And that’s where I see a hitch, I feel mainstream Many-Worlds jumps to the conclusion of what’s at first blush strongly suggested at the realization of the “recursion” in Schrödinger’s equation, the strong suggestion of a single vaster space to be completely covered or paved by maximal exploitation of the frame-change liberties included with Schrödinger’s equation. The wave-function of the multiverse.

I believe there’s space for a posture that adheres to the “observations are frame changes of Schrödinger’s equation” part while being wary of over-precipitously going forward to the maximal inductive generalization to a frame of all frames.

]]>btw To clarify, by “anthropic” I was thinking that maybe processes like photosynthesis would not work in a universe with poor fault-tolerance for QC, and perhaps other crucial biological systems would also not be feasible.

So (fairly) large-scale QC works in our Universe because otherwise we wouldn’t be here to observe it!

edit: correct spelling of anthropic

]]>**James Gallagher** postulates that “In some universes QC works, but in others it fails due to bad fault-tolerance characteristics of the Hamiltonian.”

An even stronger postulate — which is consonant with all experimental evidence and all theoretical understanding to date — might be

In all universes that are governed by relativistic renormalizable (gauge-invariant) field theories, QC fails due to bad fault-tolerance characteristics of the Hamiltonian.

If this postulate proves to be correct, then the “noisy low-degree polynomial” state-spaces of Kalai’s postulates — and of much other recent work, e.g. Swingle’s ‘S-Sourcery’ per #82 and Landau, Vazirani & Vidick per #86 — assume the role of non-renormalizable dynamics that Matthew Schwartz vividly characterizes in his free-as-in-freedom course notes for **Harvard’s PHYS 253A: quantum field theory**:

Section 23.6

Summary of non-renormalizable theoriesI want to emphasize again: non-renormalizable theories are very predictive, not just at tree-level (classically) but also through quantum effects. The quantum effects are calculable and non-analytic in momentum space.

In fact, non-renormalizable theories are so predictive that it is often better to make a non-renormalizable approximation to a renormalizable theory than to use the full renormalizable theory itself.

Examples include Einstein gravity instead of string theory, Heavy Quark Effective Theory (HQET) instead of QCD, Ginzburg-Landau theories in condensed matter, the Schrodinger equation instead of the Dirac equation, and chemistry instead of physics.

**Conclusion** Even in the event that scalable QC and BosonSampling experiments *do* someday prove to be feasible, it remains theoretically and experimentally plausible that “noisy low-degree polynomial” state-spaces suffice to describe (in Matthew Schwartz’ vivid phrasing) “all of chemistry but not all of physics.”

Indeed, researchers like Brian Swingle (#82) are even beginning to argue that the “predict all of chemistry” outcome is not just plausible, but even likely.

This is why the quantum dynamics and thermodynamics of Gil Kalai’s “noisy low-degree polynomial” state-spaces is well-worth studying.

]]>But I now do wonder if you’re relying on anthropic arguments regarding the Hamiltonian?

In some universes QC works, but in others it fails due to bad fault-tolerance characteristics of the Hamiltonian.

]]>Many (probably even most) skeptics of quantum computers will answer “YES” to this question. Certainly, this applies to Alicki, Dyakonov and me. The open question is if the state vector and Hamiltonian of the entire universe allows quantum fault-tolerance and “quantum supremacy” or not.

Both the optimists and the pessimists regarding QC need to make some extra assumptions on the nature of the quantum evolution describing our universe. (Of course, we do not have and we do not expect direct access to the Hamiltonian and state vector of the entire universe.)

]]>**James Gallagher** requests a direct reply to “is there a State Vector and Schrodinger Evolution equation which applies to the entire universe?”

One such answer, which is mathematically compatible with *both* sides of the Kalai/Harrow debate *and* is physically compatible with all existing experiments, is this one: “Yes. Moreover this universal state-vector has a PSPACE algebraic representation whose Hamiltonian evolution can be unraveled with PTIME computational resources.”

What would it take to refute this hypothesis? Scalable scattershot BosonSampling experiments whose outputs *exactly* sampled from certain permanent-related distributions would physically generate “state-vectors having a PSPACE algebraic representation whose Hamiltonian evolution *cannot* be unraveled with PTIME computational resources” (conditioned upon certain plausible complexity theory postulates, as set forth in Aaronson and Arkhipov arXiv:1011.3245 and arXiv:1309.7460).

Such BosonSampling experiments would in the opinion of many (including me) rank among the most seminal scientific advances of this or any century. So it is a very considerable accomplishment even to *conceive* such experiments (as Scott and Alex Arkhipov did), and thereby create an ongoing and exceptionally inspirational challenge to experimentalists and engineers.

Please let me say too, that *Shtetl Optimized* admirably airs diverse views on this tough subject, and Scott’s walk-the-talk respect for scientific diversity is not the least of his *many* outstanding contributions to the academic community. Good on `yah, Scott!

The beginning of the arc is anchored in student-friendly eprints like Ashtekar and Schilling “Geometrical Formulation of Quantum Mechanics” (1997, arXiv:gr-qc/9706069v1) and Brody and Hughston “Geometric Quantum Mechanics” (1999, arXiv:quant-ph/9906086). The arc extends to the present era in articles like Ashtekar’s “Introduction to Loop Quantum Gravity” (2012, arXiv:1201.4598) and Brody and Hughston “Universal Quantum Measurements” (arXiv:1505.01981v1). In aggregard, the arxiv presently offers more than 100 articles by these authors, which can be read as a compendium of post-Hilbert analyses of quantum dynamics.

The two-decade span of these articles provides insight into career options for skeptical-postulate researchers. Schilling joined the defense/intelligence/cryptography research community (not much is known about his subsequent research). Ashtekar of course has made a brilliant career in quantum gravity. Brody and Hughston have continued to work actively in fundamental physics research, while pursuing with great success parallel academic careers in “city” financial analysis.

Many more skeptic-friendly math-centric eprint-arcs can be constructed; see for example the already-vast and still-burgeoning literature relating to quantum dynamics on algebraic state-spaces (e.g., Landau, Vazirani, Vidick, Cirac, Verstraete, etc., of #86).

At present there are no textbooks that accessibly summarize this material … but it will be surprising (to me anyway) if this situation persists much longer. On the other hand, ongoing advances in category theory and algebraic geometry have scarcely begun to be assimilated into this literature, so plenty of lines of investigation remain open and largely unexplored.

**Conclusion** The mathematical methods that are associated to serious investigation of Kalai’s skeptical postulates are proving to be sufficiently interesting and powerful as to support an abundance of academic careers opportunities for young investigators.