Proyección del operador de Pauli de referencia

Acción de la cuerda de Pauli en un estado fundamental computacional

La acción de una cuerda de Pauli sobre un estado de base computacional es bastante trivial, y constituye en sí misma un único estado de base computacional. Esto es una consecuencia directa de la estructura de las matrices de Pauli, que solo tienen un único elemento distinto de cero en cada una de sus filas. Por lo tanto, su acción sobre un qubit es:

\sigma_x |0 \rangle = |1 \rangle

\sigma_x |1 \rangle = |0 \rangle

\sigma_y |0 \rangle = i|1 \rangle

\sigma_y |1 \rangle = -i|0 \rangle

\sigma_z |0 \rangle = |0 \rangle

\sigma_z |1 \rangle = -|1 \rangle

I |0 \rangle = |0 \rangle

I |1 \rangle = |1 \rangle

Cada bit de la cadena de bits que identifica la base computacional se etiquetará con un $x \in \{0, 1 \}$ e. Para que la implementación sea lo más ligera posible, representaremos las cadenas de bits con bool las variables: $0\rightarrow \textrm{False}$ y $1\rightarrow \textrm{True}$ .

Para representar la acción de cada operador de Pauli en un estado de base computacional, le asignaremos tres variables: diag, sign, imag.

diag indica si el operador es diagonal:
- $\textrm{diag}(I) = \textrm{True}$
- $\textrm{diag}(\sigma_x) = \textrm{False}$
- $\textrm{diag}(\sigma_y) = \textrm{False}$
- $\textrm{diag}(\sigma_z) = \textrm{True}$
sign Determina si se produce un cambio de signo en el elemento de la matriz asociado al 0 o al 1:
- $\textrm{sign}(I) = \textrm{False}$
- $\textrm{sign}(\sigma_x) = \textrm{False}$
- $\textrm{sign}(\sigma_y) = \textrm{True}$
- $\textrm{sign}(\sigma_z) = \textrm{True}$
imag Indica si el elemento de la matriz contiene un componente complejo:
- $\textrm{imag}(I) = \textrm{False}$
- $\textrm{imag}(\sigma_x) = \textrm{False}$
- $\textrm{imag}(\sigma_y) = \textrm{True}$
- $\textrm{imag}(\sigma_z) = \textrm{False}$

Denominamos « $\sigma \in \{ I, \sigma_x, \sigma_y \sigma_z\}$ » a un operador de Pauli arbitrario. La acción del operador de Pauli sobre un estado de base computacional puede representarse entonces mediante la operación lógica:

\sigma |x \rangle = |x == \textrm{diag}(\sigma) \rangle (-1)^{x\textrm{ and sign}(\sigma)} (i)^{\textrm{imag}(\sigma)}.

Esto se generaliza directamente a un número arbitrario de qubits.

Comprobemos si esto funciona:

def connected_element_and_amplitude_bool(
    x: bool, diag: bool, sign: bool, imag: bool
) -> tuple[bool, complex]:
    """
    Finds the connected element to computational basis state |x> under
    the action of the Pauli operator represented by (diag, sign, imag).

    Args:
        x: Value of the bit, either True or False.
        diag: Whether the Pauli operator is diagonal (I, Z)
        sigma: Whether the Pauli operator's rows differ in sign (Y, Z)
        imag: Whether the Pauli operator is purely imaginary (Y)

    Returns:
        A length-2 tuple:
            - The connected element to x, either False or True
            - The matrix element
    """
    return x == diag, (-1) ** (x and sign) * (1j) ** (imag)


sigma_indices = [0, 1, 2, 3]
sigma_string = ["I", "SX", "SZ", "SY"]
sigma_diag = [True, False, True, False]
sigma_sign = [False, False, True, True]
sigma_imag = [False, False, False, True]
qubit_values = [False, True]

for xi in sigma_indices:
    print("-------------------")
    print(sigma_string[xi])
    for x in qubit_values:
        x_p, matrix_element = connected_element_and_amplitude_bool(
            x, sigma_diag[xi], sigma_sign[xi], sigma_imag[xi]
        )
        print(
            "|"
            + str(x)
            + "> -->  |"
            + str(x_p)
            + ">    ME:"
            + str(matrix_element)
        )

Output:

-------------------
I
|False> -->  |False>    ME:(1+0j)
|True> -->  |True>    ME:(1+0j)
-------------------
SX
|False> -->  |True>    ME:(1+0j)
|True> -->  |False>    ME:(1+0j)
-------------------
SZ
|False> -->  |False>    ME:(1+0j)
|True> -->  |True>    ME:(-1+0j)
-------------------
SY
|False> -->  |True>    ME:1j
|True> -->  |False>    ME:(-0-1j)

Generamos un gran número de cadenas de bits (50 M) para un sistema de 40 qubits:

import numpy as np
from qiskit_addon_sqd.qubit import sort_and_remove_duplicates

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")
    )


n_qubits = 40
bts_matrix = random_bitstrings(50_000_000, n_qubits)

# We need to sort the bitstrings and only keep the unique ones
# NOTE: It is essential for the projection code to have the bitstrings sorted!
bts_matrix = sort_and_remove_duplicates(bts_matrix).astype("bool")

# Final subspace dimension after getting rid of duplicated bitstrings
d = bts_matrix.shape[0]

print("Total number of unique bitstrings: " + str(d))

Output:

Total number of unique bitstrings: 49998839

Funciones de proyección de Pauli del modelo de referencia SQD

La cuerda de Pauli que nos ocupa es $\sigma_z \otimes ... \otimes \sigma_z$ .

