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Q: What Are the Hardest Algorithms?
A: They’re all easy.
Just kidding. Not all quantum algorithms are easy. But, I‘ve been asked what I think are the hardest ones. Additional criteria are that the algorithms should be practical, scalable, and not textbook like Shor’s Factoring Algorithm and Grover’s Algorithm. Also, what would be the difference between running them on real hardware and on a simulator?
The List
For me, one algorithm stands above all others. Therefore, my list of the hardest algorithms includes just one: amplitude encoding. And while you may argue that amplitude encoding shouldn’t count here, after all it is input state preparation for other algorithms, it can stand alone in generative algorithms and I’m counting it. It is, without a doubt, the hardest problem I’ve solved to-date.
Is it practical?
Absolutely. Besides its aforementioned role in state preparation, which is important for many algorithms, it can be used to model human cognition. I’ve used it to compose music, but it also has applications in generating language and art. I can imagine its applicability to robotics, as well.
Is it scalable?
Maybe. This is part of the quest for quantum memory, aka QRAM, because how can we possibly encode 1,000,000…
