Shouldn't that be an OR gate? Not only does the description above say "if and only if either of the original components had the state |1>", which is an OR, but the truth table listed above shows the same thing for the flag qubit.
Of course, one could say it's an AND on the |0> states, which is just De Morgan's law, but that's pretty awkward phrasing.
the extended thesis it depends on is "No physical procedure can decide an NP-complete problem in polynomially many steps." imo thats a very strong and controversial assumption when we still dont know the limits of what quantum computers can do.
I think part of the confusion here, is that usually the extended church turing thesis means that any physical computation can be efficiently (in polynomial time) simulated by a deterministic turing machine. (And thus if quantum computers exist and BQP is a superset of P, the proposition is false). I've never seen it defined before as above. But im definitely not a complexity theorists.
"Assuming [assumptions] we show that ... can in principle solve..."
Yeah, well, you know... that doesn't sound as promising as the title.
Grover gives an at most quadratic speedup for NP-hard problems, it does not turn a non-polynomial algorithm into a polynomial one.
The paper builds on the results of "Nonlinear quantum mechanics implies polynomial-time solution for NP-complete and #P problems" by Abrams and Loyd [1], from which I quote:
> The last qubit now contains all the information that we need; however, for small s, a measurement of the last qubit will almost always return |0>, yielding no information. > We wish to distinguish between the cases s=0 and s>0.
> Step 4. Repeatedly apply the nonlinear operation to drive the states representing these two cases apart at an exponential rate: eventually, at a time determined by a polynomial function of the number of qubits n, the number of solutions s, and the rate of spreading (Lyapunov exponent) λ, the two cases will become macroscopically distinguishable.
Some people that worked on this topic told me that there seems to be some improvements on the quasi-optimal solution found, but that due to the scale of current quantum computers, it just have been tried out on small-sized problems.
Theoretically, there are some papers suggesting that there are problems in BQP (the computational model of quantum computers) outside of NP [1] and even, the PH [2] (Polynomial Hierarchy, the infinite hierarchy of composition of NP and co-NP problems), which is why we cannot still satisfactorily say whether quantum computers can or cannot solve NP-complete problems.
The Wikipedia page for BQP [3] does a good job showing what is currently known.
[1] https://arxiv.org/abs/2209.10398 [2] https://eccc.weizmann.ac.il/report/2018/107/ [3] https://en.wikipedia.org/wiki/BQP#Relationship_to_other_comp...