I was skimming the paper and came to this:
> This transformation is like an AND gate - it ignores the index qubit and places the flag qubit in the state |1> if and only if either of the original components had the state |1> for the flag qubit.
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.
if the PECTT (Physical Extended Church-Turing Thesis) is true then the current standard way of connecting classical gravity with quantum mechanics is wrong. the authors take it as evidence for full quantum gravity because the alternative is changing the Einstein equations in some arbitrary complex way. im not a physicist so this might be a bad explanation.
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.
> we still dont know the limits of what quantum computers can do.
Well, we don't know the limits of what classical computers can do too (P!=NP is not proven).
While not directly related to P!=NP, historical claims of quantum superiority were occasionally taken down by finding an efficient classical algorithm.
> the extended thesis it depends on is "No physical procedure can decide an NP-complete problem in polynomially many steps."
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.
But remember that "efficient" in terms of P and NP is about scaling. P == NP doesn't necessarily mean that a practically efficient algorithm can be found. The polynomial exponents involved may be large: O(N^1000) does eventually scale better than O(e^N), but that doesn't mean it is practically useful!
Prime factorisation is not proven to be NP-hard so the existence of a quantum polynomial algorithm (Shor) has no bearing on complexity classes other than showing that prime factorisation is in BQP (quantum polynomial). So it does challenge PECTT on 'vibes' as most experts think it is not in P, but there is no mathematical proof for it.
Grover gives an at most quadratic speedup for NP-hard problems, it does not turn a non-polynomial algorithm into a polynomial one.
Shor doesn't solve an NP hard problem. It's even possible that factoring and discrete log are in P, while P != NP.
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.
No. As far as we know, no realization of a quantum algorithm can solve NP-complete problems in polynomial many steps.
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.
I would also observe that it is a completely reasonable hypothesis that even if you can construct a problem in some higher complexity class that a quantum computer "should" be able to solve, it is possible that it would turn out that it couldn't because the complexity barrier turns out to be even more fundamental than quantum mechanics. If I understand it correctly, Scott Aaronson at least takes this possibility seriously, and points out that it is at least a possibility that this could end up being the legacy of quantum computing, along with the possibility that it could demonstrate some sort of currently-unknown fundamental limit on entanglement. And I'd add my own commentary that the experiment that shows QM breaks down and refuses to solve some problem that it "should" be able to solve would be on the instant shortlist for Nobel prizes for the experimenters, pending only extended confirmation, because it would be a big, big deal.
Semiclassical gravity is the best we can currently do for a theory of gravity without invoking speculative ideas that are currently untestable. If the paper holds up (I haven't read it), then there are several possibilities:
1. Maybe P = NP, and semiclassical gravity isn't special.
2. Maybe P = NP, and the way we'll prove it is finding an efficient way to simulate semiclassical gravity.
3. P = NP is a hypothesis about traditional theories of computation, but they don't rule out that we can build a special machine that solves them. There's a stronger hypothesis, the extended Church-Turing thesis (ECTT), that says this is impossible. Maybe the extended Church-Turing thesis is wrong, and this is how we'll show it.
4. If ECTT thesis is right, then maybe we can conduct an experiment where semiclassical gravity fails. This gives us a clue to new physics.
5. If we can't eventually conduct an experiment, then at least we learn about a new angle on complexity -- problems that can be efficiently solved this way but not by a deterministic Turing machine.
Both quantum mechanics and general relativity are thought to satisfy the ECTT, so the fact that our most experimentally successful combination of the two doesn't is of some interest. (Semiclassical gravity is thought to fail eventually, but in a way that's out of reach of current experiments.)
They use what looks like an impossible computational result to back into the idea that this is indirect evidence that gravity is quantized.
The controversy here is whether gravity is continuous (as classical Einstein models it) all the way down, or if the large scale behavior is built on a quantum foundation like everything else.
As far as I know most physicists believe gravity must be quantized, but we have no theory for it that works and plays nice with the well validated relativistic stuff at scale.
We have some candidates like string theory and loop quantum gravity but testing and refining them requires accelerators the size of the solar system or direct access to a black hole. The latter was the plot of Interstellar.
They essentially are saying semiclassical gravity (a broader theory subsuming classical) is theoretically incorrect. Like doing the konami code IRL and getting infinite money.
I'm not convinced. They aren't actually using the semiclassical Einstein equation, they are using some simplification they call Newton Schrodinger equation. They claim that this equation can lead to distinguish a state that's exponentially close to |0> from |0>. I don't follow their whole argument.
Anyway, I wouldn't be surprised if you could actually do hard computations with semiclassical Einstein equation, because they have strong self consistency - the expectation value of the stress energy tensor curves the metric, which in turn excites the quantum vacuum and causes expectation value of the stress energy tensor. But this isn't what they use in the paper. Nobody knows if this self consistency can be achieved in all configurations, and physicists working with semiclassical gravity usually do only one iteration of the self consistency.
If someone wanted to make semiclassical gravity into quantum gravity, he'll probably assume that gravity causes measurements, which would prevent these kinds of abuses where you have superposition you're probing via gravity while keeping it intact.
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.
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