Is it legal to lock the house and go out when dementia patient is sleeping in the house?

No Samantha it is not legal to leave babies or dementia unattended. Check your state law for "false imprisonment".

And for what purposes? I don't know if it's legal, but I don't think it would display common sense or precaution, not to mention that it could be considered elder abuse if the person had dementia so severely that he or she wouldn't know how to call for medical assistance if needed, might not be able to figure out how to climb out a window if a fire occurred, or if other similar life threatening emergencies occurred.

Does the person have a life alert pendant? Is there a lock box on the house exterior? Are there security cameras?

Go where, how far and for how long? There is a huge difference between a walk around the block and jumping in the car and taking off for several hours. And how advanced is the dementia, can they be trusted on their own if they should wake up????