Class for storing Hamiltonians that are quadratic in the fermionic

Inherits From: PolynomialTensor

### Used in the notebooks

Used in the tutorials

ladder operators. The operators stored in this class take the form

$$\sum{p, q} (M{pq} - \mu \delta_{pq}) a^\dagger_p a_q + \frac12 \sum_{p, q} (\Delta_{pq} a^\dagger_p a^\dagger_q + \text{h.c.}) + \text{constant} $$

where

- \$$M\$$ is a Hermitian n_qubits x n_qubits matrix.
- \$$\Delta\$$ is an antisymmetric n_qubits x n_qubits matrix.
- \$$\mu\$$ is a real number representing the chemical potential.
- \$$\delta_{pq}\$$ is the Kronecker delta symbol.


We separate the chemical potential $$\mu$$ from $$M$$ so that we can use it to adjust the expectation value of the total number of particles.

hermitian_part(ndarray): The matrix $$M$$, which represents the coefficients of the particle-number-conserving terms. This is an n_qubits x n_qubits numpy array of complex numbers. antisymmetric_part(ndarray): The matrix $$\Delta$$, which represents the coefficients of the non-particle-number-conserving terms. This is an n_qubits x n_qubits numpy array of complex numbers. constant(float, optional): A constant term in the operator. chemical_potential(float, optional): The chemical potential $$\mu$$.

antisymmetric_part The antisymmetric part.
combined_hermitian_part The Hermitian part including the chemical potential.
conserves_particle_number Whether this Hamiltonian conserves particle number.
constant The value of the constant term.
hermitian_part The Hermitian part not including the chemical potential.
n_body_tensors

n_qubits

## Methods

### add_chemical_potential

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Increase (or decrease) the chemical potential by some value.

### diagonalizing_bogoliubov_transform

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Compute the unitary that diagonalizes a quadratic Hamiltonian.

Any quadratic Hamiltonian can be rewritten in the form

$\sum_{j} \varepsilon_j b^\dagger_j b_j + \text{constant},$

where the $$b^\dagger_j$$ are a new set fermionic creation operators that satisfy the canonical anticommutation relations. The new creation operators are linear combinations of the original ladder operators. In the most general case, creation and annihilation operators are mixed together:

$\begin{pmatrix} b^\dagger_1 \\ \vdots \\ b^\dagger_N \\ \end{pmatrix} = W \begin{pmatrix} a^\dagger_1 \\ \vdots \\ a^\dagger_N \\ a_1 \\ \vdots \\ a_N \end{pmatrix},$

where $$W$$ is an $$N \times (2N)$$ matrix. However, if the Hamiltonian conserves particle number then creation operators don't need to be mixed with annihilation operators and $$W$$ only needs to be an $$N \times N$$ matrix:

$\begin{pmatrix} b^\dagger_1 \\ \vdots \\ b^\dagger_N \\ \end{pmatrix} = W \begin{pmatrix} a^\dagger_1 \\ \vdots \\ a^\dagger_N \\ \end{pmatrix},$

This method returns the matrix $$W$$.

Args
spin_sector (optional str): An optional integer specifying a spin sector to restrict to: 0 for spin-up and 1 for spin-down. Should only be specified if the Hamiltonian includes a spin degree of freedom and spin-up modes do not interact with spin-down modes. If specified, the modes are assumed to be ordered so that spin-up orbitals come before spin-down orbitals.

Returns
orbital_energies(ndarray) A one-dimensional array containing the $$\varepsilon_j$$ diagonalizing_unitary (ndarray): A matrix representing the transformation $$W$$ of the fermionic ladder operators. If the Hamiltonian conserves particle number then this is $$N \times N$$; otherwise it is $$N \times 2N$$. If spin sector is specified, then $$N$$ here represents the number of spatial orbitals rather than spin orbitals. constant(float) The constant

### diagonalizing_circuit

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Get a circuit for a unitary that diagonalizes this Hamiltonian

This circuit performs the transformation to a basis in which the Hamiltonian takes the diagonal form

$\sum_{j} \varepsilon_j b^\dagger_j b_j + \text{constant}.$

## Returns

circuit_description (list[tuple]):
A list of operations describing the circuit. Each operation
is a tuple of objects describing elementary operations that
can be performed in parallel. Each elementary operation
is either the string 'pht' indicating a particle-hole
transformation on the last fermionic mode, or a tuple of
the form \$$(i, j, \theta, \varphi)\$$,
indicating a Givens rotation
of modes \$$i\$$ and \$$j\$$ by angles \$$\theta\$$
and \$$\varphi\$$.


### ground_energy

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Return the ground energy.

### majorana_form

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Return the Majorana represention of the Hamiltonian.

Any quadratic Hamiltonian can be written in the form

$\frac{i}{2} \sum_{j, k} A_{jk} f_j f_k + \text{constant}$

where the $$f_i$$ are normalized Majorana fermion operators:

 f_j = \frac{1}{\sqrt{2} } (a^\dagger_j + a_j)

f_{j + N} = \frac{i}{\sqrt{2} } (a^\dagger_j - a_j)




and $$A$$ is a (2 * n_qubits) x (2 * n_qubits) real antisymmetric matrix. This function returns the matrix $$A$$ and the constant.

### orbital_energies

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Return the orbital energies.

Any quadratic Hamiltonian is unitarily equivalent to a Hamiltonian of the form

$\sum_{j} \varepsilon_j b^\dagger_j b_j + \text{constant}.$

We call the $$\varepsilon_j$$ the orbital energies. The eigenvalues of the Hamiltonian are sums of subsets of the orbital energies (up to the additive constant).

Args
non_negative(bool): If True, always return a list of orbital energies that are non-negative. This option is ignored if the Hamiltonian does not conserve particle number, in which case the returned orbital energies are always non-negative.

## Returns

orbital_energies(ndarray) A one-dimensional array containing the $$\varepsilon_j$$ constant(float) The constant

### projected_n_body_tensors

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Keep only selected elements.

Args
selection (Union[int, Iterable[int]): If int, keeps terms with at most (exactly, if exact is True) that many unique indices. If iterable, keeps only terms containing (all of, if exact is True) the specified indices. exact (bool): Whether or not the selection is strict.

### rotate_basis

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Rotate the orbital basis of the PolynomialTensor.

Args
rotation_matrix A square numpy array or matrix having dimensions of n_qubits by n_qubits. Assumed to be real and invertible.

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### __eq__

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Return self==value.

### __getitem__

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Look up matrix element.

Args
args Tuples indicating which coefficient to get. For instance, my_tensor[(6, 1), (8, 1), (2, 0)] returns my_tensor.n_body_tensors[1, 1, 0][6, 8, 2]

### __iter__

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Iterate over non-zero elements of PolynomialTensor.

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### __ne__

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Return self!=value.

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### __truediv__

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