Is there a metric or definition for how "quantum-friendly" a problem is?

Question

I'm looking for a way to classify computational problems based on how suitable they are for quantum computers. Specifically, is there an established metric, definition, or framework that categorizes problems from "quantum-friendly" to "quantum-unfriendly"?

Consider the following problems:

1. Random Circuit Sampling (used for demonstrating quantum supremacy)
2. Simulating Protein Folding and Drug Design
3. Breaking RSA (Shor's algorithm provides an exponential speedup)
4. Breaking AES (Grover’s algorithm provides only a quadratic speedup)
5. Problems where quantum computers have no advantage
6. Problems where quantum computers are at a disadvantage (Are there any? If so, references would be appreciated.)

This list is roughly ordered from "quantum-friendly" to "quantum-unfriendly." Such an ordering could be useful in tracking quantum computing progress. For example:

The first major milestone was achieving quantum supremacy (problem #1).

The next milestone might be demonstrating an advantage in real-world industrial optimization (problem #2).

Further down the line, we expect quantum computers to eventually break RSA, possibly in the distant future (problem #3).

Is there an existing framework that formalizes such an ordering? If not, how can we systematically categorize problems in terms of their quantum compatibility? Any references or insights would be greatly appreciated.

Answer 1

To a first order I would probably start off by counting the number of $T$ gates required for each. As explained nicely here, Clifford+$T$ is a good metric because all of the tricky classical gates (CCNOT, CSWAP, etc.) useful in the algorithms you mentioned require a handful of tough-to-implement but single-qubit $T$-gates and a bunch of easier to implement but multiple-qubit Clifford gates.

This thread from 2022 lists some references for $T$-gates in some and depths for various algorithms. The most recent (as of 2025) go-to resource is probably Dalzell et al.

In particular:

1. Random circuit sampling, up to the depth studied by Google, require no $T$ gates and use a pretty shallow depth;
2. Drug designs need probably around $10^9$ $T$-gates.
3. Breaking RSA needs roughly the same number of $T$-gates, but a much deeper circuit than drug design.
4. Breaking AES would need about $10^{25}$ $T$-gates.
5. For problems that have no advantage - if the problem does not involve any "guessing" then I can't see how a quantum algorithm could give any asymptotic advantage. Euclid's Algorithm comes to mind - I don't think a quantum computer could be advantageous at all over a classical computer in determining the greatest common divisor of two large numbers.
6. For the last case of situations where a quantum computer underperforms a classical computer, at least asymptotically BPP$\subseteq$BQP so for a BQP-capable, fully fault-tolerant quantum computer there's not such a problem. But in terms of wall-clock time then oh you betchya, quantum computers run slow. I've heard estimates of something like a fault-tolerant Toffoli gate runs slower by 9 or 10 (or more) orders of magnitude on a quantum computer than a NAND gate would run on a classical computer. This is a reason why, for example, Grover's algorithm might not give any advantages until the problem instances are huge.

We probably do have time before RSA is broken - but that doesn't mean the threat is non-existent. Even though both drug simulation and factoring, to a first order, use similar resources, I would argue that some reasons why the literature emphasizes that drug design is important include (1) drug design has economic value in industry while, apart from some interesting case studies with respect to dead Bitcoin wallets, breaking symmetric-key crypto doesn't have the same value proposition to the world, and (2) learning how to implement Shor's algorithm has been well-studied for the last 30 years, while drug design, broadly defined, may have some legs for further simplification. (It's hard to expand on this without me veering off into more "opinion-forming" - which is verboten on this forum.)