  The accurate and efficient energy estimation of quantum Hamiltonians
consisting of Pauli observables is an essential task in modern quantum
computing. We introduce a Resource-Optimized Grouping Shadow (ROGS) algorithm,
which optimally allocates measurement resources by minimizing the estimation
error bound through a novel overlapped grouping strategy and convex
optimization. Our numerical experiments demonstrate that ROGS requires
significantly fewer unique quantum circuits for accurate estimation accuracy
compared to existing methods given a fixed measurement budget, addressing a
major cost factor for compiling and executing circuits on quantum computers.
