Lower bound on operation time of composite quantum gates robust against pulse length error

Precise control of quantum systems is a cornerstone for realizing high-quality quantum technology such as quantum computing and quantum communication. The performance of control of systems often deteriorates due to systematic errors. In one-qubit control, the pulse length error (PLE) is a typical systematic error, which is often caused by deviation of the strength of the control field. A composite quantum gate (CQG) is a method for suppressing effects of such systematic errors at the cost of a long operation time. A longer operation time implies stronger decoherence, and thus a shorter CQG is preferable from the viewpoint of noise immunity. However, it has not been clear how short CQG can be implemented. This problem can be regarded as an optimization problem under constraints: optimizing the operation time while requiring the error robustness. In this paper, we find a lower bound on operation time of all CQGs with first-order robustness against the PLE, in which effects of the error are eliminated up to its first order. The derivation of this bound is based on a geometric property of robustness against the PLE. This can be used for search after high-performance CQGs.