`pygsti.tools.lindbladtools`

Utility functions relevant to Lindblad forms and projections

Module Contents

Functions

 `create_elementary_errorgen_dual`(typ, p[, q, sparse, ...]) Construct a "dual" elementary error generator matrix in the "standard" (matrix-unit) basis. `create_elementary_errorgen`(typ, p[, q, sparse]) Construct an elementary error generator as a matrix in the "standard" (matrix-unit) basis. `create_lindbladian_term_errorgen`(typ, Lm[, Ln, sparse]) Construct the superoperator for a term in the common Lindbladian expansion of an error generator.
pygsti.tools.lindbladtools.create_elementary_errorgen_dual(typ, p, q=None, sparse=False, normalization_factor='auto')

Construct a “dual” elementary error generator matrix in the “standard” (matrix-unit) basis.

The elementary error generator that is dual to the one computed by calling `create_elementary_errorgen()` with the same argument. This dual element can be used to find the coefficient of the original, or “primal” elementary generator. For example, if A = sum(c_i * E_i), where E_i are the elementary error generators given by `create_elementary_errorgen()`), then c_i = dot(D_i.conj(), A) where D_i is the dual to E_i.

There are four different types of dual elementary error generators: ‘H’ (Hamiltonian), ‘S’ (stochastic), ‘C’ (correlation), and ‘A’ (active). See arxiv:2103.01928. Each type transforms an input density matrix differently. The action of an elementary error generator L on an input density matrix rho is given by:

Hamiltonian: L(rho) = -1j/(2d^2) * [ p, rho ] Stochastic: L(rho) = 1/(d^2) p * rho * p Correlation: L(rho) = 1/(2d^2) ( p * rho * q + q * rho * p) Active: L(rho) = 1j/(2d^2) ( p * rho * q - q * rho * p)

where d is the dimension of the Hilbert space, e.g. 2 for a single qubit. Square brackets denotes the commutator and curly brackets the anticommutator. L is returned as a superoperator matrix that acts on vectorized density matrices.

Parameters

typ{‘H’,’S’,’C’,’A’}

The type of dual error generator to construct.

pnumpy.ndarray

d-dimensional basis matrix.

qnumpy.ndarray, optional

d-dimensional basis matrix; must be non-None if and only if typ is ‘C’ or ‘A’.

sparsebool, optional

Whether to construct a sparse or dense (the default) matrix.

Returns

ndarray or Scipy CSR matrix

pygsti.tools.lindbladtools.create_elementary_errorgen(typ, p, q=None, sparse=False)

Construct an elementary error generator as a matrix in the “standard” (matrix-unit) basis.

There are four different types of elementary error generators: ‘H’ (Hamiltonian), ‘S’ (stochastic), ‘C’ (correlation), and ‘A’ (active). See arxiv:2103.01928. Each type transforms an input density matrix differently. The action of an elementary error generator L on an input density matrix rho is given by:

Hamiltonian: L(rho) = -1j * [ p, rho ] Stochastic: L(rho) = p * rho * p - rho Correlation: L(rho) = p * rho * q + q * rho * p - 0.5 {{p,q}, rho} Active: L(rho) = 1j( p * rho * q - q * rho * p + 0.5 {[p,q], rho} )

Square brackets denotes the commutator and curly brackets the anticommutator. L is returned as a superoperator matrix that acts on vectorized density matrices.

Parameters

typ{‘H’,’S’,’C’,’A’}

The type of error generator to construct.

pnumpy.ndarray

d-dimensional basis matrix.

qnumpy.ndarray, optional

d-dimensional basis matrix; must be non-None if and only if typ is ‘C’ or ‘A’.

sparsebool, optional

Whether to construct a sparse or dense (the default) matrix.

Returns

ndarray or Scipy CSR matrix

pygsti.tools.lindbladtools.create_lindbladian_term_errorgen(typ, Lm, Ln=None, sparse=False)

Construct the superoperator for a term in the common Lindbladian expansion of an error generator.

Mathematically, for d-dimensional matrices Lm and Ln, this routine constructs the d^2-dimension Lindbladian matrix L whose action is given by:

L(rho) = -i [Lm, rho] ` (when `typ == ‘H’)

or

L(rho) = Ln*rho*Lm^dag - 1/2(rho*Lm^dag*Ln + Lm^dag*Ln*rho) (typ == ‘O’)

where rho is a density matrix. L is returned as a superoperator matrix that acts on a vectorized density matrices.

Parameters

typ{‘H’, ‘O’}

The type of error generator to construct.

Lmnumpy.ndarray

d-dimensional basis matrix.

Lnnumpy.ndarray, optional

d-dimensional basis matrix.

sparsebool, optional

Whether to construct a sparse or dense (the default) matrix.

Returns

ndarray or Scipy CSR matrix