If the outdoor unit is mounted on a ground based foundation, most of the sound should come from the contact point between the outdoor unit and the wall.
Only a slight adjustment of the heat pump feet should be needed to ensure that the outdoor unit does NOT come into contact with the wall.
Some sound from the compressor may leak through the insulated wall when running hard, that is a bigger problem to solve, but not as frequent I would think.

Hello,
Apologies for the English. I’m in Stockholm but my Swedish is pretty bad still!
My Neighbour had a new air to air heat pump installed a couple of years ago. The unit was faulty and the company returned with a larger unit - but used the same base. This meant that the base of the outdoor unit is pushing against their building. We are a radhus - and joining them. I am getting vibrations through the floor consistent with when the unit starts and stops and while it runs hard - usually more during the 3am- 6am times - more in deep winter. It’s causing me to need time off work due to lack of sleep, stress and exhaustion. The other neighbors have direct el and we switched off our entire house to check it wasn’t something internal. The only other thing is their ventilation - but it’s a smaller fan and in the roof, I don’t think it’s that. Is there anything I can do apart from move - they aren’t receptive to asking them to fix it - it’s not noisy and doesn’t bother them.

Hello!

Is it ground based or wall based?

EDIT: I now saw that it is pushing against the wall.

If it is wall based, you can buy 4 large rubbered damper springs, which will reduce the vibrations to a great extent.

https://www.energibutiken.se/sv/vibrationsdampare/456-vibrationsdampare-webiq.html?SubmitCurrency=1&id_currency=4&gad_source=1&gclid=CjwKCAjw0YGyBhByEiwAQmBEWvw4HlU6MilBWZ1ely6s9CCtiD_CC_EDFRYRdyfAwAmrI0TxXZyYPRoCF6sQAvD_BwE



If it is ground based, you can do the same, and also change the tension/eigenfrequency of the pillars by adding dense weights.


Edit:
Can you take some pictures of the heat pump, and especially the wall-heatpump-point-of-contact.

I hope.you can talk to the neighbour and find a solution! 

Hello,
Apologies for the English. I’m in Stockholm but my Swedish is pretty bad still!
My Neighbour had a new air to air heat pump installed a couple of years ago. The unit was faulty and the company returned with a larger unit - but used the same base. This meant that the base of the outdoor unit is pushing against their building. We are a radhus - and joining them. I am getting vibrations through the floor consistent with when the unit starts and stops and while it runs hard - usually more during the 3am- 6am times - more in deep winter. It’s causing me to need time off work due to lack of sleep, stress and exhaustion. The other neighbors have direct el and we switched off our entire house to check it wasn’t something internal. The only other thing is their ventilation - but it’s a smaller fan and in the roof, I don’t think it’s that. Is there anything I can do apart from move - they aren’t receptive to asking them to fix it - it’s not noisy and doesn’t bother them.