Data assimilation with the full covariance on a fault-tolerant quantum computer

Data assimilation (DA) is the core of numerical weather prediction, estimating an accurate state of the weather model by combining forecast and observations. DA provides a general approach and is being applied to various problems beyond weather. The state estimation is a linear solve where the background error covariance B is essential. Since B is an N × N matrix with N ~ 10^8 to 10^9 for the weather model, the full B has never been used. For decades the community has approximated B by parameterized covariance models or by low-rank ensembles with localization and thereby discards genuine structures that we know exist from large-ensemble experiments. Fault-tolerant quantum computing (FTQC) could remove this constraint, since quantum linear system solvers scale polylogarithmically in N. However, the DA equations cannot be applied naively: the observation-space form squares the block-encoding normalization factor for observations with wide footprints, and the state-space form contains B^{-1}, whose condition number ~10^8. This talk presents a combined state-observation formulation that avoids both difficulties: it contains no B^{-1}, and its condition number is governed by the observation error covariance R rather than by B. Numerical experiments with the Lorenz-96 model reproduce the exact solution with the full B and outperform localized ensemble Kalman filters. The formulation assumes that B and the observation operator are provided as block-encoded oracles, and how to construct such oracles without ever storing the matrices is the open problem. A first concrete target is an end-to-end demonstration with Lorenz-96 on a quantum simulator with only seven data qubits, which I would like to discuss with the iTHEMS community.