The number of measurements demanded by hybrid quantum-classical algorithms such as the variational quantum eigensolver (VQE) is prohibitively high for many problems of practical value. For such problems, realizing quantum advantage will require methods that dramatically reduce this cost. Previous quantum algorithms that reduce the measurement cost (e.g., quantum amplitude and phase estimation) require error rates that are too low for near-term implementation. Here we propose methods that take advantage of the available quantum coherence to maximally enhance the power of sampling on noisy quantum devices, reducing the measurement number and runtime compared to the standard sampling method of the VQE. Our scheme derives inspiration from quantum metrology, phase estimation, and the more recent “alpha- VQE” proposal, arriving at a general formulation that is robust to error and does not require ancilla qubits. The central object of this method is what we call the “engineered likelihood function” (ELF), used for carrying out Bayesian inference. We show how the ELF formalism enhances the rate of information gain in sampling as the physical hardware transitions from the regime of noisy intermediate-scale quantum computers to that of quantum error–corrected ones. This technique speeds up a central component of many quantum algorithms, with applications including chemistry, materials, finance, and beyond. Similar to the VQE, we expect small-scale implementations to be realizable on today’s quantum devices.