Computational physicists have been thinking about algorithms for simulating quantum systems essentially since computers were invented. We have decent algorithms for approximating ground states, or for systems in equilibrium (contingent on it being spin-balanced, or at half-filling, or at 0 density, ... depending on the model), or in other limited circumstances.
But lift any of those special restrictions, and simulation methods hit a sign problem [sign]. In particular, real-time evolution of quantum systems, which is what a quantum computer does by its very nature, poses in some sense the most difficult sign problem for approaches leveraging classical computing.
That's not a proof that classical algorithms can't become more capable, but it's almost certainly a question that must be answered system-by-system. The generic sign problem is NP-hard, so special-case reasoning is required.
The reason those lattice field theory computations are done that way is that they provide stochastic but polynomial-time algorithms for exactly the same kind of exponentially-large state space that appears in quantum chemistry.
> Is there a place for quantum computers if classical algorithms become more capable at simulating quantum mechanics in ways we find useful?
There is not. Our existence as a field pretty much hinges on classical computers not being able to simulate all quantum mechanical problems efficiently. We imagine that designing quantum matter: https://cognitivemedium.com/qc-a-science, https://arxiv.org/abs/1508.02595 will be very useful in the scientific and technological sense and we don't think classical computers will ever fully stand up to that task.
> Breaking crypto, unless that falls too
If classical computers can simulate quantum efficiently then using quantum computers to break crypto also falls. Simulating quantum physics and factoring are in the same complexity class: https://en.wikipedia.org/wiki/BQP
> Our existence as a field pretty much hinges on classical computers not being able to simulate all quantum mechanical problems efficiently.
I don't think this is quite accurate. It could be that many of the kinds of quantum simulations we care about can be done efficiently classically, even if the worst-case quantum simulations are classically intractable. Certainly, classical simulation algorithms are steadily improving.
Right. We are now arguing over the nuances of what would make quantum computers useful, which I address in a comment where I say "Everything matters" later in this thread.
Most people who work in this field doubt that every quantum simulation problem we care about will be classical tractable in practice, that is, non worst-case. If we believed that, we might as well give up and continue to use the robust, mature classical computers we have and will continue to have better instances of for the foreseeable future.
I agree. "Everything" matters when it comes to these applications: complexity theory, heuristics, constant factors, quantum error correction overhead, qubit quality, improvements in classical algorithms, CPU and GPU improvements e.t.c. Doesn't make sense to put too much stock in just one of these components at the cost of others.
