research level - Quantum computing and quantum control

In 2009, Bernard Chazelle published a famous algorithms paper, "Natural Algorithms," in which he applied computational complexity techniques to a control theory model of bird flocking. Control theory methods (like Lyapunov functions) had been able to show that the models eventually converged to equilibrium, but could say nothing about the rate of convergence. Chazelle obtained tight bounds on the rate of convergence.

Inspired by this, I had the idea of seeing how to apply quantum computing to quantum control, or vice versa. However, I got nowhere with this, because the situations did not seem to be analogous. Further, QComp seemed to have expressive power in finite dimensional Hilbert spaces, while QControl seemed fundamentally infinite-dimensional to me. So I could not see how to make progress.

So my question: does this seem like a reasonable research direction, and I just didn't understand the technicalities well enough? Or are control theory and quantum control fundamentally different fields, which just happen to have similar names in English? Or some third possibility?

EDIT: I have not thought about this for a few years, and I just started Googling. I found this quantum control survey paper, which, at a glance, seems to answer one of my questions: that quantum control and traditional control theory are indeed allied fields and use some similar techniques. Whether there is a research question like Chazelle's analysis of the BOIDS model that would be amenable to quantum computing techniques, I have no idea.

Answer

Well, obviously I don't know exactly what you were trying, but it's not an unreasonable direction. There is certainly a bit of interplay between the two areas. Many of the open loop techniques which have become standard practice in spin resonance (for example decoupling pulses such as WAHUHA [Waugh, J.S., Huber, L.M., Haeberlen, U. (1968) Phys. Rev. Lett., 20, 180.] etc.) are based on exactly the same Suzuki-Trotter tricks that are now used in quantum simulation algorithms.

Additionally, there have been some nice results from Steffen Glaser and others on optimal control for the basic building blocks of quantum computation, including some pretty high level operations like generating cluster states (see arXiv:0903.4066) and state transfer (see arXiv:0705.0378).

There is also a huge literature on trying to make stuff into quantum computers even when you don't have all the control knobs you might wish for (see for example Seth Lloyd's papers on Universal quantum interfaces, all of the stuff on global control, and recent papers by Daniel Burgarth and Alistair Kay on Lie algebraic control techniques).

Lastly, the two areas are very closely linked via the adiabatic model, where the efficiency is directly linked to how quickly you can adiabticly transition between two Hamiltonians.

homework and exercises - No stable closed orbits for a Newtonian gravitational field in $dneq 3$ spatial dimensions

We are supposed to show that orbits in 4D are not closed. Therefore I derived a Lagrangian in hyperspherical coordinates $$L=\frac{m}{2}(\dot{r}^2+\sin^2(\gamma)(\sin^2(\theta)r^2 \dot{\phi}^2+r^2 \dot{\theta}^2)+r^2 \dot{\gamma}^2)-V(r).$$

But we are supposed to express the Lagrangian in terms of constant generalized momenta and the variables $r,\dot{r}$. But as $\phi$ is the only cyclic coordinate after what I derived there, this seems to be fairly impossible. Does anybody of you know to calculate further constant momenta?

Answer

Hints:

1. 
   

   Prove that the angular momentum $L^{ij}:=x^ip^j-x^jp^i$ is conserved for a central force law in $d$ spatial dimensions, $i,j\in\{1,2,\ldots ,d\}.$

   
   



2. 
   

   Since the concept of closed orbits does not make sense for $d\leq 1$, let us assume from now on that $d\geq 2$.

   
   


3. 
   

   Choose a 2D plane $\pi$ through the origin that is parallel to the initial position and momentum vectors. Deduce (from the equations of motion $\dot{\bf x} \parallel {\bf p}$ and $\dot{\bf p} \parallel {\bf x}$) that the point mass continues to be confined to this 2D plane $\pi$ (known as the orbit plane) for all time $t$. Thus the problem is essentially 2+1 dimensional with radial coordinates $(r,\theta)$ and time $t$. [In other words, the ambient $d-2$ spatial dimensions are reduced to passive spectators. Interestingly, this argument essentially shows that the conclusion of Bertrand's theorem are independent of the total number $d\geq 2$ of spatial dimensions; namely the conclusion that only central potentials of the form $V(r) \propto 1/r$ or $V(r) \propto r^2$ have closed stable orbits.]

