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Quantum computing can be used to efficiently simulate quantum many-body systems. Solving such a problem is classically hard because its complexity grows exponentially with the problem size (roughly with the degree of freedoms), which is an inherent consequence of the Schroedinger-equation.

My intuitive understanding of this fact is that using quantum computers, we can essentially simulate the quantum many-body system, thus making the theoretical calculation essentially an experiment.

What about the reverse problem?

More specifically, consider the situation in which

  • we have a description of a quantum many-body system, i.e. we know a formalized set of requirements whose behavior it should follow,
  • and we are trying to find the actual system for this description?

In a practical example, we have the required properties of a chemical compound. The goal is to find a chemical formula which fulfills the requirements.

This looks for me a harder task as to calculate the physical properties of a known compound (which is, in essence, "only" a solution of the Schroedinger-equation).

For example, such a description of a practical problem in human language would be this:

I want a room temperature superconductor.

And the output would be:

The formula is: ...

Extension:

We have the dynamics, or some type of behavior of the system. In the case of chemical compounds, it could be, for example, excitation spectrum or superconductivity transition temperature. The important thing is that we have here a different direction: not from a given QM system do we want to calculate the behavior (= Schroedinger equation), but we have a wanted behavior (now: superconductivity transition temperature), we have a nearly-infinite set of possible compounds, and we are looking for the compound which fulfills the selection criteria ($T_c \geq 300K$).

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Excellent question! You are asking if we can, given a property we want in a quantum system (for example superconductivity in a chemical compound), efficiently find one example of such a compound.

First of all, it is perhaps not yet a completely settled question if we can calculate such properties in the forward direction efficiently: Given a compound, does it superconduct? It is efficiently possible to simulate the temporal evolution from a given initial state and also to calculate its energy. There are several ideas that suggest it is also efficiently possible to calculate the interesting states (either the ground state or thermal states) and I believe this is possible but I want to emphasize that some details of these are still debated or not yet investigated academically.

Assuming we can efficiently calculate if a given compound fulfills your desires, one could start an exhaustive search over a number $N$ of compounds. Such a search can be accelerated by Grover's algorithm; even for the less straight-forward case that we have here, that an unknown number of solutions exist, this is possible in a number of compound-property calculations that scales with $\sqrt{N}$.

Hence you have a quantum speedup, likely exponential in the calculation of the desired properties of one chemical compound, and subexponential in the search among compounds.

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It seems that most ways of formalizing your question would lead to a problem that's QMA-hard, and therefore we shouldn't hope for an efficient quantum algorithm to solve it. (The relationship between BQP and QMA is similar to the relationship between P and NP: it would be very surprising if there were efficient quantum algorithms for QMA-complete problems.)

The canonical QMA-complete problem is the "local hamiltonian" problem. Roughly speaking, the input to the problem is a description of the hamiltonian for a physical system acting on qubits, and the problem is to decide whether the ground state of the system has small energy. If your system of specifying constraints includes local hamiltonians as a special case, then your problem is intractable.

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Short answer for the superconducting -> formula example: no, we will not be able to do that.

Longer answer (and more optimistic)

  • We need a one-to-one correspondence between the Hamiltonian of the system we can control in the actual experiment and the theoretical one, in terms of system size (degrees of freedom that we care about) and in terms of parameters.
  • As you describe, we are currently in the situation where we would like to know how a theoretical system evolves (the solution to a known set of equations with a known set of parameters). We map the theoretical system on the experimental one, measure and effectively know the solution to the theoretical equations.
  • The reverse would be: we know the evolution we want to obtain (the theoretical system) and we want to find the experimental system that fits. We would then do an iterative optimization process: controllably change parameters in the experimental system, measure, quantify the fidelity of the final quantum state or of the whole quantum process, and systematically tweak the parameters to optimize this. I do think this is totally doable: it's simply an extension of the forward process It's almost the same experiment, only performed more times.
  • Why we can't apply this to the superconducting -> formula case? First, because of the size requirement: if we want to relate an emerging property to the details of its composition, we probably would need an all-atomic model. Second, because we cannot continuously control the experimental variables in chemical compounds with quantum accuracy.
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