I spent my childhood hacking computers. When I was seven, my cousin showed up for Thanksgiving with a box filled with computer parts and we built my first computer. I got into competitive computer gaming around age eleven, and hacking was a natural extension of these activities. Then when I was sixteen, after doing poorly at a Counterstrike tournament, I decided that I should probably apply myself to other things. Needless to say, my parents were thrilled. So that’s when I bought my first computer (instead of building my own), which for deliberate but now antediluvian reasons was a Mac. A few years later, when I was taking CS 106 at Stanford, I was the first student in the course’s history whose reason for buying a Mac was “so that I couldn’t play computer games!” And now you know the story of my childhood.

The hacker mentality is quite different than the norm and my childhood trained me to look at absolutist laws as opportunities to find loopholes (of course only when legal and socially responsible!) I’ve applied this same mentality as I’ve been doing physics and I’d like to share with you some of the loopholes that I’ve gathered.

Scharnhorst effect enables light to travel faster than in vacuum (c=299,792,458 m/s): this is about the grandaddy of all laws, that nothing can travel faster than light in a vacuum! This effect is the most controversial on my list, because it hasn’t yet been experimentally verified, but it seems obvious with the right picture in mind. Most people’s mental model for light traveling in a vacuum is of little particles/waves called photons traveling through empty space. However, the vacuum is not empty! It is filled with pairs of virtual particles which momentarily fleet into existence. Interactions with these virtual particles create a small amount of ‘resistance’ as photons zoom through the vacuum (photons get absorbed into virtual electron-positron pairs and then spit back out as photons ad infinitum.) Thus, if we could somehow reduce the rate at which virtual particles are created, photons would interact less strongly with the vacuum, and would be able to travel marginally faster than c. But this is exactly what leads to the Casimir effect: the experimentally verified fact that if you take two mirrors and put them ~10 nanometers apart, then they will attract each other because there are more virtual particles created outside the cavity than inside [low momenta virtual modes are inaccessible because the uncertainty principle requires .] This effect is extremely small, only predicting that light would travel one part in faster than c. However, it should remind us all to deeply question assumptions.

This first loophole used quantum effects to beat a relativistic bound, but the next few loopholes are purely quantum, and are mainly related to that most quantum of all limits, the Heisenberg uncertainty principle.

Smashing the standard quantum limit (SQL) with squeezed measurements: the Heisenberg uncertainty principle tells us that there is a fundamental tradeoff in nature: the more precise your information about an object’s position, the less precise your knowledge about its momentum. Or vice versa, or replace x and p with E and t, or any other conjugate variables. This uncertainty principle is oftentimes written as . For a variety of reasons, in the early days of quantum mechanics, it was hard enough to imagine creating a state with , but there was some hope because this is obtained in the ground state of a quantum harmonic oscillator. In this case, we have . However, it was harder still to imagine creating states with , these states would be said to ‘go beyond the standard quantum limit’ (SQL). Over the intervening years, not only have we figured out how to go beyond the SQL using squeezed coherent states, but this is actually essential in some of our most exciting current experiments, like LIGO.

LIGO is an incredibly ambitious experiment which has been talked about multiple times on this blog. It is trying to usher in a new era of astronomy–moving beyond detecting photons–to detecting gravitational waves, ripples in spacetime which are generated as exceptionally massive objects merge, such as when two black holes collide. The effects of these waves on our local spacetime as they travel past earth are minuscule, on the scale of , which is about one thousand times shorter than the ‘diameter’ of a proton, and is the same order of magnitude as . Remarkably, LIGO has exploited squeezed light to demonstrate sensitivities beyond the SQL. LIGO expects to start detecting gravitational waves on a frequent basis as its upgrades deemed ‘advanced LIGO’ are completed over the next few years.

Compressed sensing beats Nyquist-Shannon: let’s play a game. Imagine I’m sending you a radio signal. How often do you need to measure the signal in order to be able to reconstruct it perfectly? The Nyquist-Shannon sampling theorem is a path-breaking result which Claude Shannon proved in 1949. If you measure at least twice as often as the highest frequency, then you are guaranteed perfect recovery of the signal. This incredibly profound result laid the foundation for modern communications. Also, it is important to realize that your signal can be much more general than simply radio waves, such as with a signal of images. This theorem is a sufficient condition for reconstruction, but is it necessary? Not even close. And it took us over 50 years to understand this in generality.

