Not so Far-Fetched

While I have all but forgotten about this blog, I haven't been posting with good reason: this semester has been marked by a notable increase in work. The main event, my PhD Screening Exam, is a mere two weeks off and I am deep into the studying at this point, hardly finding time for my classes, let alone any meaningful research. Putting all that aside, there has been some exciting developments in QIP around here lately...

As you can read in this article, the USC Information Sciences Institute cut the ribbon on the first D-Wave One computer outside of the D-Wave facilities in BC. The 128-qubit, adiabatic quantum computer comes as a quite generous gift from Lockheed. The machine is in place, cooled down to a nice 20 micro-Kelvin, and being calibrated over the next couple weeks, which means that we may soon take a little field trip out to Marina and try running some experiments.

Beer wise, things are still slow as I can't find the time to brew these days. However I have received packages of hops from the tipi at my parents place in Oregon which will hopefully get used up before too long. Also, I brewed the second annual Estate Pale Ale (almost 2 months ago!), an APA brewed with wet cascade and centennial hops from my own modest hop garden here in LA.

YES

This is old news at this point, but the latest, greatest beer bar to hit LA opened up on August 1st. Mohawk Bend boasts an impressive, rotating tap list of something between 65-72 California craft brews. They have food and spirits and wine to boot, also all from California, but I patronize them for the beer alone. What's possibly the best part of it all? I can step out my door and be at MB inside 10 minutes, walking.

The opening of this theater-turned-restaurant has been long awaited.  I'm not sure of the specifics of the delays, but I do know that I've been waiting patiently for MB to open despite the big day being pushed back from early spring into the depths of summer.  Now all that is in the past and the future looks very bright.  Each time I step up to order a pint the menu has changed pretty substantially, and I have been able to wet my whistle entirely with beers I'd never tasted before.  There are a number of breweries being poured that I'd never heard of before, and even more that I'd never seen at a bar.

I could go on about all the beers, but the bottom line is that you should check this place out.  They've been packed to the jowls so don't expect an easy time ordering, let alone finding much space to sit in the bar area.  You could wait for a table, but then you're in for a 1-2 hour wait.  I'd chalk it all up to a small staff and restricted hours, which supposedly they are planning to extend soon.  When that happens, you can be sure that I'll be in there on USC game day later this fall.

On a completely separate note, I discovered a most excellent YouTube series the other day covering all the bare-bones basics of Quantum Computation.  It is a series of short, self-contained classes (in the style of Khan Academy) by Michael Nielsen, the guy who literally wrote the book on Quantum Computation. If you have any interest in how QIS works and loosely what it's all about, you should give these videos a peep. You don't need any quantum mechanics to start working through it all, though some linear algebra is pretty integral to making any headway. Any way you slice it, the stuff is very interesting and these videos lay everything out as clearly as you will find anywhere, starting with the fundamental idea of the qubit and working towards very cool results like superdense coding and quantum teleportation. It's not science fiction, it's just science.

That Makes it Easy

I just had my mind blown.  In my summer course, Random Processes, I had a homework problem regarding estimation theory and determining 2nd order statistics about linear combinations of jointly Gaussian random variables. Part of the problem is to determine the Covariance of these linear combinations. As worked away I had an incredible realization: Covariance behaves like an inner product!

Check it out, it satisfies the properties of an inner produce on the probability space:
1) It's bilinear
\[
Cov\lbrack\gamma A + \delta B,Z\rbrack = \gamma Cov\lbrack A,Z\rbrack +\delta Cov\lbrack B,Z\rbrack\]
2) It's symmetric
\[
Cov\lbrack A,B\rbrack = Cov\lbrack B,A\rbrack\]
3) It's positive semi-definite (or what I think is better said non-negative definite, but that's for another post)
\[
Var\lbrack A\rbrack =Cov\lbrack A,A\rbrack\ge 0\]
Where $Cov\lbrack A,A\rbrack = 0$ implies that $A$ is a constant random variable.

So it's not exactly an inner product--there isn't the existence of a single zero. Rather, all constant random variables behave like zero. Apparently, this defines a Quotient Space--a vector space with an subspace $N$ that forms an equivalence class with $0$--and Covariance is an inner product over such a space.

Anyway, I looked this up on Wikipedia and, sure enough, there is a little section titled "Relationship to inner products" (which pointed me towards the article on quotient spaces). I guess it's not any huge discovery. I wasn't expecting that. But I am a little peeved that Variance and Covariance were never taught this way, or that this cool perspective was never even mentioned. I feel as though I pushed my understanding and intuition of Covariance way ahead by seeing this little change of face, and it would have been a huge help when taking probability courses in the past.

