In much of Asia, kindergarten is pre-schooling, but in the USA it corresponds to grade zero which is when formal public education starts. You are actually free to skip this grade in most states (you can start your kid in first grade), but most don’t.
In the US they're separate - kindergarten corresponds to grade zero, and preschool is a year before that. Preschool is the optional one, and kindergarten is required in the vast majority of states.
> Shortly after the turn of the 20th century, proponents of educating children at an early age began lobbying for more kindergarten classes. ''The movement was never to make kindergarten compulsory,” says Barbara Beatty, an associate professor of education at Wellesley College and the author of Preschool Education in America. “It was to mandate that districts offer kindergarten.”
> Now, more than 100 years’ later, state policies still vary on how--and even if--districts must offer kindergarten.
> But even as talk of the importance of high-quality preschool education reaches unprecedented levels, some states--Alaska, Colorado, Idaho, Michigan, New Hampshire, New Jersey, New York, North Dakota, and Pennsylvania--do not require districts to establish kindergartens, according to an Education Week survey conducted for Quality Counts.
(In these states districts don’t have to offer kindergarten)
> Currently, 13 states--Arkansas, Delaware, Kentucky, Louisiana, Maryland, New Mexico, Ohio, Oklahoma, Rhode Island, South Carolina, Tennessee, Virginia, and West Virginia--and the District of Columbia require children to attend kindergarten. In Rhode Island, Tennessee, and West Virginia, the law requires that youngsters attend kindergarten even though they do not have to start school until they are 6, the age at which children customarily enter 1st grade.
(Only 13 states mandate that your kids have to attend kindergarten)