On number of parameters and operators in UCCSD in qiskit

Question

I have two questions on UCCSD in qiskit, please see the code below.

import numpy as np
from qiskit.chemistry.components.variational_forms import UCCSD
from qiskit.chemistry.drivers import PySCFDriver, UnitsType
from qiskit.chemistry import FermionicOperator
from qiskit.aqua import QuantumInstance
def get_qubit_op( atom , basis , map_type ):
    driver = PySCFDriver(atom=atom, unit=UnitsType.ANGSTROM, 
                         charge=0, spin=0, basis=basis)
    molecule = driver.run()
    repulsion_energy = molecule.nuclear_repulsion_energy
    num_particles = molecule.num_alpha + molecule.num_beta
    num_spin_orbitals = molecule.num_orbitals * 2
    ferOp = FermionicOperator(h1=molecule.one_body_integrals, h2=molecule.two_body_integrals)
    qubitOp = ferOp.mapping(map_type=map_type, threshold=0.00000001)
#     qubitOp = Z2Symmetries.two_qubit_reduction(qubitOp, num_particles)
    shift = repulsion_energy
    return qubitOp, num_particles, num_spin_orbitals, shift
atom='H .0 .0 .0; H .0 .0 0.74'
qubitOp, num_particles, num_spin_orbitals, shift = get_qubit_op( atom , basis = 'sto3g' , map_type = 'parity' )
num_qubits = qubitOp.num_qubits
print( 'num_qubits = ' , num_qubits ) 
from qiskit.chemistry.components.initial_states import HartreeFock    
init_state = HartreeFock( num_spin_orbitals , num_particles , 'parity' , two_qubit_reduction=False )
# setup the variational form for VQE
from qiskit.chemistry.components.variational_forms import UCCSD
var_form_vqe = UCCSD(
        num_orbitals=num_spin_orbitals,
        num_particles=num_particles,
        initial_state=init_state,
        qubit_mapping='parity' , 
        two_qubit_reduction = False
    )
print( 'var_form_vqe.num_parameters = ' , var_form_vqe.num_parameters )
var_form_vqe.construct_circuit([1,1,1]).draw() # Give some random para

The outputs are (without the plot of the circuit)

num_qubits =  4
var_form_vqe.num_parameters =  3

My questions are

1.
Why there are only 3 free parameters in UCCSD? Since I have 4 spin-orbs, I should expect there are at least 4^4 free parameters from the double excitation operators?

2. Is there a way to see which operators are involved in UCCSD? I plot out the circuit, but there are “Evolution^1” block involved. If I use “decompose().“, it shows all the 1 and 2 qubit operations? Would it be possible to show something in between, by which I mean for example adding barrier for the decomposed circuit, or showing which fermionic excitation operators are involved? Many thanks!

Answer 1

Since UCCSD is a particle- and spin-conserving variational form, the number of parameters does not only depend on the number of spin orbitals but also their "nature". What I mean by that is that you also need to consider these conservation rules which leads to the following scenario in your case:

H2 has 2 electrons in 2 molecular orbitals, which correspond to 4 spin orbitals (and, thus, 4 qubits since you do not apply any reductions, here). Taking into account the spin-conservation you have the following possible single excitations:

0 -> 1
2 -> 3

where I use Qiskit's notation in which spin orbitals are ordered block-wise (i.e. first all alpha orbitals, then all beta ones). If the spin was not conserved, then you would also see excitations such as 0 -> 3 and 1 -> 2.

The double excitations are a little bit simpler in this case because there is only one possibility regardless:

0, 1 -> 2, 3

However, if the example was more complex, the same reasoning would apply here, too.

Regarding your second question: if you would like to see the excitation lists as I have written them above you can take a look at the single_excitations and double_excitations attributes of the UCCSD class. I am not aware of any "partial" decompose() functionality which could otherwise provide that you describe above.