Compressed sensing was proposed between 2004-2006 by Emmanuel Candes, David Donaho and Terry Tao with important early contributions by Justin Romberg. I should note that Candes and Romberg were at Caltech during this period. The Nyquist-Shannon theorem told us that with a small amount of knowledge (a bound on the highest frequency) that we could reconstruct a signal perfectly by only measuring at a rate twice faster than the highest frequency–instead of needing to measure continuously. Compressed sensing says that with one extra assumption, assuming that only sparsely few of your frequencies are being used (call it 10 out of 1000), that you can recover your signal with high accuracy using dramatically fewer measurements. And it turns out that this assumption is valid for a huge range of applications: enabling real-time MRIs using conventional technology or more relevant to this blog, increasing our ability to distinguish quantum states via tomography.

Unlike the other topics in this blog post, I have never worked with compressed sensing, but my intuition goes like this: instead of measuring in the basis in which you are sparse (frequency for example), measure in a different basis. With high probability each of these measurements will pick up a little piece from each of the occupied modes. Then, to reconstruct your signal, you want to use the L0-“norm” to interpolate in such a way that you use the fewest frequency components possible. Computing the L0-“norm” is not efficient, so one of the major breakthroughs of compressed sensing was showing that with high probability computing the L1-norm approximates the L0 solution, and all of this can be done using a highly efficient linear program. However, I really shouldn’t be speculating because I’ve never invested much time into mastering this new tool, and I’m friends with a couple of the quantum state tomography authors, so maybe they’ll chime in?

Brahms is a cool dude. Brahms as a height map where cliffs=Gibbs phenomena=oh no! First three levels of Brahms as a Haar wavelet.

Wavelets as the mother of all bases: I previously wrote a post about the importance of choosing a convenient basis. Imagine you have an image which has a bunch of sharp contrasts, such as the outline of a person, or a horizon, or a table, basically anything. How do you store it efficiently? Due to the Gibbs phenomena, the Fourier basis is pretty awful for these applications. Here’s another motivating problem, imagine someone plays one note on an instrument. The sound is localized in both time and frequency. The Fourier basis is also pretty awful at storing/detecting this. Wavelets to the rescue! The theory of wavelets uses some beautiful math to solve the longstanding problem of finding a basis which is localized in both position and momenta space (or very close to it.) Wavelets have profound applications, some of my favorite include: modern image compression (JPEG 2000 onwards) is based on wavelets; Ingrid Daubechies and her colleagues used wavelets to detect forged paintings; recovering previously unrecoverable recordings of Brahms at the piano (I heard about this from Barry Simon, of Reed-Simon fame, who is currently teaching his last class ever); and even the FBI uses wavelets to compress images of fingerprints, obtaining a compression ratio of 20:1.

Postselection enables quantum cloning: the no-cloning theorem is well known in the field of quantum information. It says that you cannot find a machine (unitary operation U) which takes an arbitrary input state , and a known state , such that the machine maps to , and thereby cloning . This is very easy to prove using the linearity of quantum mechanics. However, there are loopholes. One of the most trivial loopholes is realizing that one can take the state and perform something called unambiguous state discrimination, which either spits out exactly which state is with some probability, or otherwise spits out “I don’t know which state.” You can postselect that the unambigious state discrimination succeeded and prepare a unitary which clones the relevant states. Peter Shor has a comment on physics stackexchange describing this. Seth Lloyd and John Preskill outlined a less trivial version of this in their recent paper which tries to circumvent firewalls by using postselected quantum teleportation.

In this blog post, I’ve only described a tiny fraction of the quantum loopholes that have been discovered. If I had more space/time, two of the next examples I would describe are beating classical correlations with quantum entanglement, in order to win at CHSH games. I would also describe weak measurements and some of the peculiar things they lead to. Beyond that, I would probably refer you to Yakir Aharonov’s amazingly fun book about quantum paradoxes.

After reading this, I hope that the next time you encounter an inviolable law of nature, you’ll apply the hacker mentality and attempt to strip it down to its essence, isolate assumptions, and potentially find a loophole. But while you’re doing this, remember that you should never argue with your mother, or with mathematics!

“What’s the hardest problem you’ve ever solved?”

Kids focus right in. Driven by a ruthless curiosity, they ask questions from which adults often shy away. Which is great, if you think you know the answer to everything a 7 year-old can possibly ask you…

Two Wednesdays ago, I was invited to participate in three Q&A sessions that quickly turned into Reddit-style AMA (ask-me-anything) sessions over Skype with four 5th grade classes and one 2nd grade class of students at Medina Elementary in Medina, Washington. When asked by the organizers what I would like the sessions to focus on, I initially thought of introducing students to the mod I helped design for Minecraft, called QCraft, which brings concepts like quantum entanglement and quantum superposition into the world of Minecraft. But then I changed my mind. I told the organizers that I would talk about anything the kids wanted to know more about. It dawned on me that maybe not all 5th graders are as excited about quantum physics as I am. Yet.

