PH.parent_hamiltonian

parent_hamiltonian

Complete implementation of the Parent Hamiltonian following:

Fumiyoshi Kobayashi and Kosuke Mitarai and Keisuke Fujii Parent Hamiltonian as a benchmark problem for variational quantum eigensolvers Physical Review A, 105, 5, 2002 https://doi.org/10.1103%2Fphysreva.105.052415

Author: Gonzalo Ferro

class tnbs.BTC_04_PH.PH.parent_hamiltonian.parent_hamiltonian.PH(amplitudes, t_invariant=False, **kwargs)

Bases: object

Class for doing Parent Hamiltonian Computations

Parameters:

  • amplitudes (list) – list with the complete amplitudes of the state of the ansatz

  • kwars (dictionary) –

    dictionary that allows the configuration of the CQPEAE algorithm:

    Implemented keys:

get_density_matrix()

Computes the density matrix asociated with the amplitudes

get_pauli_pdf()

Create pauli dataframe with all the info. Additionally pauli coefficients lower than float precision will be pruned

local_ph()

Computes the local parent hamiltonian

naive_ph()

Computes the parent hamiltonian of agiven ansatz for a full qubit interaction

save()

Saving Staff

tnbs.BTC_04_PH.PH.parent_hamiltonian.parent_hamiltonian.get_local_reduced_matrix(state, qb_pos)

Given a MPS representation of a input state and position qubit (qb_pos) computes the minimum local reduced density matrix from qb_pos qubit such that its null space can be computable: i.e. rank(local_rho) < dim(local_rho)

Parameters:

  • state (numpy array) – MPS representation of an state

  • qb_pos (int) – position of the qubit for computing the reduced density matrix

Returns:

  • local_qubits (list) – list with the qubits affected by the reduced density matrix

  • local_rho (numpy array) – array with the reduced density matrix for the input qbit position

tnbs.BTC_04_PH.PH.parent_hamiltonian.parent_hamiltonian.get_null_projectors(array)

Given an input matrix this function computes the matrix of projectors over the null space of the input matrix

Parameters:

array (numpy array) – Input matrix for computing the projectors over the null space

Returns:

h_null – Matrices with the projectors to the null space of the input array

Return type:

numpy array

tnbs.BTC_04_PH.PH.parent_hamiltonian.parent_hamiltonian.run_parent_hamiltonian(**configuration)

launch_parent_hamiltonian

For launching a VQE quantum step execution Author: Gonzalo Ferro

tnbs.BTC_04_PH.PH.parent_hamiltonian.launch_parent_hamiltonian.run_id(**configuration)

pauli

This module contains all functions needed for creating the Pauli decomposition of a Hamiltonian

Author: Gonzalo Ferro

tnbs.BTC_04_PH.PH.parent_hamiltonian.pauli.pauli_decomposition(array, dimension, jobs=-1)

Creates the Pauli Decomposition of an input matrix

Parameters:

  • array (numpy array) – input array for doing the Pauli decomposition

  • dimension (int) – dimension for creating the corresponding basis of Kronecker products of Pauli basis

  • int (jobs =) – For speed up the coefficients computations. Number of parallel computations.

Returns:

  • coefs (list) – Coefficients of the different Pauli matrices decomposition

  • strings_pauli (list) – list with the complete Pauli string decomposition

tnbs.BTC_04_PH.PH.parent_hamiltonian.pauli.pauli_inner_product(array, pauli_list, dimension)

Given an input matrix and a Kronecker Pauli matrix product computes its inner product. Under the hood it computes the Frobenius norm of the complex conjugate of the matrix by the given Kronecker Pauli matrix product.

Parameters:

  • array (numpy array) – Input array for getting the inner product

  • pauli_list (list) – list with the indices to compute the Kronecker product of the corresponding Pauli matrices.Each element of the list must be a number between 0 and 3 corresponding to the Pauli matrix following convention from pauli_operators function

  • dimension (int) – For doing the normalisation corresponding to the dimension of the matrices

Returns:

inner_product – Resulting inner product

Return type:

float

tnbs.BTC_04_PH.PH.parent_hamiltonian.pauli.pauli_operators(pauli_index)

Return the correspondent Pauli matrix.

