$ and $1591\times$ for physically
significant simulation times at precision $10^{-3}$. Performing phase
estimation at chemical accuracy, we report that the savings are similar.