   
   


4. 
   

   Deduce that the Lagrangian is $L=\frac{1}{2}m(\dot{r}^2 +r^2\dot{\theta}^2) -V(r)$.

   
   


5. 
   
   

   The momenta are $$p_{r}~=~\frac{\partial L}{\partial \dot{r}}~=~m\dot{r}$$ and $$p_{\theta}~=~\frac{\partial L}{\partial \dot{\theta}}~=~mr^2\dot{\theta}.$$

   
   


6. 
   

   Note that $\theta$ is a cyclic variable, so the corresponding momentum $p_{\theta}$ (which is the angular momentum) is conserved.

   
   


7. 
   

   Deduce that the Hamiltonian is $H=\frac{p_{r}^2}{2m}+\frac{p_{\theta}^2}{2mr^2}+ V(r)$.

   
   


8. 
   

   Interpret the angular kinetic energy term $$\frac{p_{\theta}^2}{2mr^2}~=:~V_{\rm cf}(r)$$ as a centrifugal potential term in a 1D radial world. See also this Phys.SE post. Hence the problem is essentially 1+1 dimensional $H=\frac{p_{r}^2}{2m}+V_{\rm cf}(r)+V(r)$.

   
   
   


9. 
   

   From now on we assume that the central force $F(r)$ is Newtonian gravity. Show via a $d$-dimensional Gauss' law that a Newtonian gravitational force in $d$ spatial dimensions depends on distance $r$ as $F(r)\propto r^{1-d}$. (See also e.g. the www.superstringtheory.com webpage, or B. Zwiebach, A First course in String Theory, Section 3.7.) Equivalently, the Newtonian gravitational potential is $$V(r)~\propto~\left\{\begin{array}{rcl} r^{2-d} &\text{for}& d~\neq~ 2, \\ \ln(r)&\text{for}& d~=~2. \end{array}\right. $$

   
   


10. 
    

    So from Bertrand's theorem, candidate dimensions $d$ for closed stable orbits with Newtonian gravity are:

    
    
    
    
    • $d=0$: Hooke's law (which we have already excluded via the assumption $d\geq 2$).
    
    
    • $d=3$: $1/r$ potential (the standard case).
    
    
    • $d=4$: $1/r^2$ potential (suitably re-interpreted as part of a centrifugal potential).
    
    
    
    
    

    We would like to show that the last possibility $d=4$ does not lead to closed stable orbits after all.

    
    


11. 
    

    Assume from now on that $d=4$. Notice the simplifying fact that in $d=4$, the centrifugal potential $V_{\rm cf}(r)$ and the gravitational potential $V(r)$ have precisely the same $1/r^2$ dependence!

    
    


12. 
    

    Thus if one of the repulsive centrifugal potential $V_{\rm cf}(r)$ and the attractive gravitational potential $V(r)$ dominates, it will continue to dominate, and hence closed orbits are impossible. The radial coordinate $r$ would either go monotonically to $0$ or $\infty$, depending on which potential dominates.

    
    


13. 
    
    

    However, if the repulsive centrifugal potential $V_{\rm cf}(r)$ and the attractive gravitational potential $V(r)$ just happen to cancel for one distance $r$, they would continue to cancel for all distances $r$. Newton's second law becomes $\ddot{r}=0$. Hence a closed circular orbit $\dot{r}=0$ is possible. However, this closed circular orbit is not stable against perturbations in the radial velocity $\dot{r}$, in accordance with Bertrand's theorem.

