We summarize previous field effect studies in high-T (c) cuprates and then discuss our method to smoothly tune the carrier concentration of a cuprate film over a wide range using an applied electric field. We synthesized epitaxial one-unit-cell thick films of La2-x Sr (x) CuO4 and from them fabricated electric double layer transistor devices utilizing various gate electrolytes. We were able to vary the carrier density by about 0.08 carriers per Cu atom, with the resulting change in T (c) of 30 K. The superconductor-insulator transition occurred at the critical resistance very close to the quantum resistance for pairs, R (Q) = h/(2e)(2) = 6.5 k Omega. This is suggestive of a quantum phase transition, possibly driven by quantum phase fluctuations, between a "Bose insulator" and a high-T (c) superconductor state.