By that measure of energy, burning oil or gas also has >>100% efficiency, as you only need energy to get it out of the ground, transport, and to light it. The energy stored in the oil is analogous to the "outside heat" for heat pumps.
To raise the temperature of my 300 square feet room by one degree I need the same amount of energy, no matter the source. But the ways of delivering that energy aren't the same.
If I have a simple gas burner, I need to know how much gas do I have to burn to heat my room. It will keep me warm, but it's not a good idea. Burning gas with a flame makes other gases that I don't want inside. I need to vent them, but I will lose some heat through that.
I can also get a gas furnace that condenses those byproduct gasses in a second heat exchanger for extra efficiency. Maybe seal it up better too. It needs a little electric power for ignition, so I need to take that into account. But overall, it would be more efficient. I would need less gas to heat my room.
Even though I'm using gas in both examples, the efficiency is different. It depends on how I convert that gas into heat.
I can also burn that gas in a generator to produce electric power. Then I can use that electric power for a resistive heater. Not a great idea, but it's an option.
I don't have to burn the gas myself. I can delegate generating power to a gas-fired power plant and buy power from them. And I can use a heat pump instead of a resistive heater. My heat pump will use less power than a resistive heater, so that's good, I suppose.
The question remains: how much gas do I (or a power plant) have to burn to heat my room. The answer depends on what do I use the gas for and how. That's the efficiency.
By burning oil or converting electricity to heat, 1 joule of stored chemical or electrical energy converted to heat energy will raise the heat of that something by a maximum of 1 joule. A maximum of 100% efficiency.
By using 1 joule of energy to compress cold gas into warm gas, we can raise the heat of that something by 3-4 joules. Moving heat energy around is more efficient that generating heat energy. Commercially available technology 300%-400% efficient.
This isn't correct. Here's a thought experiment to clarify:
If we burn X MJ of oil, we transfer X MJ of energy as heat to our house.
Otoh, if we set up a generator (say 40% efficient) which produces electricity from oil and dumps waste heat into our house, and use that to power a 300% efficient heat pump, we get 0.6X (generator waste heat) + 1.2X (300% * 0.4X) = 1.8X MJ of heat using the same X MJ of oil.
Thus,we get an additional 0.8X MJ of heat (80% extra heating) from the same X MJ of oil just by using a heat pump (& generator) compared to burning it. I.e we're using the energy stored inside the oil more efficiently.