Proyeksikan operator Pauli ke subruang Hilbert

Kita tunjukkan beberapa cara memproyeksikan kombinasi linear berbobot dari string Pauli ke subruang yang didefinisikan oleh himpunan bagian berukuran $d$ dari keadaan basis komputasi. Operator terproyeksi disimpan sebagai scipy.sparse.coo_matrix berukuran $d \times d$.

Sebagai contoh, kita pakai Hamiltonian model Heisenberg satu dimensi dengan kondisi batas periodik dan $L = 22$ spin:

$$H = \sum_{\langle i, j \rangle}\left( \sigma^x_i\sigma^x_j + \sigma^y_i\sigma^y_j + \sigma^z_i\sigma^z_j \right).$$

Paket ini menyediakan dua alat untuk melakukan proyeksi ini:

• qiskit_addon_sqd.qubit.matrix_elements_from_pauli(): fungsi tingkat rendah yang mengembalikan elemen matriks tak-nol dari sebuah string Pauli beserta indeks elemen tak-nol yang bersesuaian.

• qiskit_addon_sqd.qubit.project_operator_to_subspace(): a higher-level function that directly returns a scipy.sparse matrix.

Panduan ini menunjukkan cara memakai kedua alat tersebut untuk menghasilkan operator sparse yang sama.

Definisi subruang

Untuk contoh ini kita membangkitkan bitstring acak dengan panjang 22. Agar fungsi proyeksi bekerja sesuai harapan, bitstring yang mendefinisikan subruang harus unik dan terurut menaik berdasarkan representasi bilangan bulat tak bertandanya.

Hal ini bisa dilakukan dengan fungsi qiskit_addon_sqd.qubit.sort_and_remove_duplicates().

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

_needed = {"numpy": "numpy", "qiskit": "qiskit", "qiskit_addon_sqd": "qiskit-addon-sqd", "qiskit_addon_utils": "qiskit-addon-utils", "scipy": "scipy"}
_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
from qiskit_addon_sqd.qubit import sort_and_remove_duplicates

num_spins = 22
rand_seed = 22
np.random.seed(rand_seed)

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(50_000, num_spins)

# NOTE: It is essential for the projection code to have the bitstrings sorted!
bitstring_matrix = sort_and_remove_duplicates(bitstring_matrix)

print("Subspace dimension: " + str(bitstring_matrix.shape[0]))
print("Full Hilbert space dimension: " + str(2**num_spins))

Subspace dimension: 49718
Full Hilbert space dimension: 4194304

Metode pertama: elemen matriks tak-nol dan indeks

from qiskit.transpiler import CouplingMap
from qiskit_addon_utils.problem_generators import generate_xyz_hamiltonian

coupling_map = CouplingMap.from_ring(num_spins)
hamiltonian = generate_xyz_hamiltonian(coupling_map)
print(hamiltonian)

SparsePauliOp(['IIIIIIIIIIIIIIIIIIIIXX', 'IIIIIIIIIIIIIIIIIIIIYY', 'IIIIIIIIIIIIIIIIIIIIZZ', 'IIIIIIIIIIIIIIIIIIXXII', 'IIIIIIIIIIIIIIIIIIYYII', 'IIIIIIIIIIIIIIIIIIZZII', 'IIIIIIIIIIIIIIIIXXIIII', 'IIIIIIIIIIIIIIIIYYIIII', 'IIIIIIIIIIIIIIIIZZIIII', 'IIIIIIIIIIIIIIXXIIIIII', 'IIIIIIIIIIIIIIYYIIIIII', 'IIIIIIIIIIIIIIZZIIIIII', 'IIIIIIIIIIIIXXIIIIIIII', 'IIIIIIIIIIIIYYIIIIIIII', 'IIIIIIIIIIIIZZIIIIIIII', 'IIIIIIIIIIXXIIIIIIIIII', 'IIIIIIIIIIYYIIIIIIIIII', 'IIIIIIIIIIZZIIIIIIIIII', 'IIIIIIIIXXIIIIIIIIIIII', 'IIIIIIIIYYIIIIIIIIIIII', 'IIIIIIIIZZIIIIIIIIIIII', 'IIIIIIXXIIIIIIIIIIIIII', 'IIIIIIYYIIIIIIIIIIIIII', 'IIIIIIZZIIIIIIIIIIIIII', 'IIIIXXIIIIIIIIIIIIIIII', 'IIIIYYIIIIIIIIIIIIIIII', 'IIIIZZIIIIIIIIIIIIIIII', 'IIXXIIIIIIIIIIIIIIIIII', 'IIYYIIIIIIIIIIIIIIIIII', 'IIZZIIIIIIIIIIIIIIIIII', 'XXIIIIIIIIIIIIIIIIIIII', 'YYIIIIIIIIIIIIIIIIIIII', 'ZZIIIIIIIIIIIIIIIIIIII', 'XIIIIIIIIIIIIIIIIIIIIX', 'YIIIIIIIIIIIIIIIIIIIIY', 'ZIIIIIIIIIIIIIIIIIIIIZ', 'IIIIIIIIIIIIIIIIIIIXXI', 'IIIIIIIIIIIIIIIIIIIYYI', 'IIIIIIIIIIIIIIIIIIIZZI', 'IIIIIIIIIIIIIIIIIXXIII', 'IIIIIIIIIIIIIIIIIYYIII', 'IIIIIIIIIIIIIIIIIZZIII', 'IIIIIIIIIIIIIIIXXIIIII', 'IIIIIIIIIIIIIIIYYIIIII', 'IIIIIIIIIIIIIIIZZIIIII', 'IIIIIIIIIIIIIXXIIIIIII', 'IIIIIIIIIIIIIYYIIIIIII', 'IIIIIIIIIIIIIZZIIIIIII', 'IIIIIIIIIIIXXIIIIIIIII', 'IIIIIIIIIIIYYIIIIIIIII', 'IIIIIIIIIIIZZIIIIIIIII', 'IIIIIIIIIXXIIIIIIIIIII', 'IIIIIIIIIYYIIIIIIIIIII', 'IIIIIIIIIZZIIIIIIIIIII', 'IIIIIIIXXIIIIIIIIIIIII', 'IIIIIIIYYIIIIIIIIIIIII', 'IIIIIIIZZIIIIIIIIIIIII', 'IIIIIXXIIIIIIIIIIIIIII', 'IIIIIYYIIIIIIIIIIIIIII', 'IIIIIZZIIIIIIIIIIIIIII', 'IIIXXIIIIIIIIIIIIIIIII', 'IIIYYIIIIIIIIIIIIIIIII', 'IIIZZIIIIIIIIIIIIIIIII', 'IXXIIIIIIIIIIIIIIIIIII', 'IYYIIIIIIIIIIIIIIIIIII', 'IZZIIIIIIIIIIIIIIIIIII'],
              coeffs=[1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j,
 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j,
 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j,
 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j,
 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j,
 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j,
 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j,
 1.+0.j, 1.+0.j, 1.+0.j])

