Symmetrization

The py_alqorithms.symmetrization module provides functions to symmetrize a spin system by conjugating the Hamiltonian with Pauli \( \sigma^x \) operators.

Pure Spin System

For pure systems, the following two functions are provided:

• create_symmetrized_spin_circuit: Creates a quantum circuit implementing the trotterized time evolution with the Hamiltonian

  \[ \tilde H = \left( \bigotimes_i \sigma_i^x \right) H \left( \bigotimes_i \sigma_i^x \right) \]

  where \( H \) is the input spin Hamiltonian.

• apply_symmetrization_spins: Similar to the function above, but additionally applies a PauliX gate to every qubit to switch the definition of \( | 0 \rangle \) and \( | 1 \rangle \), both before and after the circuit implementing the trotterized time evolution with the symmetrized Hamiltonian.

Example

A minimal example describing the use of apply_symmetrization_spins function. The hamiltonian is given by

\[ H = \frac{1}{2} \sigma_0^z + \sigma_1^z. \]

Figure 1 illustrates the time evolution circuit of \(H\) with two trotter steps.

import py_alqorithms
from struqture_py.spins import SpinHamiltonianSystem, PauliProduct

# Parameters
algorithm = "ParityBased"
trotterization_order = 1
number_trottersteps = 2
time = 5.0

# Initizaling spin hamiltonian
hamiltonian = SpinHamiltonianSystem(2)
hamiltonian.add_operator_product(PauliProduct().z(0), 0.5)
hamiltonian.add_operator_product(PauliProduct().z(1), 1.0)

# Symmetrizing spin circuit
circuit = py_alqorithms.apply_symmetrization_spins(
    algorithm,
    hamiltonian,
    trotterization_order,
    number_trottersteps,
    time
)

Figure 1: Trotterized time evolution of spin hamiltonian

System-Bath

For system-bath algorithms, the following functions are provided:

• create_symmetrized_system_bath_circuit: Similar to create_symmetrized_spin_circuit, only applying the transformation to the system qubits. Together with the resulting circuit created by the selected algorithm, it also returns the list of qubits that constitute the system.
• apply_symmetrization_system_bath - Similar to apply_symmetrization_spins, only applying the transformation to system qubits.

Example

A minimal example describing the symmetrization of a system-bath circuit. The hamiltonian is given by

\[ H = \frac{1}{2} \sigma_{0,s}^z + \frac{1}{2} \sigma_{1,s}^x \sigma_{0,b}^x + \sigma_{1, b}^z. \]

Figure 2 illustrates a single trotter step time evolution of the system-bath hamiltonian.

import py_alqorithms
from struqture_py.spins import PauliProduct
from struqture_py.mixed_systems import MixedHamiltonianSystem, HermitianMixedProduct

# Parameters
algorithm = "ParityBased"
number_trottersteps = 1
time = 5.0
remapping = ([0,1], [2,3,4,5], None)
use_bath_as_control = False

# Initializing the system bath hamiltonian
hamiltonian = MixedHamiltonianSystem([2,2], [], [])
hamiltonian.add_operator_product(
    HermitianMixedProduct(
        [PauliProduct().z(0), PauliProduct()],
        [],
        []
    ),
    0.5
)

hamiltonian.add_operator_product(
    HermitianMixedProduct(
        [PauliProduct().x(1), PauliProduct().x(0)],
        [],
        []
    ),
    0.5
)
hamiltonian.add_operator_product(
    HermitianMixedProduct(
        [PauliProduct(), PauliProduct().z(1)],
        [],
        []
    ),
    1.0
)

# Symmetrizing system-bath circuit
circuit = py_alqorithms.apply_symmetrization_system_bath(
    algorithm,
    hamiltonian,
    number_trottersteps,
    time,
    use_bath_as_control,
    remapping
)

Figure 2: Trotterized time evolution of system-bath hamiltonian