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[Docs] Format math expressions (#738)
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* Math formatting.

* Math formatting.

* Math formatting.

* Math formatting.

* No more :math:.

* No more .. math::.

Co-authored-by: Nicholas Rubin <[email protected]>
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@rmlarose @ncrubin
rmlarose and ncrubin authored Jul 24, 2021
1 parent 906148e commit 84d169b
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69 changes: 39 additions & 30 deletions src/openfermion/circuits/gates/fermionic_simulation.py
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Original file line number Diff line number Diff line change
Expand Up @@ -184,18 +184,18 @@ def sum_of_interaction_operator_gate_generators(

@cirq.value_equality(approximate=True)
class ParityPreservingFermionicGate(cirq.Gate, metaclass=abc.ABCMeta):
r"""The Jordan-Wigner transform of :math:`\exp(-i H)` for a fermionic
Hamiltonian :math:`H`.
r"""The Jordan-Wigner transform of $\exp(-i H)$ for a fermionic
Hamiltonian $H$.
Each subclass corresponds to a set of generators :math:`\{G_i\}`
corresponding to the family of Hamiltonians :math:`\sum_i w_i G_i +
\text{h.c.}`, where the weights :math:`w_i \in \mathbb C` are specified by
Each subclass corresponds to a set of generators $\{G_i\}$
corresponding to the family of Hamiltonians $\sum_i w_i G_i +
\text{h.c.}$, where the weights $w_i \in \mathbb C$ are specified by
the instance.
The Jordan-Wigner mapping maps the fermionic modes :math:`(0, \ldots, n -
1)` to qubits :math:`(0, \ldots, n - 1)`, respectively.
The Jordan-Wigner mapping maps the fermionic modes $(0, \ldots, n -
1)$ to qubits $(0, \ldots, n - 1)$, respectively.
Each generator :math:`G_i` must be a linear combination of fermionic
Each generator $G_i$ must be a linear combination of fermionic
monomials consisting of an even number of creation/annihilation operators.
This is so that the Jordan-Wigner transform acts only on the gate's qubits,
even when the fermionic modes are offset as part of a larger Jordan-Wigner
Expand Down Expand Up @@ -231,8 +231,8 @@ def __init__(
@staticmethod
@abc.abstractmethod
def fermion_generator_components() -> Tuple['openfermion.FermionOperator']:
r"""The FermionOperators :math:`(G_i)_i` such that the gate's fermionic
generator is :math:`\sum_i w_i G_i + \text{h.c.}` where :math:`(w_i)_i`
r"""The FermionOperators $(G_i)_i$ such that the gate's fermionic
generator is $\sum_i w_i G_i + \text{h.c.}$ where $(w_i)_i$
are the gate's weights."""

@abc.abstractmethod
Expand Down Expand Up @@ -360,8 +360,8 @@ def _circuit_diagram_info_(self, args: cirq.CircuitDiagramInfoArgs