The students took the bait. They peppered me with questions for over two hours —everything from “What is a quantum physicist and how do you become one?” to “What is it like to work with a fashion designer (about my collaboration with Project Runway’s Alicia Hardesty on Project X Squared)?” and of course, “Why did you steal the cannon?” (learn more about the infamous Cannon Heist – yes kids, there is an ongoing war between the two schools and Caltech took the last (hot) shot just days ago.)”

Caltech students visited MIT during pre-frosh weekend, bearing some clever gifts.

Then they dug a little deeper: “If we have a quantum computer that knows the answer to everything, why do we need to go to school?” This question was a little tricky, so I framed the answer like this: I compared the computer to a sidekick, and the kids—the future scientists, artists and engineers —to superheroes. Sidekicks always look up to the superheroes for guidance and leadership. And then I got this question from a young girl: “If we are superheroes, what should we do with all this power?” I thought about it for a second and though my initial inclination was to go with: “You should make Angry Birds 3D!”, I went with this instead: “People often say, “Study hard so that one day you can cure cancer, figure out the theory of everything and save the world!” But I would rather see you all do things to understand the world. Sometimes you think you are saving the world when it does not need saving—it is just misunderstood. Find ways to understand one another and move to look for the value in others. Because there is always value in others, often hiding from us behind powerful emotions.” The kids listened in silence and, in that moment, I felt profoundly connected with them and their teachers.

I wasn’t expecting any more “deep” questions, until another young girl raised her hand and asked: “Can I be a quantum physicist, or is it only for the boys?” The ferocity of my answer caught me by surprise: “Of course you can! You can do anything you set your mind to and anyone who tells you otherwise, be it your teachers, your friends or even your parents, they are just wrong! In fact, you have the potential to leave all the boys in the class behind!” The applause and laughter from all the girls sounded even louder among the thunderous silence from the boys. Which is when I realized my mistake and added: “You boys can be superheroes too! Just make sure not to underestimate the girls. For your own sake.”

Why did I feel so strongly about this issue of women in science? Caltech has a notoriously bad reputation when it comes to the representation of women among our faculty and postdocs (graduate students too?) in areas such as Physics and Mathematics. IQIM has over a dozen male faculty members in its roster and only one woman: Prof. Nai-Chang Yeh. Anyone who meets Prof. Yeh quickly realizes that she is an intellectual powerhouse with boundless energy split among her research, her many students and requests for talks, conference organization and mentoring. Which is why, invariably, every one of the faculty members at IQIM feels really strongly about finding a balance and creating a more inclusive environment for women in science. This is a complex issue that requires a lot of introspection and creative ideas from all sides over the long term, but in the meantime, I just really wanted to tell the girls that I was counting on them to help with understanding our world, as much as I was counting on the boys. Quantum mechanics? They got it. Abstract math? No problem.*

It was of course inevitable that they would want to know why we created the Minecraft mod, a collaborative work between Google, MinecraftEDU and IQIM – after all, when I asked them if they had played Minecraft before, all hands shot up. Both IQIM and Google think it is important to educate younger generations about quantum computers and the complex ideas behind quantum physics; and more importantly, to meet kids where they play, in this case, inside the Minecraft game. I explained to the kids that the game was a place where they could experiment with concepts from quantum mechanics and that we were developing other resources to make sure they had a place to go to if they wanted to know more (see our animations with Jorge Cham at http://phdcomics.com/quantum).

As for the hardest problem I have ever solved? I described it in my first blog post here, An Intellectual Tornado. The kids sat listening in some sort of trance as I described the nearly perilous journey through the lands of “agony” and “self doubt” and into the valley of “grace”, the place one reaches when they learn to walk next to their worst fears, as understanding replaces fear and respect for a far superior opponent teaches true humility and instills in you a sense of adventure. By that time, I thought I was in the clear – as far as fielding difficult questions from 10 year-olds goes – but one little devil decided to ask me this simple question: “Can you explain in 2 minutes what quantum physics is?” Sure! You see kids, emptiness, what we call the quantum vacuum, underlies the emergence of spacetime through the build-up of correlations between disjoint degrees of freedom, we like to call entangled subsystems. The uniqueness of the Schmidt decomposition over generic quantum states, coupled with concentration of measure estimates over unequal bipartite decompositions gives rise to Schrodinger’s evolution and the concept of unitarity – which itself only emerges in the thermodynamic limit. In the remaining minute, let’s discuss the different interpretations of the following postulates of quantum mechanics: Let’s start with measurements…

