This past summer, my long time climbing partner David Sobek and I traveled down to Quito to test ourselves against a few of Ecuador's famous and towering volcanoes. This is a brief trip report of time there.

My goal during this trip was to see how my body reacts at elevations greater than 4800 m (the rounded-up elevation of Mont Blanc). Suffice it to say, I accomplished this goal, and have since been convinced of two things as I continue in mountaineering. One: climbing mountains for the sole purpose of "getting high" is not appealing to me. I'd much rather climb a technically interesting peak than one that is a boring and monotonous slog up to some higher elevation. Two: with proper acclimatization and rest, I react fine at elevation. The difficulty, however, is getting rest up high, which is something to work on.

Our trip consisted of five volcanoes: Pasochoa (4200 m), Corozon (4790 m), Iliniza Norte (5125 m), Cotopaxi (5897 m), and Chimborazo (6263 m). Interestingly, to climb glaciated peaks in Ecuador, you are required to go with a guide, no matter how experienced you are. Our guide, Cristian, was a treat. Despite being somewhat scrawny, you could tell he had years of experience under his belt. Indeed, whenever I asked him a question to the effect of "do you know about mountain X", he'd almost always reply something like "yes, I climbed that when I was 20 years old". On top of this, he was incredibly supportive and kind, and encouraged us to go to Bolivia for our next trip, which will probably happen.

Anyway, starting on Pasochoa, David and I rocketed up, with Cristian telling up to run the final bit because he could tell we have too much energy. The scenery here was shocking. Unlike Colorado where the mountains seem like big piles of dirt, the mountains in the Avenue of Volcanoes are lush, despite being much higher than any 14er. Here's a caracara on top of Pasochoa:

The next day, we ventured to Volcan Corozon, which, similar to Pasochoa, was really just a hike through a grassy field, albeit at over 15000 ft. I guess this was more technical, but nothing to sweat over. Here's Cristian leading us down:

Our third acclimatization hike was Volcan Iliniza Norte. This is where things started to get a little more serious, though the day was still short (I think 7 hours car to car). The trailhead is around 13000 ft, which you get to via a very bumpy but scenic drive. Before the scrambly bit on this climb, we stopped at the Nuevos Horizontes refuge to layer up. Here, there are incredibly views of the towering Iliniza Sur, which is an objective I'd like to tackle one day. Shortly after, we find ourselves on a rocky ridge with Sur in the background:

Not long after we summited, and enjoyed some fun scree-skiing on our quick descent.

We took a day off, used the zip line at our hostel, and then the next day set off for one of the two big objectives: Volcan Cotopaxi.

This would be a two day thing, beginning at the trailhead at a ridiculous 15000 ft or so. We slowly worked our way 1500 ft or so up, and settled in the Jose F. Rivas, which housed a fun little climbing gym that David and I quickly started playing in. Turns out, rock climbing at nearly 17000 ft is hard.

A nice dinner and some tea later, we sorted our packs and were off to bed. And in no time at all, it was midnight, and we were eating breakfast about to go off.

The climb up Cotopaxi is a slog. It's not particularly interesting, except for the excite of crossing a crevasse here and then, or trusting your crampons on icy volcanic rock. We made it to the top wearing respirators to fend off the SO2 at the top of this (still active) volcano. It was hard to breathe at the altitude, but it was made ten times harder by the respirator, which literally felt like breathing through a straw. Here's a summit photo, taken right as the sun was coming up:

We were psyched. On the way down, Cristian got some good photos of us hiking near the towering glacier walls and the clouds:

After getting back, we took another rest day (which again entailed just riding the zip line again and again). Then, we endured a long drive to our final objective, Chimborazo, whose summit is the closest point to the sun thanks to the equatorial bulge.

We started off toward high camp on Chimborazo mid-day, and arrived shortly after. We enjoyed the following view the whole way:

Unfortunately, I did not sleep well at 17000 ft that night, and so early morning the next day on our summit push, I felt terrible. We made it to about 5600 m before I had to throw in the towel due to exhaustion. I was bummed. Nevertheless, I also felt a sense of "not caring", in part because Chimborazo for me was always about the elevation, not the climb.

In reflection, I think climbing mountains for the purpose of "getting high" is not all that admirable. Better, in my opinion, to climb the mountains that actually mean something to you, and that challenge you in the way that you want to be challenged. For me, that means a technical challenge, like some sort of rock or mixed climbing. Thank you, Ecuador, for that incredible trip. Alpamayo, here I come.

What in a human life affords the closest thing to the feeling or experience of "foreverness"? My guess, at least at my current stage in life, is the death of a loved one or the death of a relationship, but I conjecture that these are related.

