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What are some of the most probable directions for quantum computation to go in the future and what research still needs to be done in the field?

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  • $\begingroup$ Hi, Logan. This question is way too broad and not a good fit for our site's format. Perhaps restrict your question to any one subfield of quantum computing and quantum information? $\endgroup$ – Sanchayan Dutta Feb 6 at 4:58
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One research direction that's in some ways just getting going is making the constant factors on the costs of quantum computation smaller. For example, the the first international workshop on quantum resource estimation (QRE2019) didn't happen ten years ago, it's happening later this year.

Given known techniques, if you try to run a classical computation on an error corrected quantum computer, it will be billions or trillions of times less efficient. So basically quantum algorithms start with that as a penalty factor. The constant factors are so bad that, unless we make serious improvements, no one will ever see a benefit in practice from running Grover's search algorithm.

As another example, the meat of Shor's algorithm is basically just one modular exponentiation done under superposition. Modular exponentiations are an operation your computer can do in a fraction of a second. Yet the resource estimates for running Shor's algorithm are huge.

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