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Functions for infinite size systems: High level functions
inf(so_before_list, h, so_after_list, L_ini=None, psi0_ini=None, imprecision=10**-2, D_L_max=100, D_L_max_forced=False, L_herm=True, D_psi0_max=100, D_psi0_max_forced=False)
Optimization of the lim_{N --> infinity} QFI/N over operator tilde{L} (in iMPO representation) and wave function psi0 (in iMPS representation) and check of convergence in their bond dimensions. Function for infinite size systems.
User has to provide information about the dynamics by specifying quantum channel. It is assumed that quantum channel is translationally invariant and is build from layers of quantum operations. User has to provide one defining for each layer operation as a local superoperator. Those local superoperator have to be input in order of their action on the system. Parameter encoding is a stand out quantum operation. It is assumed that parameter encoding acts only once and is unitary so the user have to provide only its generator h. Generator h have to be diagonal in computational basis, or in other words it is assumed that local superoperators are expressed in the eigenbasis of h.
Parameters:
so_before_list: list of ndarrays of a shape (d**(2*k),d**(2*k)) where k describes on how many sites particular local superoperator acts
List of local superoperators (in order) which act before unitary parameter encoding.
h: ndarray of a shape (d,d)
Generator of unitary parameter encoding. Dimension d is the dimension of local Hilbert space (dimension of physical index).
Generator h have to be diagonal in computational basis, or in other words it is assumed that local superoperators are expressed in the eigenbasis of h.
so_after_list: list of ndarrays of a shape (d**(2*k),d**(2*k)) where k describes on how many sites particular local superoperator acts
List of local superoperators (in order) which act after unitary parameter encoding.
L_ini: ndarray of a shape (D_L,D_L,d,d), optional
Initial iMPO for tilde{L}.
psi0_ini: ndarray of a shape (D_psi0,D_psi0,d), optional
Initial iMPS for psi0.
imprecision: float, optional
Expected relative imprecision of the end results.
D_L_max: integer, optional
Maximal value of D_L (D_L is bond dimension for iMPO representing tilde{L}).
D_L_max_forced: bool, optional
True if D_L_max have to be reached, otherwise False.
L_herm: bool, optional
True if Hermitian gauge have to be imposed on iMPO representing tilde{L}, otherwise False.
D_psi0_max: integer, optional
Maximal value of D_psi0 (D_psi0 is bond dimension for iMPS representing psi0).
D_psi0_max_forced: bool, optional
True if D_psi0_max have to be reached, otherwise False.
Returns:
result: float
Optimal value of figure of merit.
result_m: ndarray
Matrix describing figure of merit in function of bond dimensions of respectively tilde{L} [rows] and psi0 [columns].
L: ndarray of a shape (D_L,D_L,d,d)
Optimal tilde{L} in iMPO representation.
psi0: ndarray of a shape (D_psi0,D_psi0,d)
Optimal psi0 in iMPS representation.
inf_gen(d, ch, ch2, epsilon, symfun_L, symfun_psi0, L_ini=None, psi0_ini=None, imprecision=10**-2, D_L_max=100, D_L_max_forced=False, L_herm=True, D_psi0_max=100, D_psi0_max_forced=False)
Optimization of the figure of merit (usually interpreted as lim_{N --> infinity} QFI/N) over operator tilde{L} (in iMPO representation) and wave function psi0 (in iMPS representation) and check of convergence in their bond dimensions. Function for infinite size systems.
User has to provide information about the dynamics by specifying two channels separated by small parameter epsilon as superoperators in iMPO representation. By definition this infinite approach assumes translation invariance of the problem, other than that there are no constraints on the structure of the channel but the complexity of calculations highly depends on channel's bond dimension.
Parameters:
d: integer
Dimension of local Hilbert space (dimension of physical index).
ch: ndarray of a shape (D_ch,D_ch,d**2,d**2)
Quantum channel as superoperator in iMPO representation.
ch2: ndarray of a shape (D_ch2,D_ch2,d**2,d**2)
Quantum channel as superoperator in iMPO representation for the value of estimated parameter shifted by epsilon in relation to ch.
epsilon: float
Value of a separation between estimated parameters encoded in ch and ch2.
symfun_L: function
Function which symmetrize iMPO for tilde{L} after each step of otimization (the most simple one would be lambda x: x).
Choosing good function is key factor for successful optimization in infinite approach.
TNQMetro package features inf_L_symfun function which performs well in dephasing type problems.
symfun_psi0: function
Function which symmetrize iMPS for psi0 after each step of otimization (the most simple one would be lambda x: x).
Choosing good function is key factor for successful optimization in infinite approach.
TNQMetro package features inf_psi0_symfun function which performs well in dephasing type problems.
L_ini: ndarray of a shape (D_L,D_L,d,d), optional
Initial iMPO for tilde{L}.
psi0_ini: ndarray of a shape (D_psi0,D_psi0,d), optional
Initial iMPS for psi0.
imprecision: float, optional
Expected relative imprecision of the end results.
D_L_max: integer, optional
Maximal value of D_L (D_L is bond dimension for iMPO representing tilde{L}).
D_L_max_forced: bool, optional
True if D_L_max have to be reached, otherwise False.
L_herm: bool, optional
True if Hermitian gauge have to be imposed on iMPO representing tilde{L}, otherwise False.
D_psi0_max: integer, optional
Maximal value of D_psi0 (D_psi0 is bond dimension for iMPS representing psi0).
D_psi0_max_forced: bool, optional
True if D_psi0_max have to be reached, otherwise False.
