Getting trotterized exponential of a Hamiltonian

[einarG-1qbit]

Hello,

1. I would like to load the molecule. I guess this is done with SecondQuantizedMolecule
2. I would like to get the Hamiltonian for that molecule and get a circuit by exponentiation of the Hamiltonian and applying 1st order Trotterization. For that, I probably need get_exponentiated_qubit_operator_circuit() .
3. Next, I want to have full access to the generated circuit and be able to convert circuit gates into Clifford+T gate set or at least prepare the circuit for some conversion procedure that would be done outside Tangelo if that functionality is not present in the package.

As an exmaple, I use the molecule below.

H2 = [('H', (0, 0, 0)),('H', (0, 0, 0.74137727))]
mol_H2 = SecondQuantizedMolecule(H2, q=0, spin=0, basis="sto-3g")

The problem is that it is not obvious to me how to go from step 1 to step 2. Step 3 is also very non-obvious.

P.S. note that the resulting circuit will have multi-gates like $\exp (- i a t Y \otimes X \otimes X \otimes Y)$. Such gates are not supported in open qasm.

[alexfleury-sb]

Hi @einarG-1qbit! Thanks for your question: here is a more in-depth explanation (than yesterday's answer in slack):

1. Yes, you are right.
2. The first thing to do in that case is to convert the Hamiltonian from a fermion to a qubit basis. At the time of writing those lines, Tangelo supports the Jordan-Wigner (JW), Bravyi-Kitaev (BK), symmetry-conserving BK (scBK) and Jiang-Kalev-Mruczkiewicz-Neven (JKMN) mapping. The respective references should be included in the docstrings.
   This can be done with the fermion_to_qubit_mapping function (here an example where we want the JW mapping with the spin-up then spin-down orbital ordering).

from tangelo.toolboxes.qubit_mappings.mapping_transform import fermion_to_qubit_mapping
...
qH = fermion_to_qubit_mapping(mol_H2.fermionic_hamiltonian, "JW", mol_H2.n_active_sos, mol_H2.n_active_electrons, up_then_down=True)

After, it is indeed the function get_exponentiated_qubit_operator_circuit that transforms the Hamiltonian into a circuit via trotterization.

from tangelo.toolboxes.ansatz_generator.ansatz_utils import get_exponentiated_qubit_operator_circuit
exp_circ = get_exponentiated_qubit_operator_circuit(qH, time=1., trotter_order=1)

3. At this point, you have a tangelo.linq.Circuit object (basically it is a list of tangelo.linq.Gate). We have some helpers functions and methods to help manipulate (optimization, splitting, merging, etc.) the Circuit. However, the transpilation to a specific set of gates is not something explicitly supported for now. For this purpose, I have used pytket in the past, and the circuits can be exported/imported with the translator functions for openqasm (https://github.com/goodchemistryco/Tangelo/blob/main/tangelo/linq/translator/translate_openqasm.py). There are also functions to convert from and to the qiskit, qulacs, cirq and other backend circuit formats. There is currently a PR on that subject , and everything said above will be done from a single function called translate_circuit.

[einarG-1qbit]

Thanks Alexandre!