weylchamber.local_invariants module

Summary

Functions:

J_T_LI	Calculate value of the local-invariants functional
closest_LI	Find the closest gate that has the given Weyl chamber coordinates
g1g2g3	Calculate local invariants \((g_1, g_3, g_3)\)
g1g2g3_from_c1c2c3	Calculate local invariants from the Weyl chamber coordinates
make_LI_krotov_chi_constructor	Return a constructor for the χ’s in an LI optimization.

__all__: J_T_LI, closest_LI, g1g2g3, g1g2g3_from_c1c2c3, make_LI_krotov_chi_constructor

Reference

weylchamber.local_invariants.g1g2g3(U, ndigits=8)

Calculate local invariants \((g_1, g_3, g_3)\)

Given a two-qubit gate, calculate local invariants \((g_1, g_2, g_3)\). U must be in the canonical basis. For numerical stability, the resulting values are rounded to the given precision, cf. the ndigits parameter of the built-in round() function.

>>> print("%.2f %.2f %.2f" % g1g2g3(qutip.gates.cnot()))
0.00 0.00 1.00

Return type:Tuple[float, float, float]

weylchamber.local_invariants.g1g2g3_from_c1c2c3(c1, c2, c3, ndigits=8)

Calculate local invariants from the Weyl chamber coordinates

Calculate the local invariants \((g_1, g_2, g_3)\) from the Weyl chamber coordinates \((c_1, c_2, c_3)\), in units of π. The result is rounded to the given precision, in order to enhance numerical stability (cf. ndigits parameter of the built-in round() function)

Example

>>> CNOT = qutip.gates.cnot()
>>> print("%.2f %.2f %.2f" % g1g2g3_from_c1c2c3(*c1c2c3(CNOT)))
0.00 0.00 1.00

Return type:Tuple[float, float, float]

weylchamber.local_invariants.J_T_LI(O, U, form='g')

Calculate value of the local-invariants functional

Parameters:

• O (Union[Qobj, ndarray]) – The optimal gate
• U (Union[Qobj, ndarray]) – The achieved gate
• form (str) – form of the functional to use, ‘g’ or ‘c’

weylchamber.local_invariants.closest_LI(U, c1, c2, c3, method='leastsq', limit=1e-06)

Find the closest gate that has the given Weyl chamber coordinates

The c1, c2, c3 are given in units of π

weylchamber.local_invariants.make_LI_krotov_chi_constructor(gate, canonical_basis, unitarity_weight=0)

Return a constructor for the χ’s in an LI optimization.

Return a chi_constructor that determines the boundary condition of the backwards propagation in an optimization towards the local equivalence class of gate in Krotov’s method, based on the foward-propagtion of the Bell states.

Parameters:

• gate (qutip.Qobj) – A 4×4 quantum gate, in the canonical_basis.
• canonical_basis (list[qutip.Qobj]) – A list of four basis states that define the canonical basis \(\ket{00}\), \(\ket{01}\), \(\ket{10}\), and \(\ket{11}\) of the logical subspace.
• unitarity_weight (float) – A weight in [0, 1] that determines how much emphasis is placed on maintaining population in the logical subspace.

Returns:

a function chi_constructor(fw_states_T, *args) that receive the result of a foward propagation of the Bell states (obtained from canonical_basis via weylchamber.gates.bell_basis()), and returns a list of statex \(\ket{\chi}\) that are the boundary condition for the backward propagation in Krotov’s method. Positional arguments beyond fw_states_T are ignored.

Return type:

callable