| Safe Haskell | None |
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Quipper.Algorithms.CL.RegulatorQuantum
Description
This module implements the specialized quantum operations required in stages 1 and 4 of Hallgren’s algorithm.
The key operation for stage 1 is q_fN, implementing fN, the quasi-periodic
function used in approximating the regulator. This is the function h of
[Jozsa 2003], Sec. 9, discretized with precision 1/N and translated by a
specified jitter parameter.
The key functions for stage 4 are not yet implemented. These essentially consist of the functions fI,N, analogues of fN operating within the equivalence classes of possibly non-principal ideals I (representing other elements of the class group), as described in [Hallgren 2006, Section 5].
Synopsis
- q_normalize :: IdDistQ -> Circ (IdDistQ, IdDistQ)
- q_rho_d :: IdDistQ -> Circ (IdDistQ, IdDistQ)
- q_rho_inv_d :: IdDistQ -> Circ (IdDistQ, IdDistQ)
- q_rho_red_d :: IdRedDistQ -> Circ (IdRedDistQ, IdRedDistQ)
- q_rho_inv_red_d :: IdRedDistQ -> Circ (IdRedDistQ, IdRedDistQ)
- q_dot_prod :: IdealRedQ -> IdealRedQ -> Circ (IdealRedQ, IdealRedQ, IdealQ)
- q_dot_prod_with_dist :: IdRedDistQ -> IdRedDistQ -> Circ (IdRedDistQ, IdRedDistQ, IdDistQ)
- q_star_prod :: IdRedDistQ -> IdRedDistQ -> Circ (IdRedDistQ, IdRedDistQ, IdRedDistQ)
- q_star_square :: IdRedDistQ -> Circ (IdRedDistQ, IdRedDistQ)
- q_fN :: CLIntP -> Int -> Int -> QDInt -> IntM -> Circ (QDInt, (IdealRedQ, FPRealQ))
Basic operations on ideals
q_normalize :: IdDistQ -> Circ (IdDistQ, IdDistQ) Source #
Send I = <k,l,a,b> to l/ka I = <1,a,a,b>.
On distances, send δI to δI - log (l/ka).
q_rho_red_d :: IdRedDistQ -> Circ (IdRedDistQ, IdRedDistQ) Source #
As q_rho_d, but for reduced ideals.
q_rho_inv_red_d :: IdRedDistQ -> Circ (IdRedDistQ, IdRedDistQ) Source #
As q_rho_inv_d, but for reduced ideals.
Products of ideals
q_dot_prod :: IdealRedQ -> IdealRedQ -> Circ (IdealRedQ, IdealRedQ, IdealQ) Source #
Compute the ordinary (not necessarily reduced) product of two reduced fractional ideals.
This is I⋅J of [Jozsa 2003], Sec 7.1, following the description given in Prop. 34.
q_dot_prod_with_dist :: IdRedDistQ -> IdRedDistQ -> Circ (IdRedDistQ, IdRedDistQ, IdDistQ) Source #
Compute the dot-product of two reduced fractional ideals, all with distance.
q_star_prod :: IdRedDistQ -> IdRedDistQ -> Circ (IdRedDistQ, IdRedDistQ, IdRedDistQ) Source #
Given two reduced ideals-with-distance, compute their star-product, with distance.
This is I*J of [Jozsa 2003], Sec. 7.1, defined as the first reduced ideal-with-distance following I⋅J.
q_star_square :: IdRedDistQ -> Circ (IdRedDistQ, IdRedDistQ) Source #
Compute I * I, where I is a reduced ideal/distance pair.
The function fN
q_fN :: CLIntP -> Int -> Int -> QDInt -> IntM -> Circ (QDInt, (IdealRedQ, FPRealQ)) Source #
q_fN Δ s n l qi j
This is the function h of [Jozsa 2003], Sec. 9, discretized with precision 1/N = 2−n, and perturbed by the jitter parameter j/L.