Sparse state preparation via alias sampling #1067
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@tanujkhattar PTAL |
qualtran/bloqs/state_preparation/state_preparation_alias_sampling.py
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| In particular, we take the zero state to: | ||
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| $$ | ||
| \sum_{l=0}^{L-1} \sqrt{p_l} |\mathrm{ind}_l\rangle |\mathrm{temp}_\ell\rangle | ||
| $$ | ||
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| where $\mathrm{ind}_l$ is the index of the $l$-th non-zero coefficient, | ||
| and the probabilities $p_\ell$ are $\mu$-bit binary approximations to the true values and | ||
| where the temporary register must be treated with care, see the details in Section 5 of | ||
| reference [1] and Section III.D. of the reference [2]. | ||
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| The preparation is equivalent to [classical alias sampling] | ||
| (https://en.wikipedia.org/wiki/Alias_method): we sample `l` with probability `p[l]` by first | ||
| selecting `l` uniformly at random and then returning `ind[l]` with probability `keep[l] / 2**mu`; | ||
| otherwise returning `alt[l]`. | ||
| This bloq is nearly identical to :class:`StatePreparationByAliasSampling`, except that this loads | ||
| the non-zero coefficient indices as well from the QROM. |
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I think we should ask users to read docstring of StatePreparationAliasSampling to understand how alias sampling works and describe the reduction from sparse state preparation to dense state preparation in this docstring; similar to how its described in Ref[1].
We should explicitly define symbols for sparsity (S) and dimension n of the selection register (s.t. N = 2^n) and say how we wish to do state preparation that scales as S instead of N.
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Defined sparsity
qualtran/bloqs/state_preparation/state_preparation_alias_sampling.py
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| @classmethod | ||
| def from_lcu_probs( | ||
| cls, | ||
| lcu_probabilities: Sequence[float], |
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Should we change the input format and expect a dictionary Dict[int, float] that specifies the coefficients for the sparse indices? We can construct a dense lcu_probabilities as part of the function?
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yeah this makes more sense, I added this as the default method
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| mu = sub_bit_prec_from_epsilon(n_coeff, probability_epsilon) | ||
| selection_bitsize = bit_length(n_coeff - 1) |
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Are you sure this should be n_coeff and not n_nonzero_coeff ?
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It is n_coeff. The selection register actually stays the same as the true LCU selection register (i.e. over [0, L)). I use an extra junk register called sparse_index which iterates over [0, d) (d = sparsity), which is used to prepare the state.
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Symbolic costs for sparsity (from the |
fixes #1059