from qiskit_addon_sqd.qubit import matrix_elements_from_pauli
from scipy.sparse import coo_matrix

d = bitstring_matrix.shape[0]

# Initialize the coo_matrix object
operator_from_matrix_elements = coo_matrix((d, d), dtype="complex128")

for pauli in hamiltonian.paulis:
    matrix_elements, row_indices, col_indices = matrix_elements_from_pauli(
        bitstring_matrix, pauli
    )
    operator_from_matrix_elements += coo_matrix(
        (matrix_elements, (row_indices, col_indices)), (d, d)
    )

Implementasi tingkat tinggi

from qiskit_addon_sqd.qubit import project_operator_to_subspace

operator = project_operator_to_subspace(
    bitstring_matrix, hamiltonian, verbose=True
)

Projecting term 1 out of 66: (1+0j) * IIIIIIIIIIIIIIIIIIIIXX ...
Projecting term 2 out of 66: (1+0j) * IIIIIIIIIIIIIIIIIIIIYY ...
Projecting term 3 out of 66: (1+0j) * IIIIIIIIIIIIIIIIIIIIZZ ...
Projecting term 4 out of 66: (1+0j) * IIIIIIIIIIIIIIIIIIXXII ...
Projecting term 5 out of 66: (1+0j) * IIIIIIIIIIIIIIIIIIYYII ...
Projecting term 6 out of 66: (1+0j) * IIIIIIIIIIIIIIIIIIZZII ...
Projecting term 7 out of 66: (1+0j) * IIIIIIIIIIIIIIIIXXIIII ...
Projecting term 8 out of 66: (1+0j) * IIIIIIIIIIIIIIIIYYIIII ...
Projecting term 9 out of 66: (1+0j) * IIIIIIIIIIIIIIIIZZIIII ...
Projecting term 10 out of 66: (1+0j) * IIIIIIIIIIIIIIXXIIIIII ...
Projecting term 11 out of 66: (1+0j) * IIIIIIIIIIIIIIYYIIIIII ...
Projecting term 12 out of 66: (1+0j) * IIIIIIIIIIIIIIZZIIIIII ...
Projecting term 13 out of 66: (1+0j) * IIIIIIIIIIIIXXIIIIIIII ...
Projecting term 14 out of 66: (1+0j) * IIIIIIIIIIIIYYIIIIIIII ...
Projecting term 15 out of 66: (1+0j) * IIIIIIIIIIIIZZIIIIIIII ...
Projecting term 16 out of 66: (1+0j) * IIIIIIIIIIXXIIIIIIIIII ...
Projecting term 17 out of 66: (1+0j) * IIIIIIIIIIYYIIIIIIIIII ...
Projecting term 18 out of 66: (1+0j) * IIIIIIIIIIZZIIIIIIIIII ...
Projecting term 19 out of 66: (1+0j) * IIIIIIIIXXIIIIIIIIIIII ...
Projecting term 20 out of 66: (1+0j) * IIIIIIIIYYIIIIIIIIIIII ...
Projecting term 21 out of 66: (1+0j) * IIIIIIIIZZIIIIIIIIIIII ...
Projecting term 22 out of 66: (1+0j) * IIIIIIXXIIIIIIIIIIIIII ...
Projecting term 23 out of 66: (1+0j) * IIIIIIYYIIIIIIIIIIIIII ...
Projecting term 24 out of 66: (1+0j) * IIIIIIZZIIIIIIIIIIIIII ...
Projecting term 25 out of 66: (1+0j) * IIIIXXIIIIIIIIIIIIIIII ...
Projecting term 26 out of 66: (1+0j) * IIIIYYIIIIIIIIIIIIIIII ...
Projecting term 27 out of 66: (1+0j) * IIIIZZIIIIIIIIIIIIIIII ...
Projecting term 28 out of 66: (1+0j) * IIXXIIIIIIIIIIIIIIIIII ...
Projecting term 29 out of 66: (1+0j) * IIYYIIIIIIIIIIIIIIIIII ...
