qiu_hamiltonian_simulation.time_independent.fourier ¶

Changes of basis between the position and the momentum domain.

For a position axis of N = 2**n samples, the momentum amplitudes are the discrete Fourier transform phi(p_k) = sum_j e^(-2 pi i j k / N) psi(x_j) / sqrt(N), i.e. numpy.fft.fft(psi, norm="ortho"), with the momenta p_k in the FFT ordering of the conjugate axis. This is Qiskit's inverse quantum Fourier transform, see qiu_quantum_computing.qft.

Functions:

require_momentum_domain_axis –

Raise unless the axis is a Fourier conjugate axis in the FFT ordering.
require_position_domain_axis –

Raise unless the axis is a position domain axis.
to_momentum_basis –

Return the circuit mapping position amplitudes to momentum amplitudes.
to_position_basis –

Return the circuit mapping momentum amplitudes back to position amplitudes.

require_momentum_domain_axis ¶

require_momentum_domain_axis(axis: PhysicalAxis) -> None

Raise unless the axis is a Fourier conjugate axis in the FFT ordering.

Only then does the basis state |k> after to_momentum_basis hold the sample k of a signal on the axis.

Raises:

ValueError –

If the axis is not a Fourier domain axis, or if it is not in the FFT ordering.

Source code in packages/qiu-hamiltonian-simulation/src/qiu_hamiltonian_simulation/time_independent/fourier.py

def require_momentum_domain_axis(axis: PhysicalAxis) -> None:
    """Raise unless the axis is a Fourier conjugate axis in the `FFT` ordering.

    Only then does the basis state `|k>` after `to_momentum_basis` hold the sample
    `k` of a signal on the axis.

    Raises:
        ValueError: If the axis is not a Fourier domain axis, or if it is not in the
            `FFT` ordering.
    """
    if not axis.is_fourier_domain:
        raise ValueError(f"The axis must be a Fourier domain axis, got {axis!r}.")
    if axis.ordering != IndexOrdering.FFT:
        raise ValueError(
            f"The Fourier domain axis must be in the FFT ordering of the discrete "
            f"Fourier transform, got {axis!r}."
        )

require_position_domain_axis ¶

require_position_domain_axis(axis: PhysicalAxis) -> None

Raise unless the axis is a position domain axis.

Source code in packages/qiu-hamiltonian-simulation/src/qiu_hamiltonian_simulation/time_independent/fourier.py

def require_position_domain_axis(axis: PhysicalAxis) -> None:
    """Raise unless the axis is a position domain axis."""
    if axis.is_fourier_domain:
        raise ValueError(f"The axis must be a position domain axis, got {axis!r}.")

to_momentum_basis ¶

to_momentum_basis(num_qubits: int, method: SynthesisMethod = GATE) -> QuantumCircuit

Return the circuit mapping position amplitudes to momentum amplitudes.

Source code in packages/qiu-hamiltonian-simulation/src/qiu_hamiltonian_simulation/time_independent/fourier.py

def to_momentum_basis(
    num_qubits: int, method: SynthesisMethod = SynthesisMethod.GATE
) -> QuantumCircuit:
    """Return the circuit mapping position amplitudes to momentum amplitudes."""
    return qft_circuit(num_qubits, inverse=True, method=method)

to_position_basis ¶

to_position_basis(num_qubits: int, method: SynthesisMethod = GATE) -> QuantumCircuit

Return the circuit mapping momentum amplitudes back to position amplitudes.

Source code in packages/qiu-hamiltonian-simulation/src/qiu_hamiltonian_simulation/time_independent/fourier.py

def to_position_basis(
    num_qubits: int, method: SynthesisMethod = SynthesisMethod.GATE
) -> QuantumCircuit:
    """Return the circuit mapping momentum amplitudes back to position amplitudes."""
    return qft_circuit(num_qubits, method=method)