This paper proposes an efficient method for the simultaneous estimation of the state of a quantum system and the classical parameters that govern its evolution. This hybrid approach benefits from efficient numerical methods for the integration of stochastic master equations for the quantum system, and efficient parameter estimation methods from classical signal processing. The classical techniques use sequential Monte Carlo (SMC) methods, which aim to optimize the selection of points within the parameter space, conditioned by the measurement data obtained. We illustrate these methods using a specific example, an SMC sampler applied to a nonlinear system, the Duffing oscillator, where the evolution of the quantum state of the oscillator and three Hamiltonian parameters are estimated simultaneously.

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We discuss a general method of model selection from experimentally recorded time-trace data. This method can be used to distinguish between quantum and classical dynamical models. It can be used in postselection as well as for real-time analysis, and offers an alternative to statistical tests based on state-reconstruction methods. We examine the conditions that optimize quantum hypothesis testing, maximizing one's ability to discriminate between classical and quantum models. We set upper limits on the temperature and lower limits on the measurement efficiencies required to explore these differences, using an experiment in levitated optomechanical systems as an example.

It has been widelydocumented that the sampling and resampling steps in particle filters cannot be differentiated. The reparameterisation trick was introduced to allow the sampling step to be reformulated into a differentiable function. We extend the reparameterisation trick to include the stochastic input to resampling therefore limiting the discontinuities in the gradient calculation after this step. Knowing the gradients of the prior and likelihood allows us to run particle Markov Chain Monte Carlo (p-MCMC) and use the No-U-Turn Sampler (NUTS) as the proposal when estimating parameters. We compare the Metropolis-adjusted Langevin algorithm (MALA), Hamiltonian Monte Carlo with different number of steps and NUTS. We consider three state-space models and show that NUTS improves the mixing of the Markov chain and can produce more accurate results in less computational time.

Particle Filters (PFs) are Sequential Monte Carlo methods which are widely used to solve filtering problems of dynamic models under Non-Linear Non-Gaussian noise. Modern PF applications have demanding accuracy and run-time constraints that can be addressed through parallel computing. However, an efficient parallelization of PFs can only be achieved by effectively parallelizing the bottleneck: resampling and its constituent redistribution step. A pre-existing implementation of redistribute on Shared Memory Architectures (SMAs) achieves O((N/T) x log(N)) time complexity over T parallel cores. This redistribute implementation is, however, highly computationally intensive and cannot be effectively parallelized due to the inherently limited number of cores of SMAs. In this paper, we propose a novel parallel redistribute on OpenMP 4.5 which takes O((N/T) + logN) steps and fully exploits the computational power of SMAs. The proposed approach is up to six times faster than the O((N/T) x log(N)) one and its implementation on GPU provides a further three-time speed-up vs its equivalent on a 32-core CPU. We also show on an exemplary PF that our redistribution is no longer the bottleneck.