Projecting term 30 out of 66: (1+0j) * IIZZIIIIIIIIIIIIIIIIII ...
Projecting term 31 out of 66: (1+0j) * XXIIIIIIIIIIIIIIIIIIII ...
Projecting term 32 out of 66: (1+0j) * YYIIIIIIIIIIIIIIIIIIII ...
Projecting term 33 out of 66: (1+0j) * ZZIIIIIIIIIIIIIIIIIIII ...
Projecting term 34 out of 66: (1+0j) * XIIIIIIIIIIIIIIIIIIIIX ...
Projecting term 35 out of 66: (1+0j) * YIIIIIIIIIIIIIIIIIIIIY ...
Projecting term 36 out of 66: (1+0j) * ZIIIIIIIIIIIIIIIIIIIIZ ...
Projecting term 37 out of 66: (1+0j) * IIIIIIIIIIIIIIIIIIIXXI ...
Projecting term 38 out of 66: (1+0j) * IIIIIIIIIIIIIIIIIIIYYI ...
Projecting term 39 out of 66: (1+0j) * IIIIIIIIIIIIIIIIIIIZZI ...
Projecting term 40 out of 66: (1+0j) * IIIIIIIIIIIIIIIIIXXIII ...
Projecting term 41 out of 66: (1+0j) * IIIIIIIIIIIIIIIIIYYIII ...
Projecting term 42 out of 66: (1+0j) * IIIIIIIIIIIIIIIIIZZIII ...
Projecting term 43 out of 66: (1+0j) * IIIIIIIIIIIIIIIXXIIIII ...
Projecting term 44 out of 66: (1+0j) * IIIIIIIIIIIIIIIYYIIIII ...
Projecting term 45 out of 66: (1+0j) * IIIIIIIIIIIIIIIZZIIIII ...
Projecting term 46 out of 66: (1+0j) * IIIIIIIIIIIIIXXIIIIIII ...
Projecting term 47 out of 66: (1+0j) * IIIIIIIIIIIIIYYIIIIIII ...
Projecting term 48 out of 66: (1+0j) * IIIIIIIIIIIIIZZIIIIIII ...
Projecting term 49 out of 66: (1+0j) * IIIIIIIIIIIXXIIIIIIIII ...
Projecting term 50 out of 66: (1+0j) * IIIIIIIIIIIYYIIIIIIIII ...
Projecting term 51 out of 66: (1+0j) * IIIIIIIIIIIZZIIIIIIIII ...
Projecting term 52 out of 66: (1+0j) * IIIIIIIIIXXIIIIIIIIIII ...
Projecting term 53 out of 66: (1+0j) * IIIIIIIIIYYIIIIIIIIIII ...
Projecting term 54 out of 66: (1+0j) * IIIIIIIIIZZIIIIIIIIIII ...
Projecting term 55 out of 66: (1+0j) * IIIIIIIXXIIIIIIIIIIIII ...
Projecting term 56 out of 66: (1+0j) * IIIIIIIYYIIIIIIIIIIIII ...
Projecting term 57 out of 66: (1+0j) * IIIIIIIZZIIIIIIIIIIIII ...
Projecting term 58 out of 66: (1+0j) * IIIIIXXIIIIIIIIIIIIIII ...
Projecting term 59 out of 66: (1+0j) * IIIIIYYIIIIIIIIIIIIIII ...
Projecting term 60 out of 66: (1+0j) * IIIIIZZIIIIIIIIIIIIIII ...
Projecting term 61 out of 66: (1+0j) * IIIXXIIIIIIIIIIIIIIIII ...
Projecting term 62 out of 66: (1+0j) * IIIYYIIIIIIIIIIIIIIIII ...
Projecting term 63 out of 66: (1+0j) * IIIZZIIIIIIIIIIIIIIIII ...
Projecting term 64 out of 66: (1+0j) * IXXIIIIIIIIIIIIIIIIIII ...
Projecting term 65 out of 66: (1+0j) * IYYIIIIIIIIIIIIIIIIIII ...
Projecting term 66 out of 66: (1+0j) * IZZIIIIIIIIIIIIIIIIIII ...

Pastikan kedua implementasi menghasilkan coo_matrix yang sama:

print((operator.power(2) - operator_from_matrix_elements.power(2)).sum())

0j