Optimise `applyQFT()`

[TysonRayJones]

The current implementation of applyQuantumFourierTransform() merely effects the canonical gates of the QFT in-turn. An optimised routine is possible, whereby we merge contiguous phase gates into a single, diagonal operator. In QuEST v3, this was implemented using the (now deprecated and absolutely awful) applyPhaseFunc() routine.

In QuEST v4, an equivalent diagonal operator could be effected by a applyFullStateDiagMatr using a temporarily allocated applyFullStateDiagMatr. However, this incurs gratuitous memory allocation costs given that the elements are compile-time known.

We should instead implement the necessary diagonal operator as a bespoke routine. This requires re-deriving it (since removed last-minute from the distributed manuscript) and updating applyQuantumFourierTransform() to make use of it. The algorithm is very cute, compatible with both distribution, and density matrices, and non-contiguous arbitrarily-ordered target qubits. As such, it may be a superior solution to performing the FFT upon the constituent amplitudes directly.

This may serve as an excellent project for a student and worthy of publication 🙌

[TysonRayJones]

Worth pointing out that QFT on a non-contiguous, arbitrarily-ordered subset of qubits can make use of FFT; we simply swap active qubits to the lowest ordered positions, apply many instances of FFT in parallel upon all 2^#targets sub-arrays, and SWAP back. This works also on density matrices whereby we process each column independently. However...

• this precludes full-state QFT in distributed settings; we could not target log2(#nodes) qubits.
• this would really benefit from a fused SWAP
• this complicates threading logic; we should parallelise within one big FFT, but across multiple small FFTs.
• this requires ill-motivated use of an external library (like FFTW) for a core function, or reinvention of the wheel