ElectronicIntegrals#

class ElectronicIntegrals(alpha=None, beta=None, beta_alpha=None, *, validate=True)[source]#

Bases: LinearMixin

A container class for electronic operator coefficients (a.k.a. electronic integrals).

This class contains multiple qiskit_nature.second_q.operators.PolynomialTensor instances, dealing with the specific case of storing electronic integrals, where the up- and down-spin electronic interactions need to be handled separately. These two spins are also commonly referred to by \(\alpha\) and \(\beta\), respectively.

Specifically, this class stores three PolynomialTensor instances:

alpha: which stores the up-spin integrals
beta: which stores the down-spin integrals
beta_alpha: which stores beta-alpha-spin two-body integrals

These tensors are subject to some expectations, namely:

for alpha and beta only the following keys are allowed: "", "+-", "++--"
for beta_alpha the only allowed key is "++--"
the reported register_length attributes of all non-empty tensors must match

There are two ways of constructing the ElectronicIntegrals:

# assuming you already have your one- and two-body integrals from somewhere
h1_a, h2_aa, h1_b, h2_bb, h2_ba = ...

from qiskit_nature.second_q.operators import ElectronicIntegrals, PolynomialTensor

alpha = PolynomialTensor({"+-": h1_a, "++--": h2_aa})
beta = PolynomialTensor({"+-": h1_b, "++--": h2_bb})
beta_alpha = PolynomialTensor({"++--": h2_ba})

integrals = ElectronicIntegrals(alpha, beta, beta_alpha)

# alternatively, the following achieves the same effect:
integrals = ElectronicIntegrals.from_raw_integrals(h1_a, h2_aa, h1_b, h2_bb, h2_ba)

This class then exposes common mathematical operations performed on these tensors allowing simple manipulation of the underlying data structures.

# addition
integrals + integrals

# scalar multiplication
2.0 * integrals

This class will substitute empty beta and beta_alpha tensors with the alpha tensor when necessary. For example, this means the following will happen:

integrals_pure = ElectronicIntegrals(alpha)
integrals_mixed = ElectronicIntegrals(alpha, beta, beta_alpha)

sum = integrals_pure + integrals_mixed
print(sum.beta.is_empty())  # False
print(sum.beta_alpha.is_empty())  # False
print(sum.beta.equiv(alpha + beta))  # True
print(sum.beta_alpha.equiv(alpha + beta_alpha))  # True

The same logic holds for other mathematical operations involving multiple ElectronicIntegrals.

You can add a custom offset to be included in the operator generated from these coefficients like so:

from qiskit_nature.second_q.operators import PolynomialTensor

integrals: ElectronicIntegrals

offset = 2.5
integrals.alpha += PolynomialTensor({"": offset})

Any None-valued argument will internally be replaced by an empty PolynomialTensor (see also qiskit_nature.second_q.operators.PolynomialTensor.empty()).

Parameters:

alpha (PolynomialTensor | None) – the up-spin electronic integrals
beta (PolynomialTensor | None) – the down-spin electronic integrals
beta_alpha (PolynomialTensor | None) – the beta-alpha-spin two-body electronic integrals. This may only contain the ++-- key.
validate (bool) – when set to False, no validation will be performed. Disable this setting with care!

Raises:

KeyError – if the alpha tensor contains keys other than "", "+-", and "++--".
KeyError – if the beta tensor contains keys other than "", "+-", and "++--".
KeyError – if the beta_alpha tensor contains keys other than "++--".
ValueError – if the reported register_length attributes of the alpha-, beta-, and beta-alpha-spin tensors do not all match.

Attributes

alpha#: The up-spin electronic integrals.

alpha_beta#

The alpha-beta-spin two-body electronic integrals.

These get reconstructed from beta_alpha by transposing in the physicist’ ordering convention.

beta#: The down-spin electronic integrals.

beta_alpha#: The beta-alpha-spin two-body electronic integrals.

one_body#: Returns only the one-body integrals.

register_length#: The size of the operator that can be generated from these ElectronicIntegrals.

two_body#: Returns only the two-body integrals.

Methods

classmethod apply(function, *operands, multi=False, validate=True)[source]#

Exposes the qiskit_nature.second_q.operators.PolynomialTensor.apply() method.

This behaves identical to the apply implementation of the PolynomialTensor, applied to the alpha, beta, and beta_alpha attributes of the provided ElectronicIntegrals operands.

This method is special, because it handles the scenario in which any operand has a non-empty beta attribute, in which case the empty-beta attributes of any other operands will be filled with alpha attributes of those operands. The beta_alpha attributes will only be handled if they are non-empty in all supplied operands.

Parameters:

function (Callable[..., np.ndarray | SparseArray | complex]) – the function to apply to the internal arrays of the provided operands. This function must take numpy (or sparse) arrays as its positional arguments. The number of arguments must match the number of provided operands.
operands (ElectronicIntegrals) – a sequence of ElectronicIntegrals instances on which to operate.
multi (bool) – when set to True this indicates that the provided numpy function will return multiple new numpy arrays which will each be wrapped into an ElectronicIntegrals instance separately.
validate (bool) – when set to False, no validation will be performed. Disable this setting with care!

Returns:

A new ElectronicIntegrals.

Return type:

ElectronicIntegrals | list[ElectronicIntegrals]

classmethod einsum(einsum_map, *operands, validate=True)[source]#

Exposes the qiskit_nature.second_q.operators.PolynomialTensor.einsum() method.