Se tienen en cuenta diferentes dimensiones de subespacios mediante la división de la matriz de cadenas de bits. Mide el tiempo que tarda la proyección en el subespacio para los distintos tamaños de subespacio.

import time

from qiskit.quantum_info import Pauli
from qiskit_addon_sqd.qubit import matrix_elements_from_pauli

pauli = Pauli("Z" * n_qubits)

# Different subspace sizes to test
d_list = np.linspace(d / 1000, d, 20).astype("int")

# To store the walltime
time_array = np.zeros(20)

for i in range(20):
    int_bts_matrix = bts_matrix[: d_list[i], :]
    time_1 = time.time()
    _ = matrix_elements_from_pauli(int_bts_matrix, pauli)
    time_array[i] = time.time() - time_1
    print(f"Iteration {i} took {round(time_array[i], 6)}s")

Output:

Iteration 0 took 0.201246s
Iteration 1 took 0.348222s
Iteration 2 took 0.576333s
Iteration 3 took 0.78356s
Iteration 4 took 1.016162s
Iteration 5 took 1.305325s
Iteration 6 took 1.392751s
Iteration 7 took 1.632433s
Iteration 8 took 1.826521s
Iteration 9 took 2.02903s
Iteration 10 took 2.297458s
Iteration 11 took 2.588042s
Iteration 12 took 2.738746s
Iteration 13 took 2.906144s
Iteration 14 took 3.148833s
Iteration 15 took 3.323253s
Iteration 16 took 3.664171s
Iteration 17 took 3.680663s
Iteration 18 took 4.008313s
Iteration 19 took 4.173532s

import matplotlib.pyplot as plt

# Data for energies plot
x1 = d_list
y1 = time_array

# Plot energies
plt.title("Runtime vs subspace dimension 40 qubits")
plt.xlabel("Subspace dimension (millions)")
plt.ylabel("Wall time [s]")
plt.xticks([1e7, 2e7, 3e7, 4e7, 5e7], [str(i) for i in [10, 20, 30, 40, 50]])
plt.plot(x1, y1, marker=".", markersize=20)
plt.tight_layout()
plt.show()

Output:

Ahora hacemos lo mismo con 60 qubits:

n_qubits = 60
bts_matrix = random_bitstrings(50_000_000, n_qubits)

# We need to sort the bitstrings and just keep the unique ones
bts_matrix = sort_and_remove_duplicates(bts_matrix).astype("bool")

# Final subspace dimension after getting rid of duplicated bitstrings
d = bts_matrix.shape[0]

print("Total number of unique bitstrings: " + str(d))

Output:

Total number of unique bitstrings: 50000000

pauli = Pauli("Z" * n_qubits)

# Different subspace sizes to test
d_list = np.linspace(d / 1000, d, 20).astype("int")

# It is better to do this once
row_array = np.arange(d)

# To store the walltime
time_array = np.zeros(20)

for i in range(20):
    int_bts_matrix = bts_matrix[: d_list[i], :]
    int_row_array = row_array[: d_list[i]]
    time_1 = time.time()
    _ = matrix_elements_from_pauli(int_bts_matrix, pauli)
    time_array[i] = time.time() - time_1
    print(f"Iteration {i} took {round(time_array[i], 6)}s")

Output:

Iteration 0 took 0.236567s
Iteration 1 took 0.424116s
Iteration 2 took 0.673399s
Iteration 3 took 0.905164s
Iteration 4 took 1.168936s
Iteration 5 took 1.454204s
Iteration 6 took 1.74778s
Iteration 7 took 1.920795s
Iteration 8 took 2.259994s
Iteration 9 took 2.550674s
Iteration 10 took 2.681287s
Iteration 11 took 3.04411s
Iteration 12 took 3.293262s
Iteration 13 took 3.471247s
Iteration 14 took 3.726639s
Iteration 15 took 4.072854s
Iteration 16 took 4.221037s
Iteration 17 took 4.498535s
Iteration 18 took 4.741108s
Iteration 19 took 5.159038s

# Data for energies plot
x1 = d_list
y1 = time_array

fig, axs = plt.subplots(1, 1, figsize=(6, 6))

# Plot energies
axs.plot(x1, y1, marker=".", markersize=20)
axs.set_title("Runtime vs subspace dimension 60 qubits")
axs.set_xlabel("Subspace dimension (millions)")
plt.xticks([1e7, 2e7, 3e7, 4e7, 5e7], [str(i) for i in [10, 20, 30, 40, 50]])
axs.set_ylabel("Wall time [s]")

plt.tight_layout()
plt.show()

Output:

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