How many physical qubits are required per logical qubit in a typical Density Functional Theory implementation for a large number of atoms?
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The number of physical qubits needed for each logical qubit, depends on the error rate. For example, Table II in this paper shows that the number of physical qubits per logical qubit for various scenarios depending on whether there's an error occurring once for every $10^3$ gates, or if there's an error occurring once for very $10^6$ or $10^9$ gates:
The number of physical qubits per logical qubit in this case ranges from 313 to 15313 depending on the error rate. This paper was about quantum chemistry, but not specifically DFT, however people don't typically talk about doing DFT on quantum computers because DFT has $O(N^3)$ scaling, which means that it can be done "efficiently" on a classical computer, and quantum computers can do much more accurate calculations than DFT ones with the same "efficiency", so if you're going to use a quantum computer, you might as well use it for something more accurate than DFT. This could change if more accurate functionals are discovered.
Based on the above table, you can see that this study predicted that we would need 313 physical qubits for each logical qubit for a non-DFT quantum chemistry calculation if we had 1 error for every $10^9$ gates being performed. The paper is saying that we can't currently do the fault-tolerant (error-corrected) quantum chemistry calculation on a quantum computer, even for a situation involving just 1 qubit. We will likely need thousands of physical qubits per logical qubit, or much better error rates, to do meaningful quantum chemistry calculations on quantum computers.
