[Unitary Hack draft] quantum modular arithmetic

[khnikhil]

Partially addresses #49 as part of the Unitary Fund's 2024 hackathon (see message on #49).

[matt-lourens]

Hey @khnikhil, I really like this! The explanation to code to explanation flows nicely and I really like the references to other HierarQcal tutorials. So far I only have a few minor comments, when you're done with the draft I'll go through everything again but you're on the right track.

Comments

• Motivation paragraph: I like the tone and general idea that it tries to communicate. On the more technical side, for quantum computing the operators are unitary but not necessarily Hermitian (for example a rotation gate is not Hermitian $R_z(\theta)R_z(\theta)\ket{\psi}=R_z(2\theta)\ket{\psi}$). But the Unitarity requirement alone does restrict us to reversible computation because $U^\dagger=U^{-1}$ and $U^\dagger$ always exists (the inverse is already encoded in the structure of the matrix). So update the paragraph but keep the tone and main idea. Out of interest, and seeing if I can remember myself
  • The restriction of Unitarity ensures whatever states we create are always normalized, $\braket{0|U^\dagger U|0}=1$ and so our probabilities sum to one. But there's many reasons one can give, you can also take it as one of the postulates that the evolution of a state is unitary. The "log" of the circuit is Hermitian though, by solving the Schrodinger equation we can write $U\ket{\psi}=e^{-\frac{i}{\hbar}\hat{H}t}\ket{\psi}$, and this $\hat{H}$ is Hermitian (it's the Energy operator and we want energy to be real )
• I really like the transition from half adder to full adder!
• The picture of the full adder is a bit small, maybe you can find a bigger one?