Bitlength of "keep" and "alt" probabilities in coherent alias sampling

[christoph-riverlane]

Qualtran computes this as log_2(ε/n): https://github.com/quantumlib/Qualtran/blob/0d8793e8a7f415d6ec1e003756422d72ac81a3a1/qualtran/linalg/lcu_util.py#L161 and does not make any reference to λ (the 1-norm / subnormalisation).

However, this doesn't match the papers:

• Eq. (36) in "Encoding Electronic Spectra in Quantum Circuits with Linear T Complexity" has a different formula with dependence on λ.
• Eq. (A12) in the Tensor Hypercontraction paper also has a formula for the bit length of the keep probabilities that makes reference to λ.

Tagging @tanujkhattar whom I briefly discussed this inconsistency with at the Qualtran workshop last Friday

[tanujkhattar]

@anurudhp Since you are working on extending state preparation alias sampling, can you take this one as well?

[anurudhp]

Let me take a look