Quantum.Synthesis.RotationDecomposition

This module provides functions for decomposing a unitary n×n operator into one- and two-level unitaries.

The algorithm is adapted from Section 4.5.1 of Nielsen and Chuang. In addition to what is described in Nielsen and Chuang, our algorithm produces two-level operators that can be decomposed using only two Euler angles. The algorithm produces at most n(n−1)/2 two-level operators of type R_z(δ)R_x(γ), as well as n one-level operators of type e^iθ. Therefore, the decomposition of a unitary n×n operator yields n² real parameters, which is optimal.

Elementary rotations

data ElementaryRot a

An elementary rotation is either a combined x- and z-rotation, applied at indices j and k, or a phase change applied at index j.

• ERot_zx δ γ j k represents the operator R_z(δ)R_x(γ), applied to levels j and k.

• ERot_phase θ j represents the operator e^iθ applied to level j.

Note: when we use a list of ElementaryRots to express a sequence of operators, the operators are meant to be applied right-to-left, i.e., as in the mathematical notation for matrix multiplication. This is the opposite of the quantum circuit notation.

Constructors

ERot_zx a a Index Index

ERot_phase a Index

Instances

Show a => Show (ElementaryRot a)

matrix_of_elementary :: (Ring a, Floating a, Nat n) => ElementaryRot a -> Matrix n n (Cplx a)

Convert a symbolic elementary rotation to a concrete matrix.

matrix_of_elementaries :: (Ring a, Floating a, Nat n) => [ElementaryRot a] -> Matrix n n (Cplx a)

Convert a sequence of elementary rotations to an n×n-matrix.

Decomposition into elementary rotations

rotation_decomposition :: (Eq a, Fractional a, Floating a, Adjoint a, ArcTan2 a, Nat n) => Matrix n n (Cplx a) -> [ElementaryRot a]

Convert an n×n-matrix to a sequence of elementary rotations.

Note: the list of elementary rotations will be returned in right-to-left order, i.e., as in the mathematical notation for matrix multiplication. This is the opposite of the quantum circuit notation.

Auxiliary functions

twolevel_matrix_of_matrix :: (Ring a, Nat n) => Matrix Two Two a -> Index -> Index -> Matrix n n a

Construct a two-level n×n-matrix from a given 2×2-matrix and indices j and k.

get_phase :: ArcTan2 a => Matrix n n (Cplx a) -> Index -> ElementaryRot a

Extract the phase of the jth diagonal entry of the given matrix.

rowop :: (Eq a, Fractional a, Floating a, Adjoint a, ArcTan2 a, Nat n) => Matrix n n (Cplx a) -> (Index, Index) -> (Matrix n n (Cplx a), [ElementaryRot a])

Perform a two-level operation on rows j and k of a matrix U, such that the resulting matrix has a 0 in the (j,k)-position. Return the inverse of the two-level operation used, as well as the updated matrix.

Testing

random_unitary :: (RandomGen g, Nat n, Floating a, Random a) => g -> Matrix n n (Cplx a)

Return a "random" unitary n×n-matrix. These matrices will not quite be uniformly distributed; this function is primarily meant to generate test cases.

test :: IO ()

Generate a random matrix, decompose it, and then re-calculate the matrix from the decomposition.