Grow Your Own

A little over a year ago, as part of a gift for my mother, I flew up to Oregon and installed a hops garden at my parents home in Tumalo.  I knew that they had some spare tipi poles lying around which I could use to create a unique frame for some hop vines.

After picking a good spot on their three acres I set to work, removing a small juniper and more than a few sizable rocks from the area.  The tipi (an 18 footer) went up and the rhizomes went in the ground, complete with a gravity fed soaker hose!  I can't quite remember at this point, but I think there were 16 plants--4 Chinook, 5 Centennial, and 7 Cascade--planted between each pair of poles, with two lengths of twine running up the each triangular face of the tipi.  Last season things went alright.  The plants only grew to about 6 feet and 3 Cascade plants never even broke ground.  I'd chalk it up to planting a little late in an already short growing season.

This season, things are looking much better.  We replaced 3 Cascades that didn't make it and did a little fertilizing.  This year the soaker is running 24 hours a day, which is okay in such sandy soil and the plants seem to love it.  The result?  The biggest hop cone you've ever seen!

Clearly there is more growing to be done.  But the sides are filling in quite nicely and it's creating a neat space inside.  Hopefully this season the entire thing will fill in, but definitely in a couple years, once all the plants really mature, this will be quite the cool spot to grab some shade and enjoy an Arnold Palmer or a cool beer, if you're into that sort of thing.

From a brewing standpoint, this is also quite exciting.  I'll need to teach my parents when and how to harvest, but I think I can look forward to an impressive shipment of fresh hops to play around with this fall.  The only problem is that once the plants get to the top they will just start tangling around and create a big, knotted mess.  While this will work wonders to create shade in the middle of the day it may not be so easy to determine which hops are which during harvest.  The solution?  Say "fuck it" and just brew away!  Chinook, Centennial and Cascade can all be used as late addition hops and work well in APAs and IPAs, so I think blindly adding hops by the fistful in the last 10 minutes of the boil could yield some great, one-time-only big hoppy beers.  Talk about seasonal brewing.

Brewing Withdrawal

It's been some time since I've last brewed.  You see, I broke my ankle 3 weeks ago and have been relegated to crutches as my only means of getting around.  That being said, I've not been spending any time lifting carboys or carrying around 6 gallons of hot wort.  The last time I brewed was on May 7th!  So much for the summer plans of brewing weekly.

At this point, the kegs are running dangerously low and I need to heal up so that I can crank out some Bitter and Porter to fill the pipeline again.  The upshot is that the beer I made on May 7th, an American Pale Ale, hasn't been touched since I pitched the yeast on 5/8.  With a little lifting help that beer will be in a keg by day's end.  As always, this is one of my favorite parts of the whole beer-making process because I'll have an entire hydrometer to sample and get a first glimpse of how the beer turned out.

Now more about the beer...

Summer Pale Ale, brewed on May 7th, 2011

Grain:
8.25# 2-Row
1.68# Dark Munich
1.16# Crystal 40° L

Hops:
12 AAUs Cascade (60 min)
1 oz Centennial (7 min)
0.5 oz Cascade (3 min)
0.5 oz Cascade (0 min)

Yeast:
Wyeast 1056 - American Ale

Mashed at 152° for 60 minutes with a thickness of 1.66 qts/lb.  LA has moderately hard water so I used 3.5 gallons of distilled water mixed with 5 gallons of de-chlorinated tap water.  I hit my mash temp on the nose and I chalk it up to good pre-heating of my mash tun.  I had a problem hitting my mash temps and then realized that preheating takes more than just a couple pitchers of hot water 10 minutes before the strike.  I pre-heat for 45 minutes, adding water as fast as I can heat it on the stove.

After chilling the wort I read an OG of 1.054, one point short of where I was aiming.  Started yeast in 3/4 pitcher of chilled wort and pitched the next day.  Over the first 4 days of fermentation the water bath measured between 66°-67°.

So it's been chugging away for just short of 6 weeks.  At this point in writing the beer is kegged, under pressure, and cooling in the fridge.  Final gravity came out to a dry 1.009.  Down from 1.054, that puts the ABV at ~5.9%.  The color is a nice, golden copper.  So far so good in keeping to the style guidelines.  There was a bit of stirred up yeast in the beer at the end of the siphon, but it's just due to the 1056 and it's nothing 2 weeks of chilling won't clear up nicely.