    
    

general relativity - Correct derivation of Einstein's equations from the Hilbert action

I have been trying to understand general relativity from a first-principles perspective in my spare time, and I have been unable to find a convincing derivation of the Einstein equations. The most complete one I can find is the one on Wikipedia, but it has a big mathematical gap that I can't figure out. Namely, when computing the variation of the Riemann curvature tensor, the author assumes that the variation operator is a derivation, i.e. satisfies the product rule for derivatives. This seems to be false, because the variation in question is not itself an ordinary derivative, but rather the Euler-Lagrange "derivative", whose definition for a function of the (inverse) metric and its first two partials (like the Riemann tensor) is

$$ \frac{\delta \mathcal{L}(g^{ij}, \partial_k g^{ij}, \partial_l \partial_k g^{ij})}{\delta g^{ij}} = \frac{\partial \mathcal{L}}{\partial g^{ij}} - \partial_k \frac{\partial \mathcal{L}}{\partial(\partial_k g^{ij})} + \partial_l \partial_k \frac{\partial \mathcal{L}}{\partial(\partial_l \partial_k g^{ij})}. $$

The second and third terms do not satisfy the product rule. It appears almost as though in the linked derivation the author is taking simple partials with respect to the inverse metric, which is entirely wrong. And yet, that derivation is linked to Carroll's textbook, so it must have some credibility. I don't have the textbook, so I can't check whether it explains this logic more completely. Therefore I turn to Physics.SE. What's going on here?

Answer

After pondering Michael Seifert's answer, I have realized what the full resolution of my problem is. The issue is that the expression $\delta \mathcal{L}$, which is defined to be

$$ \delta \mathcal{L} = \frac{\partial \mathcal{L}}{\partial g^{ij}} \delta g^{ij} + \frac{\partial \mathcal{L}}{\partial (\partial_k g^{ij})} \partial_k (\delta g^{ij}) + \frac{\partial \mathcal{L}}{\partial (\partial_l \partial_k g^{ij})} \partial_l \partial_k (\delta g^{ij}),$$

cannot be confused with $\frac{\delta\mathcal{L}}{\delta g^{ij}}$, unlike with differentials. This is because we don't have the linear approximation

$$ \delta \mathcal{L} = \frac{\delta\mathcal{L}}{\delta g^{ij}} \delta g^{ij} $$

as, again, we do have for differentials, but rather

$$ \delta \mathcal{L} = \frac{\delta\mathcal{L}}{\delta g^{ij}} \delta g^{ij} + \partial_i f^i, $$

for some vector $f^i$. This difference is what prevents $\frac{\delta \mathcal{L}}{\delta g^{ij}}$ from being a derivation. Doing the whole computation with the operator $\delta$ rather than the functional derivative $\frac{\delta}{\delta g^{ij}}$ works out just fine. This is actually what is depicted on the Wikipedia page; I simply assumed that the $\delta$-differential notation was a shorthand.

particle physics - Did the researchers at Fermilab find a fifth force?

Please consider the publication

Invariant Mass Distribution of Jet Pairs Produced in Association with a W boson in $p\bar{p}$ Collisions at $\sqrt{s} = 1.96$ TeV

by the CDF-Collaboration, already with huge media attention. I must admit that I am puzzled, astonished and excited at the same time. Not being an expert, it would be really nice if someone could shed light on the following question they posed:

Did we find a fifth fundamental force that cannot be explained by the standard model?

What do you think?

PS.: A part of figure 1 in the publication that is re-printed over and over again:

Answer

The likely answer is No, the whole signal is just an artifact of the difficult statistical manipulations. A Weizmann Institute physicist has noted that the peak has been shifted by one bin and the whole discovery could therefore be due to a pile-up effect or jet energy calibration; a small shift of the jet energy removes the effect. See also

http://motls.blogspot.com/2011/04/fermilab-cdf-new-force-press-conference.html

for more details. Additional doubts about the valid statistical procedures were mentioned on Tommaso Dorigo's blog, too.