In the light of that otherwise deep, dark, sinking, forever-gone feeling, I wrote a song. Its name refers to the uncertainty of the future and forever as we humans proceed To God Knows Where.

Given any complexity class \[\mathsf{C}\], the ``random certificates with bounded two-sided error'' operator \[\mathsf{BP}\] is defined as follows: A language \[L \in \mathsf{BP} \cdot \mathsf{C}\] if and only if there exists a polynomial \[p\] and a language \[V \in \mathsf{C}\] such that for all \[x \in \Sigma^*\],

- \[x \in L \implies Pr_r[(x,r) \in V] \geq 3/4\]

- \[x \not\in L \implies Pr_r[(x,r) \not\in V] \geq 3/4\]

Here, it is assumed that \[r \in \Sigma^{p(|x|)}\] is drawn uniformly.

An obvious quirk of this definition is the 3/4, and this really bothered me the first time I ever came across the definitions of \[\mathsf{BPP}\] and \[\mathsf{BQP}\]. Of course, I now know that for \[\mathsf{BPP}\] and \[\mathsf{BQP}\] the 3/4 (or sometimes 2/3) bounds are totally incidental, in the sense that there can be boosted to \[1 - 2^{-t(n)}\] for any polynomial \[t\]. This follows by basically taking a majority vote and applying the Chernoff bound.

A similar thing is true more generally for the class \[\mathsf{BP} \cdot \mathsf{C}\]. In particular, if \[\mathsf{C}\] is closed under (polynomial) majority reductions, then it follows that probability amplification is possible in \[\mathsf{BP} \cdot \mathsf{C}\].

Probability amplification, by the way, is formally defined as follows: A class \[\mathsf{BP} \cdot \mathsf{C}\] admits of probability amplification if and only if for all polynomials \[t\], there is a polynomial \[p\] such that for all \[x \in \Sigma^*\]:

- \[x \in L \implies Pr_r[(x,r) \in V] \geq 1 - 2^{-t(|x|)}\]

- \[x \not\in L \implies Pr_r[(x,r) \not\in V] \geq 1 - 2^{-t(|x|)}\],

where \[r \in \Sigma^{p(|x|)}\] is drawn uniformly.

The easily provable claim, then, is that if \[\mathsf{C}\] is closed under majority reductions, then \[\mathsf{BP} \cdot \mathsf{C}\] admits of probability amplification.

Now, what is interesting about this is if the MAJORITY function is all that essential here. Is it necessary that \[\mathsf{C}\] be able to compute MAJORITY (in the sense of being closed under majority reductions) in order for \[\mathsf{BP} \cdot \mathsf{C}\] to admit of probability amplification?

I find it really interesting that the answer is, in fact, no. The counterexample is \[\mathsf{BP} \cdot \mathsf{AC}^0\]. \[\mathsf{AC}^0\] can't compute MAJORITY, but yet it is known that \[\mathsf{BP} \cdot \mathsf{AC}^0 = \mathsf{AC}^0\]. Therefore, the randomness does nothing for you in \[\mathsf{AC}^0\]. In particular, \[\mathsf{BP} \cdot \mathsf{AC}^0 = \mathsf{AC}^0\] implies \[\mathsf{BP} \cdot \mathsf{AC}^0\] admits of probability amplification, because the probabilities can be made unity.

This begs the question whether a class \[\mathsf{BP} \cdot \mathsf{C}\] admits of probability amplification if and only if either

- \[\mathsf{BP} \cdot \mathsf{C} = \mathsf{C}\] (i.e., there is a way to derandomize \[\mathsf{BP} \cdot \mathsf{C}\])

- or \[\mathsf{C}\] is closed under majority reductions.

Excited to think about this a bit more! If you see a counterexample, then please let me know.

Posted in:
Complexity

I had the pleasure of lecturing at the Quetzal Quantum Accelerator Program this year, and decided, after a bit of deliberating with the organizers, to speak on one of the topics in quantum mechanics that is closest to my heart: Bell's Theorem, with an eye toward nonlocal games.

The talk was geared toward high school and early undergraduates, but inevitably sometimes you just have to do the quantum mechanics (though I stand behind my belief that conveying Bell nonlocality is *way* easier than explaining Shor's algorithm, especially to people with little to no number theory in their back pockets). Anyway, I think I did a good job conveying the basic idea behind Bell, but would still appreciate feedback from others. My talk is available on YouTube here. (Feel free to skip over the minor technical issues I had at the start!) Oh, and shoutout to Sofia's great introductory talk on quantum mechanics!