PiGate: A gate for working exactly with multiples of pi

[ScottWe]

Summary

This pull request introduces a gate which allows for working exactly with multiples of pi. This commit is the first step towards verifying circuits with rotations parameterized by multiples of pi.

Details

Currently, the parameterized equivalence checker in src/python/verifier does not support constant parameters. This is because, given an arbitrary real (or rational number) x, it is not possible to represent sin(x) exactly using only addition, multiplication, subtraction, division, and square roots (or even n-th roots). For special values of x, this is possible though.

This pull request introduces a new named gate, called pi(n), which corresponds to the constant angle pi/n. The implementation in this pull request handles many values of n which appear in quantum computing, such as when n is a power of two. More precisely, the techniques in this pull request can be extended to any n such that the regular n-gon is constructible (see the Gauss-Wantzel Theorem). The cases which I do not support (e.g., 17*2^k and 257*2^k) are unlikely to appear in real quantum algorithms and have horrendous formulas.

Future Work

In a follow-up pull request, I plan to integrate this with the OpenQASM 2 parser.

[xumingkuan]

LGTM, thanks!