Parameters:

pauli_index (int) – Number for selecting the Pauli matrix: 0 -> identity, 1-> X, 2-> Y, 3 -> Z

Returns:

2x2 Pauli Matrix

Return type:

pauli = np.array

tnbs.BTC_04_PH.PH.parent_hamiltonian.pauli.pauli_strings(pauli_list)

Provides the Pauli strings of the input list.

Parameters:

pauli_list (list) – list with the ints of the Pauli matrices. See pauli_operators for the mapping

Returns:

string_pauli – correspondent Pauli strings of the input

Return type:

str

contractions

Implementation of reduced density matrix computations and tensor contractions

Author: Gonzalo Ferro

tnbs.BTC_04_PH.PH.parent_hamiltonian.contractions.contract_indices(tensor1, tensor2, contraction1=[], contraction2=[])

Compute the contraction of 2 input tensors for the input contraction indices.The computation is done by, transposing indices, reshapin and doing matrix multiplication. Tensor legs can be of different dimension. BE AWARE: Order in contraction indices it is very important: contraction1 = [1, 2, 5] contraction2 = [2, 0, 6] -> Contractions will be: [1-2, 2-0, 5-6] But if contraction2 = [0, 2, 6] then contraction will be: [1-0, 2-2,5-6] If both contraction indices are empty then the tensors will be glued

Parameters:

  • tensor1 (numpy array) – first tensor

  • tensor2 (numpy array) – second tensor

  • contraction1 (list) – contraction indices for first tensor

  • contraction2 (list) – contraction indices for second tensor

Returns:

rho – Desired reduced density operator in matrix form

Return type:

numpy array

tnbs.BTC_04_PH.PH.parent_hamiltonian.contractions.contract_indices_old(tensor1, tensor2, contraction1, contraction2)

Compute the contraction of 2 input tensors for the input contraction indices.The computation is done by, transposing indices, reshaping and doing matrix multiplication

Parameters:

  • tensor1 (numpy array) – first tensor

  • tensor2 (numpy array) – second tensor

  • contraction1 (list) – contraction indices for first tensor

  • contraction2 (list) – contraction indices for second tensor

Returns:

rho – Desired reduced density operator in matrix form

Return type:

numpy array

tnbs.BTC_04_PH.PH.parent_hamiltonian.contractions.contract_indices_string(tensor1, tensor2, contraction_indices)

Compute the contraction of 2 input tensors for the input contraction indices.The computation is done by, transposing indices, reshaping and doing matrix multiplication

Parameters:

  • tensor1 (numpy array) – array in MPS format of the state for computing reduced density matrix

  • tensor2 (numpy array) – array in MPS format of the state for computing reduced density matrix

  • contraction_indices (list) – Indices of the MPS state that will be contracted to compute the correspondent reduced density matrix

Returns:

rho – Desired reduced density operator in matrix form

Return type:

numpy array

tnbs.BTC_04_PH.PH.parent_hamiltonian.contractions.reduced_matrix(state, free_indices, contraction_indices)

Compute the reduced density matrix for the input contraction indices The computation is done by, transposing indices, reshaping and doing matrix multiplication

Parameters:

  • state (numpy array) – array in MPS format of the state for computing reduced density matrix

  • free_indices (list) – Free indices of the MPS state (this is the qubit that will NOT be traced out)

  • contraction_indices (list) – Indices of the MPS state that will be contracted to compute the correspondent reduced density matrix

Returns:

rho – Desired reduced density operator in matrix form

Return type:

numpy array

tnbs.BTC_04_PH.PH.parent_hamiltonian.contractions.reduced_matrix_string(state, free_indices, contraction_indices)

Compute the reduced density matrix for the input contraction indices The computation is done by using np.einsum. Limited to 52 indices.

Parameters:

  • state (numpy array) – array in MPS format of the state for computing reduced density matrix

  • free_indices (list) – Free indices of the MPS state (this is the qubit that will NOT be traced out)

  • contraction_indices (list) – Indices of the MPS state that will be contracted to compute the correspondent reduced density matrix

Returns:

matriz_rho – Desired reduced density operator in matrix form

Return type:

numpy array