This behaves identical to the einsum implementation of the PolynomialTensor, applied to the alpha, beta, and beta_alpha attributes of the provided ElectronicIntegrals operands.

Parameters:

einsum_map (dict[str, tuple[str, ...]]) – a dictionary, mapping from numpy.einsum() subscripts to a tuple of strings. These strings correspond to the keys of matrices to be extracted from the provided ElectronicIntegrals operands. The last string in this tuple indicates the key under which to store the result in the returned ElectronicIntegrals.
operands (ElectronicIntegrals) – a sequence of ElectronicIntegrals instances on which to operate.
validate (bool) – when set to False, no validation will be performed. Disable this setting with care!

Returns:

A new ElectronicIntegrals.

Return type:

ElectronicIntegrals

equiv(other)[source]#

Check equivalence of first ElectronicIntegrals with other

Parameters:: other (object) – second ElectronicIntegrals object to be compared with the first.
Returns:: True when ElectronicIntegrals objects are equivalent, False when not.
Return type:: bool

classmethod from_raw_integrals(h1_a, h2_aa=None, h1_b=None, h2_bb=None, h2_ba=None, *, validate=True, auto_index_order=True)[source]#

Loads the provided integral matrices into an ElectronicIntegrals instance.

When auto_index_order is enabled, qiskit_nature.second_q.operators.tensor_ordering.find_index_order() will be used to determine the index ordering of the h2_aa matrix, based on which the two-body matrices will automatically be transformed to the physicist’ order, which is required by the qiskit_nature.second_q.operators.PolynomialTensor.

Parameters:

h1_a (np.ndarray | SparseArray) – the alpha-spin one-body integrals.
h2_aa (np.ndarray | SparseArray | None) – the alpha-alpha-spin two-body integrals.
h1_b (np.ndarray | SparseArray | None) – the beta-spin one-body integrals.
h2_bb (np.ndarray | SparseArray | None) – the beta-beta-spin two-body integrals.
h2_ba (np.ndarray | SparseArray | None) – the beta-alpha-spin two-body integrals.
validate (bool) – whether or not to validate the integral matrices. Disable this setting with care!
auto_index_order (bool) – whether or not to automatically convert the matrices to physicists’ order.

Raises:

QiskitNatureError – if auto_index_order=True, upon encountering an invalid qiskit_nature.second_q.operators.tensor_ordering.IndexType.

Returns:

The resulting ElectronicIntegrals.

Return type:

ElectronicIntegrals

second_q_coeffs()[source]#

Constructs the total PolynomialTensor contained the second-quantized coefficients.

This function constructs the spin-orbital basis tensor as a qiskit_nature.second_q.operators.PolynomialTensor, by arranging the alpha and beta attributes in a block-ordered fashion (up-spin integrals cover the first part, down-spin integrals the second part of the resulting register space).

If the beta and/or beta_alpha attributes are empty, the alpha data will be used in their place.

Returns:: The PolynomialTensor representing the entire system.
Return type:: PolynomialTensor

split(function, indices_or_sections, *, validate=True)[source]#

Exposes the qiskit_nature.second_q.operators.PolynomialTensor.split() method.

This behaves identical to the split implementation of the PolynomialTensor, applied to the alpha, beta, and beta_alpha attributes of the provided ElectronicIntegrals operands.

Note

When splitting arrays this will likely lead to array shapes which would fail the shape validation check. This is considered an advanced use case which is why the user is left to disable this check themselves, to ensure they know what they are doing.

Parameters:

function (Callable[..., np.ndarray | SparseArray | Number]) – the splitting function to use. This function must take a single numpy (or sparse) array as its first input followed by a sequence of indices to split on. You should use functools.partial if you need to provide keyword arguments (e.g. partial(np.split, axis=-1)). Common methods to use here are numpy.hsplit() and numpy.vsplit().
indices_or_sections (int | Sequence[int]) – a single index or sequence of indices to split on.
validate (bool) – when set to False, no validation will be performed. Disable this setting with care!

Returns:

The new ElectronicIntegrals instances.

Return type:

list[ElectronicIntegrals]

classmethod stack(function, operands, *, validate=True)[source]#

Exposes the qiskit_nature.second_q.operators.PolynomialTensor.stack() method.

This behaves identical to the stack implementation of the PolynomialTensor, applied to the alpha, beta, and beta_alpha attributes of the provided ElectronicIntegrals operands.

Note

When stacking arrays this will likely lead to array shapes which would fail the shape validation check. This is considered an advanced use case which is why the user is left to disable this check themselves, to ensure they know what they are doing.

Parameters:

function (Callable[..., np.ndarray | SparseArray | Number]) – the stacking function to apply to the internal arrays of the provided operands. This function must take a sequence of numpy (or sparse) arrays as its first argument. You should use functools.partial if you need to provide keyword arguments (e.g. partial(np.stack, axis=-1)). Common methods to use here are numpy.hstack() and numpy.vstack().
operands (Sequence[ElectronicIntegrals]) – a sequence of ElectronicIntegrals instances on which to operate.
validate (bool) – when set to False, no validation will be performed. Disable this setting with care!

Returns:

A new ElectronicIntegrals.

Return type:

ElectronicIntegrals

trace_spin()[source]#

Returns a PolynomialTensor where the spin components have been traced out.

This will sum the alpha and beta components, tracing out the spin.

Returns:: A PolynomialTensor with the spin traced out.
Return type:: PolynomialTensor