The taste?  A big hop flavor!  The ounce of Cascade in the last 3 minutes of the boil give it those classic citrus notes.  The Centennial works well to add some complexity.  The beer is crisply bitter, but not overwhelming.  I could see it walking the line with an IPA, especially with it's subtle alcohol flavor, and the calculated IBUs--46 by Tinseth--are at the high end of the spectrum for a Pale Ale.  All the same I think I can easily tack on the old "West Coast" title and call it a day.  The malt profile is simple.  The Munich works nicely to deepen the base flavor, and the crystal 40 is just doing it's crystal 40 thing--offering a caramel sweetness.

I'm quite excited about this beer.  It's the same recipe as the "Estate Pale Ale" I made last fall to experiment with wet-hopping using home grown hops.  I used the same bittering addition and just dumped in a jumble of fresh Columbus, Cascade and Centennial continuously throughout the last 10 minutes.  I think this beer will see an improvement in the hop character--the flavor hops are more organized.  Very present, but more orderly so one can more easily identify the different hops.

I'm looking forward to the next couple weeks.  With any luck, I'll get some help and I will be able to sneak in a brew day.  In the meantime, it'll take some willpower to let this pale rest and get ready.  It's about time that there were some fresh beer on tap around here.

Bad Memory

Throughout my entire schooling, I've never been very good at remembering the minutia.  This, I believe, is why I was never any good at history despite how much I enjoy it.  I also think that this gives rise to my success in mathematics and physics: I never really remember anything, but rather how to derive it.  In working over the derivations I piece together a larger world in which I can swim about and see the landscape that disparate topics create when viewed from afar.  With this in mind, I recently came across a very nice, straight-forward derivation of Schrödinger's equation that I'd like to share, plus my LaTeX could use some practice...

For Uniform Dynamics, time evolution is described by a unitary operator:
\[
|\psi(t)\rangle = U(t,t_0)|\psi(t_0)\rangle
\]And we can consider the derivative with respect to time of a state $|\psi\rangle$ at time $t$ as an operator $G$ acting on $|\psi(t)\rangle$:
\[
\begin{align}
\frac{d}{dt}|\psi(t)\rangle & = G|\psi(t)\rangle \\
\ & = \frac{d}{dt}U(t,t_0)|\psi(t_0)\rangle \\
\ & = \frac{d}{dt}U(t,t_0)\lbrack U(t,t_0)^\dagger U(t,t_0) \rbrack |\psi(t_0)\rangle \\
\end{align}
\]So, because $U(t,t_0)|\psi(t_0)\rangle=|\psi(t)\rangle$ we can say that $G = \frac{d}{dt}U(t,t_0)U(t,t_0)^\dagger$, and, keeping normalization constant,
\[\begin{align}
\frac{d}{dt}\langle\psi|\psi\rangle & = 0 \\
\ & = \lbrack\frac{d}{dt}\langle\psi(t)|\rbrack|\psi(t)\rangle + \langle\psi(t)|\lbrack\frac{d}{dt}|\psi(t)\rangle\rbrack \\
\ & = \langle\psi(t)|G^\dagger|\psi(t)\rangle + \langle\psi(t)|G|\psi(t)\rangle \\
\ & = \langle\psi(t)|G^\dagger + G|\psi(t)\rangle
\end{align}\]Therefore $G^\dagger = -G$, meaning that $G$ is Anti-Hermitian and we may write it as $i$ times a Hermitian operator, say $H$:
\[H = i\hbar G\]Here the $\hbar$ is just a constant which could be absorbed into $H$ but we need it for convention and for keeping the units squared away. Using this knowledge paired with our first consideration of the time rate of change of a given state we have ourselves what we're looking for:
\[\begin{align}
\frac{d}{dt}|\psi(t)\rangle & = G|\psi(t)\rangle \\
\ i\hbar\frac{d}{dt}|\psi(t)\rangle & = H|\psi(t)\rangle
\end{align}\]We can call $H$ the Hamiltonian, and it works out that it has units of energy, as well as the basis states being the eigenstates with energy eigenvalues. Perfect! It is working through guided derivations such as this one that really helps me keep the pieces organized, rather than just memorizing an equation and how to use it.

LaTeX!

So I just figured out how to use LaTeX in Blogger, check it out:

\[
\begin{align}
\nabla \times \vec{\mathbf{B}} -\, \frac1c\, \frac{\partial\vec{\mathbf{E}}}
{\partial t} & = \frac{4\pi}{c}\vec{\mathbf{j}} \\
\nabla \cdot \vec{\mathbf{E}} & = 4 \pi \rho \\
\nabla \times \vec{\mathbf{E}}\, +\, \frac1c\, \frac{\partial\vec{\mathbf{B}}}{\partial t} & = \vec{\mathbf{0}} \\
\nabla \cdot \vec{\mathbf{B}} & = 0
\end{align}
\]