Reaching out to elementary school kids is just one way we can make science come alive, and many of us here at IQIM look forward to sharing with kids of any age our love for adventuring far and wide to understand the world around us. In case you are an expert in anything, or just passionate about something, I highly recommend engaging the next generation through visits to classrooms and Skype sessions across state lines. Because, sometimes, you get something like this from their teacher:

Hello Dr. Michalakis,

My class was lucky enough to be able to participate in one of the Skype chats you did with Medina Elementary this morning. My students returned to the classroom with so many questions, wonderings, concerns, and ideas that we could spend the remainder of the year discussing them all.

Your ability to thoughtfully answer EVERY single question posed to you was amazing. I was so impressed and inspired by your responses that I am tempted to actually spend the remainder of the year discussing quantum mechanics J.

I particularly appreciated your point that our efforts should focus on trying to “understand the world” rather than “save” the world. I work each day to try and inspire curiosity and wonder in my students. You accomplished more towards my goal in about 40 minutes than I probably have all year. For that I am grateful.

All the best,
A.T.

* Several of my female classmates at MIT (where I did my undergraduate degree in Math with Computer Science) had a clarity of thought and a sense of perseverance that Seal Team Six would be envious of. So I would go to them for help with my hardest homework.

It was the summer of 2008. I was 22 years old, and it was my second week working in the crude oil and natural gas options pit at the New York Mercantile Exchange (NYMEX.) My head was throbbing after two consecutive weeks of disorientation. It was like being born into a new world, but without the neuroplasticity of a young human. And then the crowd erupted. “Yeeeehawwww. YeEEEeeHaaaWWWWW. Go get ’em cowboy.”

It seemed that everyone on the sprawling trading floor had started playing Wild Wild West and I had no idea why. After at least thirty seconds, the hollers started to move across the trading floor. They moved away 100 meters or so and then doubled back towards me. After a few meters, he finally got it, and I’m sure he learned a life lesson. Don’t be the biggest jerk in a room filled with traders, and especially, never wear triple-popped pastel-colored Lacoste shirts. This young aspiring trader had been “spurred.”

In other words, someone had made paper spurs out of trading receipts and taped them to his shoes. Go get ’em cowboy.

I was one academic quarter away from finishing a master’s degree in statistics at Stanford University and I had accepted a full time job working in the algorithmic trading group at DRW Trading. I was doing a summer internship before finishing my degree, and after three months of working in the algorithmic trading group in Chicago, I had volunteered to work at the NYMEX. Most ‘algo’ traders didn’t want this job, because it was far-removed from our mental mathematical monasteries, but I knew I would learn a tremendous amount, so I jumped at the opportunity. And by learn, I mean, get ripped calves and triceps, because my job was to stand in place for seven straight hours updating our mathematical models on a bulky tablet PC as trades occurred.

I have no vested interests in the world of high-frequency trading (HFT). I’m currently a PhD student in the quantum information group at Caltech and I have no intentions of returning to finance. I found the work enjoyable, but not as thrilling as thinking about the beginning of the universe (what else is?) However, I do feel like the current discussion about HFT is lop-sided and I’m hoping that I can broaden the perspective by telling a few short stories.

What are the main attacks against HFT? Three of them include the evilness of: front-running markets, making money out of nothing, and instability. It’s easy to point to extreme examples of algorithmic traders abusing markets, and they regularly do, but my argument is that HFT has simply computerized age-old tactics. In this process, these tactics have become more benign and markets more stable.

Front-running markets: large oil producing nations, such as Mexico, often want to hedge their exposure to changing market prices. They do this by purchasing options. This allows them to lock in a minimum sale price, for a fee of a few dollars per barrel. During my time at the NYMEX, I distinctly remember a broker shouting into the pit: “what’s the price on DEC9 puts.” A trader doesn’t want to give away whether they want to buy or sell, because if the other traders know, then they can artificially move the price. In this particular case, this broker was known to sometimes implement parts of Mexico’s oil hedge. The other traders in the pit suspected this was a trade for Mexico because of his anxious tone, some recent geopolitical news, and the expiration date of these options.

