Add fermionic transitions to the pool of configurations

Here we demonstrate the functionalities to augment the pool of electronic configurations obtained by the action of transition operators on each electronic configuration.

We demonstrate how to add single-electron hops of the type:

$$c^\dagger_{p\sigma} c_{q\sigma} |\textbf{x} \rangle$$

for $p, q = 1, ..., N_\textrm{orb}$ and $\sigma \in \{ \uparrow, \downarrow\}$, and for all $|\textbf{x} \rangle$ in the batch of electronic configurations. We start by generating a batch of random electronic configurations:

# Added by doQumentation — installs the packages this notebook needs if they are missing
import importlib.util

_needed = {"numpy": "numpy", "qiskit_addon_sqd": "qiskit-addon-sqd"}
_missing = [pip for module, pip in _needed.items()
            if importlib.util.find_spec(module) is None]
# One at a time, so a package that fails to install does not block the others
for _pip in _missing:
    %pip install -q {_pip}
if not _missing:
    print("\u2713 All required packages are installed")

import numpy as np

n_qubits = 8
n_orb = n_qubits // 2

rand_seed = 22
np.random.seed(rand_seed)

# Generate some random bitstrings for testing
def random_bitstrings(n_samples, n_qubits):
    return (
        np.round(np.random.rand(n_samples, n_qubits))
        .astype("int")
        .astype("bool")
    )

bitstring_matrix = random_bitstrings(100, n_qubits)

The excitation operators are specified inside a numpy array whose length is equal to the number of fermionic nodes (or qubits). Each element of the array must be a string that can take values:

• 'I': Identity
• '+': Creation operator
• '-': Annihilation operator

We generate all possible single-electron transitions (amongst same spin species).

transitions_single = np.array(
    [
        ["I" for i in range(2 * n_orb)]
        for j in range(4 * (n_orb**2 - n_orb) // 2 + 1)
    ]
)
count = 1
for i in range(n_orb):
    for j in range(i + 1, n_orb):
        # spin up
        transitions_single[count, i] = "+"
        transitions_single[count, j] = "-"
        count += 1
        transitions_single[count, i] = "-"
        transitions_single[count, j] = "+"
        count += 1

        # spin down
        transitions_single[count, i + n_orb] = "+"
        transitions_single[count, j + n_orb] = "-"
        count += 1
        transitions_single[count, i + n_orb] = "-"
        transitions_single[count, j + n_orb] = "+"
        count += 1

print(transitions_single)

[['I' 'I' 'I' 'I' 'I' 'I' 'I' 'I']
 ['+' '-' 'I' 'I' 'I' 'I' 'I' 'I']
 ['-' '+' 'I' 'I' 'I' 'I' 'I' 'I']
 ['I' 'I' 'I' 'I' '+' '-' 'I' 'I']
 ['I' 'I' 'I' 'I' '-' '+' 'I' 'I']
 ['+' 'I' '-' 'I' 'I' 'I' 'I' 'I']
 ['-' 'I' '+' 'I' 'I' 'I' 'I' 'I']
 ['I' 'I' 'I' 'I' '+' 'I' '-' 'I']
 ['I' 'I' 'I' 'I' '-' 'I' '+' 'I']
 ['+' 'I' 'I' '-' 'I' 'I' 'I' 'I']
 ['-' 'I' 'I' '+' 'I' 'I' 'I' 'I']
 ['I' 'I' 'I' 'I' '+' 'I' 'I' '-']
 ['I' 'I' 'I' 'I' '-' 'I' 'I' '+']
 ['I' '+' '-' 'I' 'I' 'I' 'I' 'I']
 ['I' '-' '+' 'I' 'I' 'I' 'I' 'I']
 ['I' 'I' 'I' 'I' 'I' '+' '-' 'I']
 ['I' 'I' 'I' 'I' 'I' '-' '+' 'I']
 ['I' '+' 'I' '-' 'I' 'I' 'I' 'I']
 ['I' '-' 'I' '+' 'I' 'I' 'I' 'I']
 ['I' 'I' 'I' 'I' 'I' '+' 'I' '-']
 ['I' 'I' 'I' 'I' 'I' '-' 'I' '+']
 ['I' 'I' '+' '-' 'I' 'I' 'I' 'I']
 ['I' 'I' '-' '+' 'I' 'I' 'I' 'I']
 ['I' 'I' 'I' 'I' 'I' 'I' '+' '-']
 ['I' 'I' 'I' 'I' 'I' 'I' '-' '+']]

Now we apply the transition operators to the configurations in bitstring_matrix:

from qiskit_addon_sqd.fermion import enlarge_batch_from_transitions

bitstring_matrix_aug = enlarge_batch_from_transitions(
    bitstring_matrix, transitions_single
)

print(bitstring_matrix_aug.shape)
print(bitstring_matrix_aug)

(723, 8)
[[False False False ... False False  True]
 [False  True False ...  True False False]
 [ True  True  True ...  True False  True]
 ...
 [ True False  True ... False False  True]
 [ True  True  True ...  True False  True]
 [False False False ... False False  True]]