An anesthesia provider can be a physician (anesthesiologist), or a CRNA (nurse anesthetist). A CRNA is a highly-trained individual, as programs are highly competitive. First an applicant must be an ICU nurse (an elite breed), and must be excellent within even that field. Then, there are 2-3 additional years of training.
This is quite necessary because the anesthesia provider is not just responsible for sleep, but for keeping the patient alive during surgery. In the best situation, anesthesiologists and nurse anesthetists work together and back each other up; having only one or the other is more likely in rural or outpatient settings.