Some confident traders took a risk and faded the market. They ended up making between $1-2 million dollars from these trades, relative to what the fair price was at that moment. I mention relative to the fair price, because Mexico ultimately received the better end of this trade. The price of oil dropped in 2009, and Mexico executed its options enabling it to sell its oil at a higher than market price. Mexico spent $1.5 billion to hedge its oil exposure in 2009.

This was an example of humans anticipating the direction of a trade and capturing millions of dollars in profit as a result. It really is profit as long as the traders can redistribute their exposure at the ‘fair’ market price before markets move too far. The analogous strategy in HFT is called “front-running the market” which was highlighted in the New York Times’ recent article “the wolf hunters of Wall Street.” The HFT version involves analyzing the prices on dozens of exchanges simultaneously, and once an order is published in the order book of one exchange, then using this demand to adjust its orders on the other exchanges. This needs to be done within a few microseconds in order to be successful. This is the computerized version of anticipating demand and fading prices accordingly. These tactics as I described them are in a grey area, but they rapidly become illegal.

Making money from nothing: arbitrage opportunities have existed for as long as humans have been trading. I’m sure an ancient trader received quite the rush when he realized for the first time that he could buy gold in one marketplace and then sell it in another, for a profit. This is only worth the trader’s efforts if he makes a profit after all expenses have been taken into consideration. One of the simplest examples in modern terms is called triangle arbitrage, and it usually involves three pairs of currencies. Currency pairs are ratios; such as USD/AUD, which tells you, how many Australian dollars you receive for one US dollar. Imagine that there is a moment in time when the product of ratios is 1.01. Then, a trader can take her USD, buy AUD, then use her AUD to buy CAD, and then use her CAD to buy USD. As long as the underlying prices didn’t change while she carried out these three trades, she would capture one cent of profit per trade.

After a few trades like this, the prices will equilibrate and the ratio will be restored to one. This is an example of “making money out of nothing.” Clever people have been trading on arbitrage since ancient times and it is a fundamental source of liquidity. It guarantees that the price you pay in Sydney is the same as the price you pay in New York. It also means that if you’re willing to overpay by a penny per share, then you’re guaranteed a computer will find this opportunity and your order will be filled immediately. The main difference now is that once a computer has been programmed to look for a certain type of arbitrage, then the human mind can no longer compete. This is one of the original arenas where the term “high-frequency” was used. Whoever has the fastest machines, is the one who will capture the profit.

Instability: I believe that the arguments against HFT of this type have the most credibility. The concern here is that exceptional leverage creates opportunity for catastrophe. Imaginations ran wild after the Flash Crash of 2010, and even if imaginations outstripped reality, we learned much about the potential instabilities of HFT. A few questions were posed, and we are still debating the answers. What happens if market makers stop trading in unison? What happens if a programming error leads to billions of dollars in mistaken trades? Do feedback loops between algo strategies lead to artificial prices? These are reasonable questions, which are grounded in examples, and future regulation coupled with monitoring should add stability where it’s feasible.

The culture in wealth driven industries today is appalling. However, it’s no worse in HFT than in finance more broadly and many other industries. It’s important that we dissociate our disgust in a broad culture of greed from debates about the merit of HFT. Black boxes are easy targets for blame because they don’t defend themselves. But that doesn’t mean they aren’t useful when implemented properly.

Are we better off with HFT? I’d argue a resounding yes. The primary function of markets is to allocate capital efficiently. Three of the strongest measures of the efficacy of markets lie in “bid-ask” spreads, volume and volatility. If spreads are low and volume is high, then participants are essentially guaranteed access to capital at as close to the “fair price” as possible. There is huge academic literature on how HFT has impacted spreads and volume but the majority of it indicates that spreads have lowered and volume has increased. However, as alluded to above, all of these points are subtle–but in my opinion, it’s clear that HFT has increased the efficiency of markets (it turns out that computers can sometimes be helpful.) Estimates of HFT’s impact on volatility haven’t been nearly as favorable but I’d also argue these studies are more debatable. Basically, correlation is not causation, and it just so happens that our rapidly developing world is probably more volatile than the pre-HFT world of the last Millennia.

We could regulate away HFT, but we wouldn’t be able to get rid of the underlying problems people point to unless we got rid of markets altogether. As with any new industry, there are aspects of HFT that should be better monitored and regulated, but we should have level-heads and diverse data points as we continue this discussion. As with most important problems, I believe the ultimate solution here lies in educating the public. Or in other words, this is my plug for Python classes for all children!!

I promise that I’ll repent by writing something that involves actual quantum things within the next two weeks!