I-project ang mga Pauli operator sa mga Hilbert subspace

Ipinapakita namin ang iba't ibang paraan ng pag-project ng weighted linear combination ng mga Pauli string sa isang subspace na tinutukoy ng subset na may sukat na $d$ ng mga computational basis state. Ang na-project na operator ay iniimbak bilang isang $d \times d$ scipy.sparse.coo_matrix.

Bilang halimbawa, tinitingnan natin ang Hamiltonian ng one-dimensional na Heisenberg model na may periodic boundary condition at $L = 22$ na mga spin:

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

Nagbibigay ang package na ito ng dalawang tool para gawin ang projection na ito:

• qiskit_addon_sqd.qubit.matrix_elements_from_pauli(): isang lower-level na function na nagbabalik ng mga non-zero na matrix element ng isang Pauli string at ng mga kaukulang index ng mga non-zero na element.

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

Ipinapakita ng gabay na ito kung paano gamitin ang dalawang tool na ito para makabuo ng parehong sparse operator.

Kahulugan ng subspace

Para sa halimbawang ito, gumagawa tayo ng mga random na bitstring na may haba na 22. Para gumana nang tama ang mga projection function, mahalagang ang mga bitstring na tumutukoy sa subspace ay natatangi at nakaayos nang pataas ayon sa kanilang unsigned integer na representasyon.

Magagawa ito gamit ang function na 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

Unang paraan: mga nonzero na matrix element at index

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)
    )

Higher-level na implementasyon

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 ...

Tingnan kung pareho ang nilalabas ng dalawang implementasyon na coo_matrix:

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

0j