Bayesian Quantum Amplitude Estimation

We present BAE, a problem-tailored and noise-aware Bayesian algorithm for quantum amplitude estimation. In a fault tolerant scenario, BAE is capable of saturating the Heisenberg limit; if device noise is present, BAE can dynamically characterize it and self-adapt. We further propose aBAE, an annealed variant of BAE drawing on methods from statistical inference, to enhance robustness. Our proposals are parallelizable in both quantum and classical components, offer tools for fast noise model assessment, and can leverage preexisting information. Additionally, they accommodate experimental limitations and preferred cost trade-offs. We propose a robust benchmark for amplitude estimation algorithms and use it to test BAE against other approaches, demonstrating its competitive performance in both noisy and noiseless scenarios. In both cases, it achieves lower error than any other algorithm as a function of the cost. In the presence of decoherence, it is capable of learning when other algorithms fail.