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It is often said that one of the early applications for Quantum Computers will be drug discovery.
Q: Roughly speaking, How many qubits will be needed to study (or simulate) a molecule such as: $C_{29}H_{31}N_{7}O$ (Imatinib, sold under the brand name Gleevec & Glivec ) ?
Or in general, if a molecule has 100 atoms, how many qubits will be needed to simulate it ?
My quick answer: something between 4 and 4000. ^_^
The number of qubits in an electronic structure calculation depends on at least three things:
For perfect accuracy, you'd need an infinite number of orbitals to fully represent your system. Most quantum-computing studies today adopt the "minimal" basis set needed to accommodate all the shells. So, hydrogens get two orbitals (2 for 1s) and carbons and nitrogens get ten orbitals (2 for 1s, 2 for 2s, and 6 for 2p). See for example this paper, which claims (accurately, I think) to model the largest molecule to date on a quantum computer: ethylene $C_2 H_4$ with 28 orbitals (and in this case, 28 qubits). So, imatinib $C_{29} H_{31} N_{7} O$ would need something like $N=29*10 + 31*2 + 7*10 + 1*10=432$ orbitals. This is of course the bare minimum number; for practically useful calculations, you'd want a more sophisticated basis set which has something in between 432 and infinity orbitals...
The most common way of mapping your chemistry problem onto qubits is with second quantization, assigning each orbital to its own qubit. (So, $432$ orbitals $= 432$ qubits.) This is the easiest conceptually, since it means "0" on a qubit corresponds to an empty orbital, and "1" indicates it's filled by an electron. But as long as you are simulating a closed system where the number of electrons remains fixed, this one-to-one mapping is excessive (eg. a molecule with two electrons will use the states $|1100\rangle$, $|1001\rangle$, etc. but you're never going to use the state $|0000\rangle$). So in principle you could devise a qubit mapping which maps each valid configuration onto its own computational basis state, thus requiring $\log_2 {N \choose \eta}$ qubits, where $\eta$ is the number of electrons. Imatinib has $\eta = 29*6 + 31*1 + 7*7 + 1*8 = 262$ electrons, so this compact qubit mapping would require only about $414$ qubits. Do note that this absolute-minimum mapping will horribly complicate every other part of your algorithm and is not likely worth the handful of qubits saved...
So far I've been assuming you want to do a typical Quantum Phase Estimation or Variational Quantum Eigensolver experiment, and it sounds rather pessimistic. You'd need something like ten times as many qubits as our largest quantum computers today to model imatinib? The fact is that it's a whole lot worse - if you want to do drug discovery, you won't learn a whole lot just by simulating the drug alone. You need to see how it interacts with macromolecules, which are much, much larger. To do the full electronic structure of both tyrosine kinase and imatinib together, you'd need far into the thousands of qubits. Never mind all the solvent interactions!
Fortunately you would never do that. In traditional molecular modeling, one severely limits the size of the system being explored quantum-ly. For example, you might simulate most of the enzyme with classical force fields, treating only the active site and its ligand with density functional theory. Or, if you have a quantum computer, you might choose to use Density Matrix Embedding Theory to fragment your complex into fragments, each approximated by an active space using just four qubits, then classically merge all the fragments to obtain a total energy calculation. See for example this paper, which offers genuine quantum computer-generated results ranking different ligand interactions in β-secretase. Accuracies are...a work in progress, but the concept at least is proven!
