  Markov Chain Monte Carlo (MCMC) is a powerful method for drawing samples from
non-standard probability distributions and is utilized across many fields and
disciplines. Methods such as Metropolis-Adjusted Langevin (MALA) and
Hamiltonian Monte Carlo (HMC), which use gradient information to explore the
target distribution, are popular variants of MCMC. The Sequential Monte Carlo
(SMC) sampler is an alternative sampling method which, unlike MCMC, can readily
utilise parallel computing architectures and also has tuning parameters not
available to MCMC. One such parameter is the L-kernel which can be used to
minimise the variance of the estimates from an SMC sampler. In this letter, we
show how the proposal used in the No-U-Turn Sampler (NUTS), an advanced variant
of HMC, can be incorporated into an SMC sampler to improve the efficiency of
the exploration of the target space. We also show how the SMC sampler can be
optimized using both a near-optimal L-kernel and a Hamiltonian proposal
