I-benchmark ang Pauli operator projection

Aksyon ng Pauli string sa isang computational basis state

Ang aksyon ng isang Pauli string sa isang computational basis state ay medyo simple, at ito mismo ay isang computational basis state. Direktang bunga ito ng istruktura ng mga Pauli matrix, na may iisang nonzero na elemento lamang sa bawat row nila. Kaya, ang aksyon nila sa isang qubit ay:

---

$$\sigma_x |0 \rangle = |1 \rangle$$ $$\sigma_x |1 \rangle = |0 \rangle$$

---

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

---

$$\sigma_z |0 \rangle = |0 \rangle$$ $$\sigma_z |1 \rangle = -|1 \rangle$$

---

$$I |0 \rangle = |0 \rangle$$ $$I |1 \rangle = |1 \rangle$$

---

Ang bawat bit sa bitstring na nagla-label sa computational basis ay lalagyan ng label na $x \in \{0, 1 \}$. Para panatilihing magaan hangga't maaari ang implementasyon, irerepresenta natin ang mga bitstring gamit ang mga variable na bool: $0\rightarrow \textrm{False}$ at $1\rightarrow \textrm{True}$.

Para irepresenta ang aksyon ng bawat Pauli operator sa isang computational basis state, magtatalaga tayo rito ng tatlong variable: diag, sign, imag.

• Ang diag ay nagpapakita kung diagonal ang operator:

  • $\textrm{diag}(I) = \textrm{True}$

  • $\textrm{diag}(\sigma_x) = \textrm{False}$

  • $\textrm{diag}(\sigma_y) = \textrm{False}$

  • $\textrm{diag}(\sigma_z) = \textrm{True}$

• Tinutukoy ng sign kung may pagbabago ng sign sa matrix element na konektado sa 0 o 1:

  • $\textrm{sign}(I) = \textrm{False}$

  • $\textrm{sign}(\sigma_x) = \textrm{False}$

  • $\textrm{sign}(\sigma_y) = \textrm{True}$

  • $\textrm{sign}(\sigma_z) = \textrm{True}$

• Tinutukoy ng imag kung may complex na component ang matrix element:

  • $\textrm{imag}(I) = \textrm{False}$

  • $\textrm{imag}(\sigma_x) = \textrm{False}$

  • $\textrm{imag}(\sigma_y) = \textrm{True}$

  • $\textrm{imag}(\sigma_z) = \textrm{False}$

Nilalagyan natin ng label ang isang arbitraryong Pauli operator bilang $\sigma \in \{ I, \sigma_x, \sigma_y \sigma_z\}$. Ang aksyon ng Pauli operator sa isang computational basis state ay maaari nang irepresenta ng logic operation:

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

Direkta itong nagagawang general para sa arbitraryong bilang ng mga qubit. Suriin natin kung gumagana ito:

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

_needed = {"matplotlib": "matplotlib", "numpy": "numpy", "qiskit": "qiskit", "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")

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)
        )

-------------------
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)

Gumagawa tayo ng malaking bilang ng mga bitstring (50 M) para sa isang 40-qubit na sistema:

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))

Total number of unique bitstrings: 49998839

I-benchmark ang mga function ng SQD Pauli projection

Ang Pauli string na pinag-aaralan ay $\sigma_z \otimes ... \otimes \sigma_z$.

Isinasaalang-alang ang iba't ibang dimensyon ng subspace sa pamamagitan ng pag-slice sa matrix ng mga bitstring. Sinusukat namin ang oras ng subspace projection para sa iba't ibang laki ng subspace.

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")

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()

Gagawin naman natin ngayon ang pareho para sa 60 qubit:

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))

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")

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()