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Description
The wave function encodes our knowledge of a quantum system prior to a measurement. By incorporating posterior information, one can retrospectively improve the estimate of unknown outcomes of earlier measurements. For a continuously monitored system, this inference problem is described by a Hidden Markov model whose forward and backward dynamics are governed by stochastic master equations. As quantum technologies continue to scale toward increasingly macroscopic systems, solving these equations becomes computationally prohibitive due to the curse of dimensionality associated with large state spaces.
In this contribution, we present a Gaussian-sum approach that circumvents this limitation. The generally non-Gaussian Wigner function is approximated by a quasi-mixture of Gaussian functions, each evolving according to linear equations of motion. For homodyne and heterodyne detection, the evolution of each Gaussian function can be solved independently and is therefore ideally suited for parallel computation. This enables efficient smoothing of higher-order moments in macroscopic bosonic systems.