Returns:
result: float
Optimal value of figure of merit.
result_m: ndarray
Matrix describing figure of merit in function of bond dimensions of respectively tilde{L} [rows] and psi0 [columns].
L: ndarray of a shape (D_L,D_L,d,d)
Optimal tilde{L} in iMPO representation.
psi0: ndarray of a shape (D_psi0,D_psi0,d)
Optimal psi0 in iMPS representation.
inf_state(so_before_list, h, so_after_list, rho0, L_ini=None, imprecision=10**-2, D_L_max=100, D_L_max_forced=False, L_herm=True)
Optimization of the lim_{N --> infinity} QFI/N over operator tilde{L} (in iMPO representation) and check of convergence in its bond dimension. Function for infinite size systems and fixed state of the system.
User has to provide information about the dynamics by specifying quantum channel. It is assumed that quantum channel is translationally invariant and is build from layers of quantum operations. User has to provide one defining for each layer operation as a local superoperator. Those local superoperator have to be input in order of their action on the system. Parameter encoding is a stand out quantum operation. It is assumed that parameter encoding acts only once and is unitary so the user have to provide only its generator h. Generator h have to be diagonal in computational basis, or in other words it is assumed that local superoperators are expressed in the eigenbasis of h.
Parameters:
so_before_list: list of ndarrays of a shape (d**(2*k),d**(2*k)) where k describes on how many sites particular local superoperator acts
List of local superoperators (in order) which act before unitary parameter encoding.
h: ndarray of a shape (d,d)
Generator of unitary parameter encoding. Dimension d is the dimension of local Hilbert space (dimension of physical index).
Generator h have to be diagonal in computational basis, or in other words it is assumed that local superoperators are expressed in the eigenbasis of h.
so_after_list: list of ndarrays of a shape (d**(2*k),d**(2*k)) where k describes on how many sites particular local superoperator acts
List of local superoperators (in order) which act after unitary parameter encoding.
rho0: ndarray of a shape (D_rho0,D_rho0,d,d)
Density matrix describing initial state of the system in iMPO representation.
L_ini: ndarray of a shape (D_L,D_L,d,d), optional
Initial iMPO for tilde{L}.
imprecision: float, optional
Expected relative imprecision of the end results.
D_L_max: integer, optional
Maximal value of D_L (D_L is bond dimension for iMPO representing tilde{L}).
D_L_max_forced: bool, optional
True if D_L_max have to be reached, otherwise False.
L_herm: bool, optional
True if Hermitian gauge have to be imposed on iMPO representing tilde{L}, otherwise False.
Returns:
result: float
Optimal value of figure of merit.
result_v: ndarray
Vector describing figure of merit in function of bond dimensions of tilde{L}.
L: ndarray of a shape (D_L,D_L,d,d)
Optimal tilde{L} in iMPO representation.
inf_state_gen(d, rho, rho2, epsilon, symfun_L, L_ini=None, imprecision=10**-2, D_L_max=100, D_L_max_forced=False, L_herm=True)
Optimization of the figure of merit (usually interpreted as lim_{N --> infinity} QFI/N) over operator tilde{L} (in iMPO representation) and check of convergence in its bond dimension. Function for infinite size systems and fixed state of the system.
User has to provide information about the dynamics by specifying two channels separated by small parameter epsilon as superoperators in iMPO representation. By definition this infinite approach assumes translation invariance of the problem, other than that there are no constraints on the structure of the channel but the complexity of calculations highly depends on channel's bond dimension.
Parameters:
d: integer
Dimension of local Hilbert space (dimension of physical index).
rho: ndarray of a shape (D_rho,D_rho,d,d)
Density matrix at the output of quantum channel in iMPO representation.
rho2: ndarray of a shape (D_rho2,D_rho2,d,d)
Density matrix at the output of quantum channel in iMPO representation for the value of estimated parameter shifted by epsilon in relation to rho.
epsilon: float
Value of a separation between estimated parameters encoded in rho and rho2.
symfun_L: function
Function which symmetrize iMPO for tilde{L} after each step of otimization (the most simple one would be lambda x: x).
Choosing good function is key factor for successful optimization in infinite approach.
TNQMetro package features inf_L_symfun function which performs well in dephasing type problems.
L_ini: ndarray of a shape (D_L,D_L,d,d), optional
Initial iMPO for tilde{L}.
imprecision: float, optional
Expected relative imprecision of the end results.
D_L_max: integer, optional
Maximal value of D_L (D_L is bond dimension for iMPO representing tilde{L}).
D_L_max_forced: bool, optional
True if D_L_max have to be reached, otherwise False.
L_herm: bool, optional
True if Hermitian gauge have to be imposed on iMPO representing tilde{L}, otherwise False.
Returns:
result: float
Optimal value of figure of merit.
result_v: ndarray
Vector describing figure of merit in function of bond dimensions of tilde{L}.
L: ndarray of a shape (D_L,D_L,d,d)
Optimal tilde{L} in iMPO representation.
inf_L_symfun(l)
Symmetrize function for iMPO representing tilde{L} which performs well in dephasing type problems.
Parameters:
l: ndarray of a shape (D_L,D_L,d,d)
iMPO for tilde{L}.
Returns:
l: ndarray of a shape (D_L,D_L,d,d)
Symmetrize iMPO for tilde{L}.
inf_psi0_symfun(p)
Symmetrize function for iMPS representing psi0 which performs well in dephasing type problems.
Parameters:
p: ndarray of a shape (D_psi0,D_psi0,d)
iMPS for psi0.
Returns:
p: ndarray of a shape (D_psi0,D_psi0,d)
Symmetrize iMPS for psi0.
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