Algorithms like Dantzig’s or Grobner’s work fine on “native” linear/polynomial programming problems. But if you take a hard computational problem [or a random problem which is hard-on-average], and translate that into the appropriate format, they don’t offer any speedup. In fact, it is harder to solve the problems that way, since any simulation introduces some overhead. Likewise, we should expect quantum simulation algorithms to fast on “natural” quantum systems, but not very useful on systems which are designed to do hard computations by using the rules of quantum mechanics.

]]>I thought you would be interested.

Hoping to see your take on it.

]]>**Ben Standeven** remarks [correctly but narrowly] “Using subvarieties of a Hilbert space means using the Nullstellensatz to calculate observables; an EXPSPACE-complete problem in general. So you aren’t really getting any speedup this way.”

As Captain Picard’s borgified *persona* Locutus once remarked: **“A\(\ \)narrow vision, Number\(\ \)One!“**

**Question** Shall we abandon Dantzig’s simplex algorithm, because it is EXPTIME in general? Shall we abandon Kohn-Sham methods because — let’s face it! — our understanding of *how* these quantum simulation methods work is highly imperfect?

**The `t\(\ \)Hooft Alternative** Or should we embrace these methods, and seek to extend them, per the `t\(\ \)Hooftian maxim (quoted in #91) “We should not be deterred by ~~ no go theorems ~~*no speedup theorems*“.

**Summary** Corporations like Lockheed Martin are exhibiting an experience-grounded and mathematically well-founded `t\(\ \)Hooftian appreciation that transformational enterprises can be founded — indeed commonly *have* been founded — upon computational hardware that affords “only” polynomial speedups, and dynamical simulation algorithms whose workings formally are EXPTIME or even mysterious outright.

**Conclusion** Yet another wonderful title for a 21st century STEAM-saga would be *The `t\(\ \)Hooft Alternative.*

Please keep in mind that, if this *weren’t* the case—i.e., if there *were* an empirical way to distinguish Bohmian mechanics from “standard” QM—then it’s extremely likely that Bohmian mechanics would’ve already been ruled out by experiments, which have confirmed QM (insofar as it talks about a few particles at a time) to staggering precision.

- Any chance you might comment on Nick Bostrom’s Superintelligence book?

Yes, there’s a chance. 🙂 I read it and found it thought-provoking, but it would take some time to gather the provoked thoughts.

]]>**Itai** remarks “It’s not that I like so much Bohmian mechanics , but I want to distinguish what is real criticism and what is not.”

It’s not clear what traits distinguish “real” criticism, *e.g.* from “complex” criticism … but five weighty considerations are:

**(1) No new good algorithms** have arisen from Bohmian dynamics.

**(2) No broadly useful simulations** have been based upon Bohmian dynamics.

**(3) No new scientific instruments** have been **designed** with the aid of insights from Bohmian dynamics.

**(4) No substantial scientific discoveries** have been stimulated by insights from Bohmian dynamics.

**(5) No prosperous mathematical disciplines** have arisen as abstractions of Bohmian dynamics.

These reasons help us understand why algorithm/application-oriented quantum dynamicists like Linus Pauling, John Marcus, John Pople and Walter Kohn have won Nobel Prizes … and David Bohm didn’t.

Looking ahead, at least three-of-four 2014 Fields Medalists are doing work that broadly relates to post-Hilbert dynamics in the `t Hooftian sense (of #91 and #95)

**•\(\ \)Maryam\(\ \)Mirzakhani:** hyperbolic geometry \(\Leftrightarrow\) ergodic dynamical flows.

**•\(\ \)Martin\(\ \)Hairer:** stochastic trajectories \(\Leftrightarrow\) Lindblad-Carmichael unravellings on varietal state-spaces.

**•\(\ \)Artur\(\ \)Avila:** nonlinear stochastic PDE \(\Leftrightarrow\) quantum metrology triangles and quantum transport dynamics

The overlap of cutting-edge technology with cutting-edge mathematics is responsible for the burgeoning vigor of `t Hooftian post-Hilbert dynamics.

In contrast, it’s not evident to most folks (including Scott, t` Hooft, and me too) that Bohmian dynamics is providing comparably substantial inspiration to young mathematicians.

**Question** Why should young researchers focus on Bohmian dynamics, when so many other transformational 21st century research and enterprise opportunities are presenting themselves, across the entire STEAM spectrum? The world wonders!

**Summary** The prospects of post-Hilbert dynamics are bright … but (seemingly) not Bohmian.

Ah, OK. Your plan won’t work, because the classical machine is only allowed to make polynomially many queries to the computation graph of the oracle; but the oracle’s computation could be arbitrarily long.

@Itai #97:

The proof you’re looking for is just the one you quoted. Since matrix and wave formulations are equivalent, it does not matter whether the Bohm particle is guided by a wave or by a matrix; its trajectories are the same either way. Even if the proof is wrong, this would simply mean that there are two versions of Bohmian mechanics; one with a wave and one with a matrix.

@Sidles: #96?:

obviously it is neither necessary, nor feasible, nor even desirable that everyone think alike in regard to pilot waves…

I should have noticed this before; but using subvarieties of a Hilbert space means using the Nullstellensatz to calculate observables; an EXPSPACE-complete problem in general. So you aren’t really getting any speedup this way.

]]>You might want to put a hold on that Nobel Prize for right now.

]]>Matrix and wave formulation of QM was proved to be equivalent , I’m not sure who did a “correct proof” yet , Schrodinger and Pauli did not complete a correct proof, and maybe Von Newman ultimately proof the equivalence – never seen the full proof , got my references here http://www.lajpe.org/may08/09_Carlos_Madrid.pdf

I hope he had no mistake in the proof too as with the no-go theorem about hidden variables ).

But, I have never seen same equivalent proof to the Bohm wave theory ( he has no distinction between observables and states , and the only “operator” is the Hamiltonian ).

So , Unless such a proof exists , you can not “attack” pilot wave with arguments from matrix mechanics such as unitary evolution on Hilbert space and so on.

It’s not that I like so much Bohmian mechanics , but I want to distinguish what is real criticism and what is not.

]]>