measurements

Helpers.

fidelity(rho, sigma)

Fidelity between two states.

Parameters:

Name	Type	Description	Default
rho	Qarray	state.	required
sigma	Qarray	state.	required

Returns:

Type	Description
float	Fidelity between rho and sigma.

Source code in jaxquantum/core/measurements.py

def fidelity(rho: Qarray, sigma: Qarray) -> float:
    """Fidelity between two states.

    Args:
        rho: state.
        sigma: state.

    Returns:
        Fidelity between rho and sigma.
    """
    rho = rho.to_dm()
    sigma = sigma.to_dm()

    sqrt_rho = powm(rho, 0.5)

    return ((powm(sqrt_rho @ sigma @ sqrt_rho, 0.5)).tr()) ** 2

get_logical_basis(n_qubits)

Compute a complete operator basis of a system composed of logical qubits.

        Args:
            n_qubits: number of qubits

        Returns:
            List containing the complete operator basis.

Source code in jaxquantum/core/measurements.py

def get_logical_basis(n_qubits: int) -> Qarray:
    """Compute a complete operator basis of a system composed of logical
    qubits.

                Args:
                    n_qubits: number of qubits

                Returns:
                    List containing the complete operator basis.
    """
    if n_qubits < 1:
        raise ValueError("n_qubits must be at least 1.")

    n_qubits -= 1

    operators = Qarray.from_list(
        [identity(2) / 2, sigmax() / 2, sigmay() / 2, sigmaz() / 2]
    )

    if n_qubits == 0:
        return operators

    sub_basis = get_logical_basis(n_qubits)
    basis = []

    sub_basis_size = sub_basis.bdims[-1]

    for i in range(4):
        for j in range(sub_basis_size):
            basis.append(operators[i] ^ sub_basis[j])

    return Qarray.from_list(basis)

get_physical_basis(qubits)

Compute a complete operator basis of a QEC code on a physical system specified by a number of qubits.

    Args:
        qubits: list of qubit codes, must have
        common_gates and params attributes.

    Returns:
        List containing the complete operator basis.

Source code in jaxquantum/core/measurements.py

def get_physical_basis(qubits: List) -> Qarray:
    """Compute a complete operator basis of a QEC code on a
    physical system specified by a number of qubits.

            Args:
                qubits: list of qubit codes, must have
                common_gates and params attributes.

            Returns:
                List containing the complete operator basis.
    """

    qubit = qubits[0]
    qubits = qubits[1:]
    try:
        operators = Qarray.from_list(
            [
                identity(qubit.params["N"]),
                qubit.common_gates["X"],
                qubit.common_gates["Y"],
                qubit.common_gates["Z"],
            ]
        )
    except KeyError:
        print("QEC code must have common_gates for all three axes.")
    except AttributeError:
        print("QEC code must have common_gates and params attribute.")

    if len(qubits) == 0:
        return operators

    sub_basis = get_physical_basis(qubits)
    basis = []

    sub_basis_size = sub_basis.bdims[-1]

    for i in range(4):
        for j in range(sub_basis_size):
            basis.append(operators[i] ^ sub_basis[j])

    return Qarray.from_list(basis)

overlap(rho, sigma)

Overlap between two states or operators.

Parameters:

Name	Type	Description	Default
rho	Qarray	state/operator.	required
sigma	Qarray	state/operator.	required

Returns:

Type	Description
Array	Overlap between rho and sigma.

Source code in jaxquantum/core/measurements.py

def overlap(rho: Qarray, sigma: Qarray) -> Array:
    """Overlap between two states or operators.

    Args:
        rho: state/operator.
        sigma: state/operator.

    Returns:
        Overlap between rho and sigma.
    """

    if rho.is_vec() and sigma.is_vec():
        return jnp.abs(((rho.to_ket().dag() @ sigma.to_ket()).trace())) ** 2
    elif rho.is_vec():
        rho = rho.to_ket()
        res = (rho.dag() @ sigma @ rho).data
        return res.squeeze(-1).squeeze(-1)
    elif sigma.is_vec():
        sigma = sigma.to_ket()
        res = (sigma.dag() @ rho @ sigma).data
        return res.squeeze(-1).squeeze(-1)
    else:
        return (rho.dag() @ sigma).trace()

quantum_state_tomography(rho, physical_basis, logical_basis)

Perform quantum state tomography to retrieve the density matrix in the logical basis.

Args:
    rho: state expressed in the physical Hilbert space basis.
    physical_basis: list of logical operators expressed in the physical
    Hilbert space basis forming a complete logical operator basis.
    logical_basis: list of logical operators expressed in the
    logical Hilbert space basis forming a complete operator basis.


Returns:
    Density matrix of state rho expressed in the logical basis.

Source code in jaxquantum/core/measurements.py

def quantum_state_tomography(
    rho: Qarray, physical_basis: Qarray, logical_basis: Qarray
) -> Qarray:
    """Perform quantum state tomography to retrieve the density matrix in
    the logical basis.

        Args:
            rho: state expressed in the physical Hilbert space basis.
            physical_basis: list of logical operators expressed in the physical
            Hilbert space basis forming a complete logical operator basis.
            logical_basis: list of logical operators expressed in the
            logical Hilbert space basis forming a complete operator basis.


        Returns:
            Density matrix of state rho expressed in the logical basis.
    """
    dm = jnp.zeros_like(logical_basis[0].data)
    rho = rho.to_dm()

    if physical_basis.bdims[-1] != logical_basis.bdims[-1]:
        raise ValueError(
            f"The two bases should have the same size for the "
            f"last batch dimension. Received "
            f"{physical_basis.bdims} and {logical_basis.bdims} "
            f"instead."
        )

    space_size = physical_basis.bdims[-1]

    for i in tqdm(range(space_size), total=space_size):
        p_i = (rho @ physical_basis[i]).trace()
        dm += p_i * logical_basis[i].data

    return Qarray.create(dm, dims=logical_basis.dims, bdims=physical_basis[0].bdims)