In #quantumInformation often implementation estimates arise by counting options available for each of a number of qubits.
For example, if each #qubit is in a definite state |0〉 or |1〉 then there are 2ᴸ such configurations for 𝐿 qubits. If we are able to at will apply 𝑋,𝑌 or 𝑍 operations to each qubit then there are 4ᴸ such operations.
(If you do something on some qubits and nada on others then it's still a distinct operation jointly, so there are 4, not 3, options for each qubit)
How much is that?
A quick upper bound can be obtained by taking 2³=8<10 .
So for example for 𝐿=9 qubits if we want to run a protocol starting from every single configuration of definite qubit states (eg process tomography) then we will have to run it no more than thousand 10³ times. And so quickly you get an idea about what is much and what is too much.
More general operations can be decomposed into 𝑋,𝑌 or 𝑍 (and nothing, so the identity). During my PhD I was working with Marcel Goihl who was running a lot of numerics for a #quantum model of insulation arising from disorder. We were looking for charges that would get stuck and thus not participate in conductance. For transport through 𝐿=9 qubits we'd have 4⁹=2¹⁸=10⁶ million contributions that we'd need to calculate.
For 𝐿=13 this becomes 2²⁶<10⁹ contributions.
Marcel got curious that this is quite many and how much energy his calculations used on an #hpc cluster. Running a typical idea check quickly amounted to the yearly budget of a statistical German person.
He now works at a CO2 offsetting company and before graduating wrote with Ryan Sweke the proposal to keep track of what's the #enviroment cost of our calculations in #physics
https://scientific-conduct.github.io
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