Measurement as Cold Diffusion:Pricing the Shot Frontier

Pipelines that reconstruct a quantum state's full Born distribution are usually built by importing classical Gaussian diffusion onto the device, which assumes away the cost that dominates them: the distribution must be read out by measurement, and estimating it over $2^n$ outcomes converges only as $1/\sqrt{m}$ in the shot budget $m$. We instead take the measurement to be the forward process, making the shot count the diffusion time. We prove that its control parameter is the only one among standard cold-diffusion corruptions that is a genuine resource—costing $\Theta (2^n)$ shots to hold the error fixed, while the Gaussian, blur and mask knobs are free—with a predicted constant confirmed to $2.1\%$ and the scaling verified over a $256\times$ range in $D$. A learned prior bends that frontier, flattening it and saving up to $47.6\times$ of inference-time readout only for fixed-complexity data and only once the prior's acquisition cost is amortized; on ground-state Born distributions of an Ising chain the flattening fails. The corruption is also not interchangeable: restorers move between multinomial and Poisson readout at a cost of $0$–$3\%$ but lose $57$–$229\%$ moving from Gaussian or blur onto real readout. Finally, the prior does not appear able to be quantum. We characterize the class a postselecting circuit restorer realizes—the projectivization of a linear map, closed under coherent composition—and find classical controls with fewer parameters beating it at every scale, with logged curves showing it at a floor rather than out of steps. We are explicit that this is a characterization, not an impossibility proof. On ibm_kawasaki the idealized forward model tracks the device only while shots bind, yet a restorer trained solely on it still halves device error at the state-preparation floor.