It is unlikely that a heat pump could heat water that quickly to be "tankless."

The problem is not a brief surge of startup energy; it is the total sustained power you need to heat the water for the shower.

Let's say you want to run a single shower at 2.5gpm, the water comes into the home at 55F (12.8C, pretty temperate!) and you want your shower at 100F (37.8C), so a 25C rise. 2.5gpm = 9.463 L/min = 158mL/sec. Water's heat capacity is 4.184* J/mL/degC, so the sustained thermal power you need to deliver is:

4.184 x 158 x 25 = 16.5kJ/s = 16.5kW

At 240V, 16.5kW would be almost 70A.

What makes a heat pump heater more efficient is that it multiplies the electrical power input into a larger thermal power output, by "borrowing" thermal energy from the surroundings. So you could get (for example) 15kW thermal power from 5kW electrical power. In practice the heater will be tanked, rather than tankless, for two reasons:

1. The effective power will probably be less than that.

2. That was an estimate to run just one shower and you might want to run more than that!

So the tank lets you buffer the power draw over time; there will be some thermal loss over time, but insulation is cheap, and it's hard to fight a 3-5x amplification from the heat pump.

* Yes, yes, per gram/K but 1mL of water is close enough to 1g for this.

> there will be some thermal loss over time, but insulation is cheap

FWIW, one US govt source attributes ~20% of a typical household's domestic energy usage to water heating, though the source doesn't attribute it to thermal loss or initial heating.

>They use a tankless water heater so electrical use is ~zero when nobody is using hot water, which is most of the time. If you use a tank - heating ~40 gallons of water 24/7/365 - that's hard to compete with by just being more efficient.

You need to compare this against the average heating days for their location.

Water heaters are a "incandescent lightbulbs are 100% efficient in the winter" situation.