The next step for technology is the quantum age, yet there is much theoretical work needed to be done before this becomes a reality. Super-Golden gates may be part of the solution. Quantum computers harness the quirks of quantum mechanics in order to perform extremely powerful computations and, if successfully created, could solve many problems previously deemed impossible, therefore impacting fields ranging from cancer research to artificial intelligence. Quantum computers are made up of building blocks known as gates, each of which alters the information travelling across it and, when combined, make a circuit. Super-Golden gates are a finite set of such gates that, when combined together with very few elements, can form circuits that approximate all other possible gates. This is crucial in the manufacturing process as it would be unrealistic to build every gate needed from scratch; hence, we want to construct them by combining others we already know how to create. To turn this physical problem into a mathematical one we require number theory, the study of integers. The project studies the number theory behind Super-Golden gates as described by Parzanchevski and Sarnak in their paper published in 2017 and will upon exploiting the ideas, implement them into a program. The program will output the circuit created from our Super-Golden gates to approximate any inputted gate. Further research into other possible finite gate sets must be done as Super-Golden gates leave gaps when trying to improve our approximations in the aim of producing a more accurate computer.