@ahelwer: just a tiny correction but isn’t it
i² = −1
for the imaginary number i?
Oh yes, thank you for finding that! Corrected.
Thanks for writing this!
Wondering if you’re planning a second part with a higher dimensional state, demonstrating entanglement or more sophisticated behaviour?
Asiking since I feel like this particular example doesn’t answer yet “why bother using quantum computing for that”? In essence, here we just multiply initial state by the evolution matrix, which is a trivial task for classical computers (when the size isn’t exponential).
For instance, when justifying use for Shor’s or Grover’s algorithm it’s easy to understand the speedup since the problems are fairly abstract and it’s clear how compexity grows with size. Wonder if in a similar manner, it’s possible to come up with some made-up system of N electrons (or other particles, if it would make Hamiltonian simpler/closer to reality), that would be relatively easy to solve analytically (or just reason about qualitatevely), but straighforward construction of Hamiltonian matrix would be infeasible?
And then once could map the Hamiltonian onto quantum gates and that demonstrate that the results predicated by quantum computer match the analytical solution.
Glad you enjoyed it! I submitted another of my quantum-themed blog posts here: https://lobste.rs/s/ohmr6o/walking_faster_than_light_tightrope
You’re correct that classical computers are perfectly capable of executing the simulation method described in the post, and indeed this is a simulation method used on classical computers in the real world. However, this method reaches its classical limit for systems with 40-50 atoms; beyond that other approximations must be used. Quantum computers would (theoretically) encounter no such limit, and could simulate systems scaling more or less linearly with the size of the quantum computer itself.
It’s a common problem with introductory quantum computing presentations: by necessity they must be simple, and simplicity requires small problem sizes, so anything in an introductory presentation will be easily simulated by a classical computer. I suspect any presentation involving systems that would actually be difficult to execute on a classical computer would be extremely math-heavy (requiring dense notation to wrangle the large problem size) and thus have very limited appeal.
Your idea of a problem which is easy to solve analytically might exist, I’ve heard things involving the Ising model are like this; I’ll have to ask my coworkers about it!
Was also discussing that with a coworker today and he wondered, perhaps known statistical properties of quantum systems would be a good candidate for demonstration?
E.g. we know certain properties of Bose-Einstein condensates analytically and experimentally, now what if we try to infer these properties via ‘quantum brute force’, that is solving equations explicitly on quantum computer?
Yeah, and to be fair even something with, say 10 particles even if it’s possible to solve numerically by classical computer (e.g. if one just uses the 1024 x 1024 matrix, which is manageable for modern cpus), the solution would still be sort of blackboxy and hardly comprehendible for a human (at least, to the target audience who wants to learn why quantum simulations would be cool).
That’s why I feel it’s important to have some pedagogical made up example/approximation that allows for some analytical solution and scales well with number of atoms/particles involved. I understand though that it might be harder than it seems (e.g. like in QFT where numerically simulating vacuum is a big achievent and still requires supercomputers). Would be interesting if with some tweaks or unphysical assumptions it was possible to simplify Hamiltonian and make it tractable, just for the sake of demonstrating quantum advantage :)
Oh, cool, Ising model was on my reading list for a while; should bump it up. Thanks!