電子配置のプールにフェルミオン遷移を追加する

ここでは、各電子配置に遷移演算子を作用させて得られる電子配置のプールを拡張する機能を紹介します。

次の形式の単一電子ホッピングを追加する方法を示します。

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

$p, q = 1, ..., N_\textrm{orb}$、$\sigma \in \{ \uparrow, \downarrow\}$ とし、電子配置のバッチに含まれるすべての $|\textbf{x} \rangle$ に対して適用します。 まず、ランダムな電子配置のバッチを生成します。

# 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)

励起演算子は、フェルミオンノード (またはクビット) の数と同じ長さの numpy 配列の中で指定します。配列の各要素は、次の値をとることができる文字列でなければなりません。

• 'I': Identity

• '+': 生成演算子

• '-': 消滅演算子

(同じスピン種の間で) 可能なすべての単一電子遷移を生成します。

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' '-' '+']]

次に、bitstring_matrix 内の電子配置に遷移演算子を適用します。

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]]