Mom is starting to unlock the front door and go outside unsupervised. How do I keep her in the house while no one else is home?

No-one can & should sit in a chair wrapped in cotton wool all day.

So what issues are present that you don't think Mom is safe to cross the street & visit her friend?

I have the same question as Beatty: why shouldn't your mother leave the house and go to visit her friend (assuming she knows to stay out of doors and socially distanced, of course)?

Please don't think we're challenging you - there may be very good reasons for all we know, it's just you haven't said what they are, and they might make a difference to what you can do about it too.

I will assume here Mom has a Dementia or she is a stubborn person. If Dementia, Mom should not be left alone or locked in. The key dead bolts inside a home are against fire code. My cousin had them placed on his doors but someone was with my Uncle 24/7. You may have to make a decision to hire a caregiver or time for an AL if she can afford it.

You haven't really provided a lot of information.
Your profile says:
"I have been Tube Feeding my mother, recovering from throat cancer, at my home untill she is able to eat oraly, have the tube removed, so she can return to her appartment which won't let her back untill she is off the tube.
I must now return to work and need someone to continue my effort.
I can better explain talking to someone rather than using the computer."

Is there a reason she must remain in the house while you work?
Can you not hire aides to stay with her while you work?
Does she have dementia?

The last question others have asked, mainly because you said she "wanders." If she has dementia, it isn't likely she will be able to return to her apartment, to live alone. If she had surgery, it could be post-surgical cognitive issues. Sometimes that resolves, but you need to know.

You really can't and shouldn't lock someone in their home (or yours.) It isn't safe, for many reasons. If she needs to be monitored, for ANY reason, then you need to hire aides to assist/watch her.

Definitely look into hiring a companion to stay with her when you're at work and no one is home. Someone who will be willing to take her to out and not just sit on the couch and look at their phone when on duty. Hire privately by advertising on a caregiver website. You'll always get better people when hiring privately and the wages are negotiable.
If mom is fine at night when the family is home then this suggestion will probably be the right answer to your problem.
Installing locks on the doors that only open with a key won't help if she had dementia and is determined to get out. In fact it could even be more dangerous.
I worked for a family who had the grandmother with dementia living with them and she was alone during the day while the adults were at work and the kids at school. They installed key-only locks to keep her from wandering off. She was crawling out of the first-floor windows. She'd call 911 emergency and tell them she was locked in the house and couldn't get out. The family found me after the police explained to them about her being alone and locked in the house. They were nice about it and understanding, but made it clear about that being unacceptable and dangerous. Don't do that. You can put a lock like that on the bedroom door to stop her wandering around at night when everyone's asleep, but not anywhere else. Please consider hiring companion services. There are organizations like the United Way who even have companionship volunteers for free.

You can't. It is time to have caregiver with her when family cannot be there. Doesn't even matter if she has dementia. She goes outside and for some reason you don't want her doing that. Someone has to physically be there to stop it or be with her to walk outside.

I assume your mother has dementia and requires supervision (otherwise she would certainly have free will to do as she pleases).
If this is the case, then leaving her unattended in the home may be just as hazardous (if not more so) with cooking appliances and the like.
The simple answer is to install deadlocks (can't be opened without a key) but that would have serious safety implications in an emergency (not to mention being unfair to your mother).
With my dad (who is a notorious wanderer), he can freely access the backyard without being able to leave the property; but out on the streets by himself would be a hazard to road users - he has no traffic sense - not to mention that he would soon get completely lost. My only option: where I go, he comes with me (not so bad, it gets him plenty of exercise).
If your mother's only purpose is to reach out to her friend, why not arrange for the friend to visit? Or perhaps you could say you'll take her over before/after you leave/return. Alternatively, could your mother meet her needs by calling on the telephone for a daily chat (or Skype..., yeah, not likely)?
Failing any of these, you may need to get her a carer while you are out.

Since you don't have in your profile that mom has any kind of dementia, I have to agree with Countrymouse and Beatty, when they ask why shouldn't she be able to go visit her friend if she wants. I'm sure she's probably bored while you are at work and just wants some company. I can't say I blame her.
If there is a good reason why she shouldn't be doing this, then I guess you'll have to hire some full-time help to be with her while you're at work.

I suppose that your intention is keeping her safe but she must be lonely.

Hire someone to be with her to supervise or consider placement so she can socialize with others.

Are you at home? or are you asking about when the house is empty of everyone else?
Is she mentally with it or does she have dementia at all?
How long is everyone out daily or during the week?
Does she have access to cooking for herself?