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Compile an arbitrary unitary matrix

Introduction of my notes

Peter Shor, once reply to a comment by a reviewer, which criticize the reliability of quantum algorithm, by saying that:

Just like photons are both wave and particles, quantum computers are both analog and digital. We can use the digital part to do error correction-- Peter Shor

The motivation of taking this note, is from a homework of quantum computation class PHY 245 in UCLA by professor Hurtson.

Which ask us to create arbitrary single qubit unitary gate by THT gate sequence THTH . The problem raise my interest so I study more on Solovay-Kitaev theorem, by watching the youtube video by Preskill and take some notes about it.

Just like the way perter shor explain error correction, there is also a mix between analog and digital in the realm of compilation. Solovay-Kitaev theorem treat circuit compilation as a digital process, without any information of the hamiltonion.

In a nutshell, I take this note trying to understand the following two problem:

1.What kind of gate subset is complete to generate arbitrary unitary gate?

2.What is the time complexity to compile the finite gate set?

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