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If condensed matter theorists have the Ising model, gravitational physicists have the Schwarzschild solution, and quantum foundation theorists have the Bell inequalities, then theoretical computer scientists have satisfiability, or SAT. In the world of computer science (and particularly computational complexity), many discussions inevitably circle back to SAT. In fact, SAT isn’t just something that theoretical computer scientists study. Satisfiability has a rich history with statistical physics, a field which wields powerful tools to probe the properties of SAT. As such, SAT is a problem which touches several fields, which makes it a breeding ground for cross-disciplinary ideas.
To be a scientist means to explore. You need to start from what is known and jump out into the void, investigating new ideas. In this regard, the scientist is an explorer, a person searching for new truths in a world without a map. To be more precise, a scientist uncovers the new map as they learn.
If you want to learn a topic today, the resources are much more plentiful than even a few decades ago. The internet has given us wonderful resources to learn from, including some which leverage internet technologies to provide animations and teach topics in a much more interactive way. This is particularly true for mathematics and physics, which have been entrenched in dry textbooks that are a chore to read for much too long.
Hilbert space is big.
No, not big like the how the Earth is big compared to you. Rather, Hilbert space is astronomically big. Actually, that’s not quite right either. It’s bigger than that. I guess the best adverb I can use is that it’s mathematically big. In a Hilbert space, you tend to have a lot of room to maneuver. (To read more about that, check out my essay, “The Curse of Dimensionality”.)
In the last essay, we saw how to pick a random unit vector in a N dimensional space. This wasn’t for nothing, and now we are going to put that knowledge to use to ask the following question:
If I have a line of N qubits and randomly pick a state from the Hilbert space, what kind of entanglement will it have?