  Precise control of quantum systems is one of the most important milestones
for achieving practical quantum technologies, such as computation, sensing, and
communication. Several factors deteriorate the control precision and thus their
suppression is strongly demanded. One of the dominant factors is systematic
errors, which are caused by discord between an expected parameter in control
and its actual value. Error-robust control sequences, known as composite
pulses, have been invented in the field of nuclear magnetic resonance (NMR).
These sequences mainly focus on the suppression of errors in one-qubit control.
The one-qubit control, which is the most fundamental in a wide range of quantum
technologies, often suffers from detuning error. As there are many possible
control sequences robust against the detuning error, it will practically be
important to find ``optimal" robust controls with respect to several cost
functions such as time required for operation, and pulse-area during the
operation, which corresponds to the energy necessary for control. In this
paper, we utilize the Pontryagin's maximum principle (PMP), a tool for solving
optimization problems under inequality constraints, to solve the time and
pulse-area optimization problems. We analytically obtain pulse-area optimal
controls robust against the detuning error. Moreover, we found that
short-CORPSE, which is the shortest known composite pulse so far, is a probable
candidate of the time optimal solution according to the PMP. We evaluate the
performance of the pulse-area optimal robust control and the short-CORPSE,
comparing with that of the direct operation.
