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qml.qchem.taylor_hamiltonian

taylor_hamiltonian(pes, max_deg=4, min_deg=3, mapping='binary', n_states=2, wire_map=None, tol=1e-12)[source]

Returns Taylor vibrational Hamiltonian.

The Taylor vibrational Hamiltonian is defined in terms of kinetic \(T\) and potential \(V\) components as:

\[H = T + V.\]

The kinetic term is defined in terms of momentum \(p\) operator as

\[T = \sum_{i\geq j} K_{ij} p_i p_j,\]

where the \(K\) matrix is defined in terms of vibrational frequencies, \(\omega\), and mode localization unitary matrix, \(U\), as:

\[K_{ij} = \sum_{k=1}^M \frac{\omega_k}{2} U_{ki} U_{kj}.\]

The potential term is defined in terms of the normal coordinate operator \(q\) as:

\[V(q_1,\cdots,q_M) = V_0 + \sum_{i=1}^M V_1^{(i)}(q_i) + \sum_{i>j} V_2^{(i,j)}(q_i,q_j) + \sum_{i<j<k} V_3^{(i,j,k)}(q_i,q_j,q_k) + \cdots,\]

where \(V_n\) represents the \(n\)-mode component of the potential energy surface computed along the normal coordinate. The \(V_n\) terms are defined as:

\[\begin{split}V_0 &\equiv V(q_1=0,\cdots,q_M=0) \\ V_1^{(i)}(q_i) &\equiv V(0,\cdots,0,q_i,0,\cdots,0) - V_0 \\ V_2^{(i,j)}(q_i,q_j) &\equiv V(0,\cdots,q_i,\cdots,q_j,\cdots,0) - V_1^{(i)}(q_i) - V_1^{(j)}(q_j) - V_0 \\ \nonumber \vdots\end{split}\]

These terms are then used in a multi-dimensional polynomial fit with a polynomial specified by min_deg and max_deg to get \(n\)-mode Taylor coefficients. For instance, the one-mode Taylor coefficient \(\Phi\) is related to the one-mode potential energy surface data as:

\[V_1^{(j)}(q_j) \approx \Phi^{(2)}_j q_j^2 + \Phi^{(3)}_j q_j^3 + ...\]

Similarly, the two-mode and three-mode Taylor coefficients are computed if the two-mode and three-mode potential energy surface data, \(V_2^{(j, k)}(q_j, q_k)\) and \(V_3^{(j, k, l)}(q_j, q_k, q_l)\), are provided.

This real space form of the vibrational Hamiltonian can be represented in the bosonic basis by using equations defined in Eqs. 6, 7 of arXiv:1703.09313:

\[\hat q_i = \frac{1}{\sqrt{2}}(b_i^\dagger + b_i), \quad \hat p_i = \frac{1}{\sqrt{2}}(b_i^\dagger - b_i),\]

where \(b^\dagger\) and \(b\) are bosonic creation and annihilation operators, respectively.

The bosonic Hamiltonian is then converted to a qubit operator with a selected mapping method to obtain a linear combination as:

\[H = \sum_{i} c_i P_i,\]

where \(P\) is a tensor product of Pauli operators and \(c\) is a constant.

Parameters:

  • pes (VibrationalPES) – object containing the vibrational potential energy surface data

  • max_deg (int) – maximum degree of the polynomial used to compute the coefficients

  • min_deg (int) – minimum degree of the polynomial used to compute the coefficients

  • mapping (str) – Method used to map to qubit basis. Input values can be "binary" or "unary". Default is "binary".

  • n_states (int) – maximum number of allowed bosonic states

  • wire_map (dict) – A dictionary defining how to map the states of the Bose operator to qubit wires. If None, integers used to label the bosonic states will be used as wire labels. Defaults to None.

  • tol (float) – tolerance for discarding the imaginary part of the coefficients during mapping

Returns:

the Taylor Hamiltonian

Return type:

Operator

Example

>>> freqs = np.array([0.0249722])
>>> pes_onemode = np.array([[0.08477, 0.01437, 0.00000, 0.00937, 0.03414]])
>>> pes_object = qml.qchem.VibrationalPES(freqs=freqs, pes_data=[pes_onemode], localized=False)
>>> ham = qml.qchem.taylor_hamiltonian(pes_object)
>>> print(ham)
0.026123120450329353 * I(0) + -0.01325338030021957 * Z(0) + -0.0032539545260859464 * X(0)
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