Decompositions

Module name: thewalrus.decompositions

This module implements common shared matrix decompositions that are used to perform gate decompositions.

For mathematical details of these decompositions see

Houde et al. Matrix decompositions in Quantum Optics: Takagi/Autonne, Bloch-Messiah/Euler, Iwasawa, and Williamson [42] Summary ——-

williamson(V[, rtol, atol])	Williamson decomposition of positive-definite (real) symmetric matrix.
symplectic_eigenvals(cov)	Returns the symplectic eigenvalues of a covariance matrix.
blochmessiah(S)	Returns the Bloch-Messiah decomposition of a symplectic matrix S = O @ D @ Q
takagi(A[, svd_order])	Autonne-Takagi decomposition of a complex symmetric (not Hermitian!) matrix.
pre_iwasawa(S)	Pre-Iwasawa decomposition of a symplectic matrix.
iwasawa(S)	Iwasawa decomposition of a symplectic matrix.

Code details

blochmessiah(S)

Returns the Bloch-Messiah decomposition of a symplectic matrix S = O @ D @ Q

where O and Q are orthogonal symplectic matrices and D is a positive-definite diagonal matrix of the form diag(d1,d2,…,dn,d1^-1, d2^-1,…,dn^-1).

See Houde et al. Matrix decompositions in Quantum Optics: Takagi/Autonne, Bloch-Messiah/Euler, Iwasawa, and Williamson

Parameters:

S (array[float]) – 2N x 2N real symplectic matrix

Returns:

orthogonal symplectic matrix O

array[float], : diagonal matrix D array[float]) : orthogonal symplectic matrix Q

Return type:

tuple(array[float],

iwasawa(S)

Iwasawa decomposition of a symplectic matrix. See Arvind et al. The Real Symplectic Groups in Quantum Mechanics and Optics and Houde et al. Matrix decompositions in Quantum Optics: Takagi/Autonne, Bloch-Messiah/Euler, Iwasawa, and Williamson

Parameters:

S (array) – the symplectic matrix

Returns:

(E,D,F) symplectic matrices such that E @ D @ F = S, EE = np.block([[AA, np.zeros(N,N)],[CC, np.linalg.inv(A.T)]]) with A.T @ C == C.T @ A, and AA upper trinagular with ones in the diagonal DD is diagonal and symplectic, FF is symplectic orthogonal.

Return type:

tuple[array, array, array]

pre_iwasawa(S)

Pre-Iwasawa decomposition of a symplectic matrix. See Arvind et al. The Real Symplectic Groups in Quantum Mechanics and Optics and Houde et al. Matrix decompositions in Quantum Optics: Takagi/Autonne, Bloch-Messiah/Euler, Iwasawa, and Williamson

Parameters:

S (array) – the symplectic matrix

Returns:

(E,D,F) symplectic matrices such that E @ D @ F = S and, E = np.block([[np.eye(N), np.zeros(N,N)],[X, np.eye(N)]]) with X == X.T, D is block diagonal with the top left block being the inverse of the bottom right block, F is symplectic orthogonal.

Return type:

tuple[array, array, array]

symplectic_eigenvals(cov)

Returns the symplectic eigenvalues of a covariance matrix.

Parameters:

cov (array) – a covariance matrix

Returns:

symplectic eigenvalues

Return type:

(array)

takagi(A, svd_order=True)

Autonne-Takagi decomposition of a complex symmetric (not Hermitian!) matrix. Note that the input matrix is internally symmetrized by taking its upper triangular part. If the input matrix is indeed symmetric this leaves it unchanged.

See Houde et al. Matrix decompositions in Quantum Optics: Takagi/Autonne, Bloch-Messiah/Euler, Iwasawa, and Williamson

Parameters:

• A (array) – square, symmetric matrix

• svd_order (boolean) – whether to return result by ordering the singular values of A in descending (True) or ascending (False) order.

Returns:

(r, U), where r are the singular values, and U is the Autonne-Takagi unitary, such that \(A = U \diag(r) U^T\).

Return type:

tuple[array, array]

williamson(V, rtol=1e-05, atol=1e-08)

Williamson decomposition of positive-definite (real) symmetric matrix.

See Houde et al. Matrix decompositions in Quantum Optics: Takagi/Autonne, Bloch-Messiah/Euler, Iwasawa, and Williamson

Parameters:

• V (array[float]) – positive definite symmetric (real) matrix

• rtol (float) – the relative tolerance parameter used in np.allclose

• atol (float) – the absolute tolerance parameter used in np.allclose

Returns:

(Db, S) where Db is a diagonal matrix

and S is a symplectic matrix such that \(V = S Db S^T\)

Return type:

tuple[array,array]

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