How can I start a business on an H-1B?

A H-1B visa holder can have a passive interest in a company, wherein they would own shares but not work. If the H-1B visa holder wants to work for the entity, there are options where that may be available, but the organization would need to be able to demonstrate that it has the authority to control the work of the H-1B worker, including potentially firing the individual. This limits the scope of what may be possible, although I have had success in such filings, including a case where the H-1B worker owned 100% of the share/membership certificates in the company but had a non-shareholder board that was given control over the employee.
This filing was approximately 3 years ago, and I have not had a client seek to push the envelope in this manner since then. I have had success with other case filings where the H-1B was a minority shareholder and the company was able to successfully sponsor the person for a H-1B visa. The challenge often rests with ensuring that the role that the individual will be sponsored for aligns with the individual's educational background and demonstrating actual control over the H-1B workers' activities.
In the event the investment is sufficient to demonstrate the ability to file an EB-5 case, and your investment is in an entity that qualifies for unreserved classification, you may be entitled to file a concurrent adjustment of status alongside the I-526 petition.