Is there any real world problem where I can see quantum computing being better than classical computing?

Question

Is there any paper or piece of code showing, on a REAL quantum computer, that a specific real world problem (possibly related to computer science, machine learning or finance and possibly NOT related to cryptography) is solved more efficiently than a classical computer ? In general, I'd be happy with any problem where a Quantum Computer can bring value, which does not need to be necessarily speedup. In fact, for me it's enough if the quantum version is better under any point of view (stability, robustness, energy consumption, model expressiveness ...) as long as this superiority is somehow showed and documented true actual comparisons and experiments.

All the resources I looked at never make any comparison with the classical version and often claim quantum computing is powerful just showing it can solve a specific problem, without telling much about the classical counterpart.

Answer 1

You probably did not look good enough. Every fundamental quantum algorithm is always presented and compared to a classical counterpart. There is simply no well-known quantum algorithm that was ever given without complexity analysis.

Currently, most of the QC field is theoretical and not realizable in practice. But there are many useful and valuable experiments that are routinely performed on real quantum devices.

If we leave out cryptography and Shor's algorithm which has exponential speedup over classical factoring methods, there are:

1. Grover's search with quadratic speedup. It can be executed on a real quantum computer. Two working logical qubits are enough to experimentally demonstrate that Grover can find a preimage of a function in one query! Whereas classical search would take from 1 to 4 queries. You can run it yourself on a cloud. I'm not attaching any references because it is basic textbook stuff. The code that you can run is here.
2. Adiabatic quantum search with quadratic speedup. Similar to Grover search but with adiabatic computation. Complexity analysis is, of course, included.
3. Quantum teleportation. In network communication, they teleport quantum information. This has been done many times. Some scientists even managed to teleport quantum information to a space satellite. This is important for quantum internet and communication.
4. Quantum chemistry. Small molecule systems are routinely solved on actual quantum hardware. So far, no practical advantages over classical methods. But this is a question of time. We'll get there someday.
5. Optimization with D-Wave quantum annealers. The practical advantage over classical solvers is not clear so far. But I'm sure in several years we will get there too. In 2021 they demonstrated the quantum advantage over path-integrated MC.
6. You can create time crystals. See here. For some reason, physics people are really excited about it. If you or I don't understand why it is important, it doesn't mean it has no value.
7. Quantum optics for optimization. Coherent Ising machines were experimentally tested and the speedup was experimentally observed.
8. Finally, machine learning... IBM has done some work and proved a quantum advantage in quantum kernel methods.

These are things off the top of my head. I'm sure there is plenty of quantum stuff that is currently running on the near-term hardware.

Answer 2

D-wave has been used to solve multiple optimization problems already, see examples here: https://www.dwavesys.com/solutions-and-products/manufacturing-logistics/ The only issue with this as an answer is that D-Wave's devices are arguably not real Quantum Computers, but they are hardly classical computers either so I think this still applies.