[GQSP] Negative powers and Hamiltonian simulation

[anurudhp]

No description provided.

[anurudhp]

@tanujkhattar I've implemented the hamiltonian simulation now, along with the call graph. I added unittests with randomly generated select and prepare oracles. I'm still debugging why the expected block doesn't match e^iHt. In the meantime could you review the rest of the code?

[tanujkhattar]

I'll take a look soon, thanks for the update

[anurudhp]

I've narrowed down the issue to this: as the $e^{it\cos\theta}$ approximation is very accurate, it forces $Q = 0$, and therefore all the QSP angles (thetas) are also 0. This gives us the trivial final unitary $U^d$ (d = degree), and not $P(U)$.

Mathematically this makes sense as the only polynomial s.t. $|P(e^{i\theta})| = 1$ is $P(z) = z^k$. But in practice we just have a really good approximation and $Q \approx 0$, which gets treated as $Q = 0$. I'm not sure how to fix this.

@tanujkhattar

[tanujkhattar]

Left a round of comments for code organization and cleanup while I think more about the hamiltonian simulation issue. Thanks for doing this!

[tanujkhattar]

Let's create a new subpackage for qualtran/bloqs/qsp/ and move the generalized qsp code inside the new folder. We can have two files right now

1. qualtran/bloqs/qsp/generalized_qsp.py - This should have GeneralizedQSP, gqsp_phase_factors, gqsp_complementary_polynomial (rename qsp to gqsp in the last 2 functions)
2. qualtran/bloqs/qsp/polynomial_approximations.py - This file can have functions which return coefficeints for commonly used polynomial approximations. For example: approx_exp_cos(precision, scaling_const) and approx_exp_sin(precision, scaling_const), which returns coefficients of the Jacobi-anger polynomial approximation which can be called directly from the HamiltonianSimulationViaGQSP Bloq.

For HamiltonianSimulationByGQSP, we can create a new subpackage for bloqs/hamiltonian_simulation/ and move HamiltonianSimulationByGQSP there. But this can be done once the above refactorings are complete to avoid merge conflicts.

[tanujkhattar]

For GeneralizedQSP, add a bloq_example, autogenerate the notebook from docstrings (you can add manually add more cells and they'll be preserve during the autogeneration process), add bloq_autotester tests and assert_decompose_is_consistent_with_t_complexity.

Does it make sense to show some kind of symbolic analysis in GQSP? The current API won't support it since we expect P and Q to be tuples of complex numbers and we currently don't have a way to specify sequences of variable symbolic lengths but we can file an issue to track this if there's a nice example where users would benefit from seeing the cost of applying GQSP as in terms of symbolic parameters they care about (for example hamiltonian simulation).

[anurudhp]

Being refactored in #848, #851

[anurudhp]

Split into #802, #810, #848, #851