Randomized Quantization with Exact Error Distribution

We design a randomized scalar quantization scheme, where the quantization error is independent of the source and follows any given unimodal distribution (e.g. Gaussian distribution) exactly. We characterize the optimal encoding length of the quantization, and show that our scheme is optimal. This can also be regarded as a one-shot channel simulation setting, where the channel to be simulated is an additive noise channel. Potential applications include neural compression and coupling from the past.