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I recently read this report from BCG, which stated:

For scientists trying to design a compound that will attach itself to, and modify, a target disease pathway, the critical first step is to determine the electronic structure of the molecule. But modeling the structure of a molecule of an everyday drug such as penicillin, which has 41 atoms at ground state, requires a classical computer with some $10^{86}$ bits—more transistors than there are atoms in the observable universe. Such a machine is a physical impossibility. But for quantum computers, this type of simulation is well within the realm of possibility, requiring a processor with 286 quantum bits, or qubits.

Along with this resource estimate for penicillin, I've also seen similar mentions of the number of qubits required to model the ground state of caffeine (160 qubits). Given that the above report offers no reference(s) (probably in the name of business intelligence) and much Internet searching and looking into the quantum chemistry literature has come up short, my question is: Where are these resource estimates coming from – is there a journal article that published these numbers? I would really like to identify the methodology and assumptions used in making these estimates.

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    $\begingroup$ $\log_210^{86}=285.6$, so I guess the question is where did the statement that penicillin "requires a classical computer with some $10^{86}$ bits" come from. $\endgroup$
    – Mark S
    Aug 11 '19 at 3:03
  • $\begingroup$ @MarkS I was really hoping it would be more nuanced than that, but I fear you’re right. I’ll pull this thread and see if I can find more details. $\endgroup$
    – Greenstick
    Aug 11 '19 at 3:19
  • $\begingroup$ No need to pull the question! It’s a good one. $\endgroup$
    – Mark S
    Aug 11 '19 at 12:18
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I'm not sure if the 286 qubit estimate has ever been fully explained, but we can backwards reason about how to get to the figure.

First off, accuracy of quantum chemistry simulations via Trotterization is a function of the basis set (in both classical and quantum simulations). The basis set is kinda like a coordinatization the electron orbitals. There are a lot of different types of basis sets, like sto-ng or coupled cluster bases, each with varying performance.

Sto-ng is the minimal basis set, which requires the fewest number of qubits / classical simulation time at the cost of accuracy. For this, each typical electron orbital is assigned a qubit. So, doing a quick calculation (with orbitals pulled from here):

  • Carbon: 4 orbitals, 16 atoms
  • Hydrogen: 1 orbital, 18 atoms
  • Nitrogen: 5 orbitals, 2 atoms
  • Oxygen: 5 orbitals, 4 atoms
  • Sulfur: 9 orbitals, 1 atom

In total, this adds up to 64 + 18 + 10 + 20 + 9 = 121 orbitals. Now, we also need to account for spin up/spin down orbitals, so that's at least 242 electrons, for our least accurate simulations.

It's likely that the estimate of 286 is a coupled cluster estimate or sto-ng estimate with additional orbitals thrown in (for example, we might also want to simulate the possibility that electrons become excited beyond the valence orbitals).

As for the classical estimate, the comment by Mark S is entirely right: on a classical computer, we'd need to store the potential combinations where the electrons sit. In a naive computation, this is simply $ 2^n $ where $n$ is the number of spin-orbitals. There are some optimizations that can be made to reduce this cost, but this exponential scaling is what really prevents ab initio techniques from being applied more widely (think: an extra electron will double the storage cost of your computation... yikes!!)

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  • $\begingroup$ Thanks, this is just about what I was looking for! $\endgroup$
    – Greenstick
    Aug 27 '20 at 18:07
  • $\begingroup$ Sorry, I did Chemistry around 20 years back. What do you mean by "potential combinations where the electrons sit". Can you share some reference which I can read to understand this? Is it that in 286 orbitals, an electron can be in any orbital? I am also not able to figure out how you came up with (2 to the power n) value. $\endgroup$ Jan 25 at 10:06
  • $\begingroup$ I created this question to understand the math but unfortunately am not able to comprehend the answers - quantumcomputing.stackexchange.com/questions/15673/… $\endgroup$ Jan 25 at 11:18
  • $\begingroup$ The logic is that you can use the orbital model (i.e. 1p, 2p, 2s, ....), each of which has a certain capacity. Then, you can write the electrons occupancies as a binary string. For example, set 0101 to mean that you have an electron and the 2nd and 4th orbital $\endgroup$
    – C. Kang
    Jan 25 at 21:43
  • $\begingroup$ The easiest way to store all potential configurations is to store all potential binary combinations from 0000....0 to 11...1 ($2^n$), but when $n$ large this computation becomes impossible. Instead, quantum computers can store these configurations natively, so they require few qubits (each additional qubit adds an additional orbital, whereas you'd need exponentially many classical bits to add an orbital) $\endgroup$
    – C. Kang
    Jan 25 at 21:44
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They are just estimates. But they are not arbitrary estimates, but are based on a reference algorithm.

For example, a simple algorithm for a particle: Grover's algorithm is constructed with two qbits and can be used to find the correct answer to four quantum states of a particle, in a single step.

Eventually, a scientific publication would be more accurate in saying: "We have simulated a caffeine molecule using 90 qbits, by the Y-Topological Algorithm."

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  • $\begingroup$ Right, but estimates should have a basis nonetheless. For example, are they imagining a simulation with a VQE? What type of hardware is being assumed? I would guess they’re referencing an implementation on a NISQ device, but whether they mean EC/logical qubits is nonetheless not clear. Further, while we may be able to put much of the articles that pop up when you search the Internet down marketing hype, the BCG report is different in that it’s intended to guide corporate decision making and the authors specifically note that they reviewed around 130 journal articles for their synthesis. $\endgroup$
    – Greenstick
    Aug 11 '19 at 2:49
  • $\begingroup$ There is no standard. Reviewing many publications is an old scientific technique: objectivity is a shared subjectivity; that is, 'axiom', and the axioms are simply "a tacit agreement". I like to use the pre-standard: "configuration (hardware) A versus configuration B; what is the ratio between the two of the time spent looking for the prime factors for the same number X (where X is a number greater than 10^100000 ...) " $\endgroup$
    – YerkoBits
    Aug 12 '19 at 3:17

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