What are use cases for learning about unknown quantum states and processes?

Question

I've seen numerous research papers about learning unknown quantum states or unitary operations from multiple copies of them. This includes fields such as quantum machine learning and tomography.

I’m curious about the practical situations where they arise. Why and when do people encounter scenarios involving multiple copies of unknown quantum states or unknown unitary processes? Wouldn't the fact that we have already generated certain states imply that we know what unitary operations we applied?

Answer 1

The question about the unknown state is answered easily in the context of quantum simulation or in any algorithm in which you know the generator of the computation, but not the target state. In these cases, it is very important to find ways of predicting many properties of the state with as few measurements as possible, this is why people are looking quite thoroughly into methods for state tomography or tomography-like measurements.

As for process learning, the motivation is a little more subtle, but goes into the same direction, in principle. Suppose you perform a quantum experiment for which you know the setup, but not the outcome. In the end, you always have to measure to receive data out of the experiment. In these cases, you might want to get an efficient representation of the (unitary or possibly non-unitary) process to make the quantum-classical interface as simply accessible as possible. The same holds for quantum simulation again. If you know a compilation of the circuit you want to execute, this does not mean that you have access to the operator that is the product of all gates. For the first use case, you might want to look into this paper and for the second for instance this one.

Answer 2

Here's a few examples:

1. You want to precisely reconstruct the states generated by your experimental apparatus. Even though you might have built the apparatus yourself, there might be many sources of noise and imperfections that make it deviate from your theoretical model, and therefore having methods to learn or reconstruct states that don't depend on such a model is useful. Or on a more practical level, you want to prove to people (and yourself) that your experimental apparatus indeed does what you built it to do.
2. You don't have precise control over the quantum channel corresponding to the physical operation you built. This is of course, to some degree, always the case, depending on the precision required by whatever you are doing. Methods to learn the process can be helpful to improve your model of it.
3. If you know how to use a specific piece of equipment to reconstruct the input states fed into it, you can then use that characterise piece of equipment to characterise the states produced by other uncharacterized ones.
4. In the context of quantum machine learning, a common task is to compute (possibly after a training procedure etc) some function of classical data encoded into the quantum states. This is again a form of learning about the state itself, as you anyway reconstuct something about the state to then derive what you want about classical info encoded into it.