  The quantum locking technique, which uses additional short low-loss
sub-cavities, is effective in reducing quantum noise in space gravitational
wave antenna DECIGO. However, the quantum noise of the main interferometer
depends on the control systems in the sub-cavities. Here we demonstrate a new
method to optimize the quantum noise independently of the feedback gain by
completing the square in the quantum locking system. We successfully
demonstrate in simulations that this method is effective in optimizing the
homodyne angle to the best quantum-noise-limited sensitivity.