An adjustment of jet energy by 3 percent is enough to make the signal totally insignificant. Animation by Tommaso Tabarelli de Fatis, a member of CMS.

The text above also reviews some theoretical literature and clarifies that if the signal happens to be real - and the D0 Collaboration is going to release its own verdict about the phenomenon in a few weeks - the most likely explanations are

1. 
   

   a new, fifth force - one mediated by a Z' boson (with the mass of 144 GeV or so) which is the particle that was decaying to two jets in those events. It must be leptophobic - (almost) no interactions with the leptons - and such Z' bosons, messengers of new $U(1)$ groups similar to the electroweak Z' boson but independent from them, are predicted by a very large fraction of grand unified theories and/or string theory models; articles by Alon Faraggi and many others are listed above; the simplest (but not the only) way to get a new leptophobic $U(1)$ is to obtain it as a piece of a color $SU(4)$ broken to $SU(3)\times U(1)$

   
   


2. 
   

   a technipion, a particle analogous to a pion in technicolor theories that break the electroweak symmetry by similar composite particles, and not necessarily scalar ones (so there is no true Higgs particle in those theories); a paper by Kenneth Lane et al. is linked above, too; technicolor has been thought to be nearly dead for years - the optimistic proposal to interpret the bump as a technicolor effect doesn't solve all the detailed problems with technicolor theories

   
   


3. 
   

   a stop squark in a supersymmetric theory - but it must be an R-parity-violating version of supersymmetry (e.g. because the new superpartners were not produced in pairs) which is unattractive for many other reasons (in simple terms, R-parity-violating theories are only consistent with proton stability given immense new assumptions, and they don't produce natural well-behaved dark-matter candidates); an article is linked to by the weblog above, too

   
   

quantum mechanics - Which electron is first ionized $n=2,ell=1, m=?$

Eletrons in atoms are described by n,l, m, s quantum numbers? For noble gas 1s2 2s2 2p6 which is the first electron that will be ionized n=2, l=1, m=?, s = ?

How does heat transfer between two atoms in solid material?

Been looking at heat equation and it's derivation, according to Wikipedia it uses 2 mathematical assumptions. My problem is that although it all seems OK, what is the physics of heat transfer in solids? so far haven't seen answer to the following questions regarding heat equation:

1. 
   

   Given 2 adjacent atoms, how does one measure the hot and the colds one? ( theoretically at least, no need for a physical thermometer)

   
   



2. 
   

   How does heat transfers from hot to the cold one? ( most likely by photons, but do electrons from one atom get knocked out and knockout the electrons of the other? is that considered a temperature increase or just change of charges?)

   
   


3. 
   

   If this transfer occurs with discreet amount of energy being moved from the hot one to the cold one, what are the intervals (time wise) between each transfer? (In other words what's the time interval between photons being emitted within a solid?)

   
   

Answer

Temperature is not a concept that has a lot of utility at the level of single atoms because it represents the mean kinetic energy of a group of particles (to within a coefficient).

You can define it, it just doesn't help much.

At the level of two atoms you revert to a more fundamental model such as the forces between them.

One atom transfers energy to another through electromagnetic forces between them. When that energy manifests as randomized kinetic energy at the microscopic scale we refer to it as "heat" at the macroscopic scale.

In a solid it is usually reasonable to treat the forces between individual pairs of atoms as being spring-like (i.e. they obey $F_{i,j} = -k(r_{i,j} - r_0)$). Starting from there you can build various models of solid behavior. For instance the Einstein model of a crystal.

quantum mechanics - Coherences in the density matrix

It is said that the off-diagonal elements of density matrix are "coherence". When a system interacts with its environment the off-diagonal elements decay and the final density matrix is the diagonal one, a statistical mixture. This process is called decoherence.

We know that every density matrix can be diagonalized in some basis.

What would decoherence be when the density matrix is diagonal in some basis?