class InteractionOperatorFermionicGate(ParityPreservingFermionicGate):
r"""The Jordan-Wigner transform of :math:`\exp(-i H)` for a fermionic
Hamiltonian :math:`H`, where :math:`H` is an interaction operator.
r"""The Jordan-Wigner transform of $\exp(-i H)$ for a fermionic
Hamiltonian $H$, where $H$ is an interaction operator.
See openfermion.ParityPreservingFermionicGate and
openfermion.InteractionOperator for more details.
Expand Down Expand Up @@ -401,27 +401,30 @@ class QuadraticFermionicSimulationGate(InteractionOperatorFermionicGate,
cirq.TwoQubitGate, cirq.EigenGate):
r"""``(w0 |10⟩⟨01| + h.c.) + w1 |11⟩⟨11|`` interaction.
With weights :math:`(w_0, w_1)` and exponent :math:`t`, this gate's matrix
With weights $(w_0, w_1)$ and exponent $t$, this gate's matrix
is defined as
.. math::
$$
e^{-i t H},
$$
where
.. math::
$$
H = \left(w_0 \left| 10 \right\rangle\left\langle 01 \right| +
\text{h.c.}\right) -
w_1 \left| 11 \right\rangle \left\langle 11 \right|.
$$
This corresponds to the Jordan-Wigner transform of
.. math::
$$
H = (w_0 a^{\dagger}_i a_{i+1} + \text{h.c.}) +
w_1 a_{i}^{\dagger} a_{i+1}^{\dagger} a_{i} a_{i+1},
$$
where :math:`a_i` and :math:`a_{i+1}` are the annihilation operators for
the fermionic modes :math:`i` and :math:`(i+1)`, respectively mapped to the
where $a_i$ and $a_{i+1}$ are the annihilation operators for
the fermionic modes $i$ and $(i+1)$, respectively mapped to the
first and second qubits on which this gate acts.
Args:
Expand Down Expand Up @@ -532,34 +535,37 @@ class CubicFermionicSimulationGate(InteractionOperatorFermionicGate,
cirq.ThreeQubitGate, cirq.EigenGate):
r"""``w0|110⟩⟨101| + w1|110⟩⟨011| + w2|101⟩⟨011|`` + h.c. interaction.
With weights :math:`(w_0, w_1, w_2)` and exponent :math:`t`, this gate's
With weights $(w_0, w_1, w_2)$ and exponent $t$, this gate's
matrix is defined as
.. math::
$$
e^{-i t H},
$$
where
.. math::
$$
H = \left(w_0 \left| 110 \right\rangle\left\langle 101 \right| +
\text{h.c.}\right) +
\left(w_1 \left| 110 \right\rangle\left\langle 011 \right| +
\text{h.c.}\right) +
\left(w_2 \left| 101 \right\rangle\left\langle 011 \right| +
\text{h.c.}\right)
$$
This corresponds to the Jordan-Wigner transform of
.. math::
$$
H = -\left(w_0 a^{\dagger}_i a^{\dagger}_{i+1} a_{i} a_{i+2} +
\text{h.c.}\right) -
\left(w_1 a^{\dagger}_i a^{\dagger}_{i+1} a_{i+1} a_{i+2} +
\text{h.c.}\right) -
\left(w_2 a^{\dagger}_i a^{\dagger}_{i+2} a_{i+1} a_{i+2} +
\text{h.c.}\right),
$$
where :math:`a_i`, :math:`a_{i+1}`, :math:`a_{i+2}` are the annihilation
operators for the fermionic modes :math:`i`, :math:`(i+1)` :math:`(i+2)`,
where $a_i$, $a_{i+1}$, $a_{i+2}$ are the annihilation
operators for the fermionic modes $i$, $(i+1)$ $(i+2)$,
respectively mapped to the three qubits on which this gate acts.
Args:
Expand Down Expand Up @@ -681,34 +687,37 @@ class QuarticFermionicSimulationGate(InteractionOperatorFermionicGate,
cirq.EigenGate):
r"""Rotates Hamming-weight 2 states into their bitwise complements.
With weights :math:`(w_0, w_1, w_2)` and exponent :math:`t`, this gate's
With weights $(w_0, w_1, w_2)$ and exponent $t$, this gate's
matrix is defined as
.. math::
$$
e^{-i t H},
$$
where
.. math::
$$
H = \left(w_0 \left| 1001 \right\rangle\left\langle 0110 \right| +
\text{h.c.}\right) +
\left(w_1 \left| 1010 \right\rangle\left\langle 0101 \right| +
\text{h.c.}\right) +
\left(w_2 \left| 1100 \right\rangle\left\langle 0011 \right| +
\text{h.c.}\right)
$$
This corresponds to the Jordan-Wigner transform of
.. math::
$$
H = -\left(w_0 a^{\dagger}_i a^{\dagger}_{i+3} a_{i+1} a_{i+2} +
\text{h.c.}\right) -
\left(w_1 a^{\dagger}_i a^{\dagger}_{i+2} a_{i+1} a_{i+3} +
\text{h.c.}\right) -
\left(w_2 a^{\dagger}_i a^{\dagger}_{i+1} a_{i+2} a_{i+3} +
\text{h.c.}\right),
$$
where :math:`a_i`, ..., :math:`a_{i+3}` are the annihilation operators for
the fermionic modes :math:`i`, ..., :math:`(i+3)`, respectively
where $a_i$, ..., $a_{i+3}$ are the annihilation operators for
the fermionic modes $i$, ..., $(i+3)$, respectively
mapped to the four qubits on which this gate acts.
Expand Down
34 changes: 17 additions & 17 deletions src/openfermion/circuits/low_rank.py
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Original file line number Diff line number Diff line change
Expand Up @@ -22,26 +22,26 @@ def get_chemist_two_body_coefficients(two_body_coefficients, spin_basis=True):
r"""Convert two-body operator coefficients to low rank tensor.
The input is a two-body fermionic Hamiltonian expressed as
:math:`\sum_{pqrs} h_{pqrs} a^\dagger_p a^\dagger_q a_r a_s`
$\sum_{pqrs} h_{pqrs} a^\dagger_p a^\dagger_q a_r a_s$
We will convert this to the chemistry convention expressing it as
:math:`\sum_{pqrs} g_{pqrs} a^\dagger_p a_q a^\dagger_r a_s`
$\sum_{pqrs} g_{pqrs} a^\dagger_p a_q a^\dagger_r a_s$
but without the spin degree of freedom.
In the process of performing this conversion, constants and one-body
terms come out, which will be returned as well.
Args:
two_body_coefficients (ndarray): an N x N x N x N
numpy array giving the :math:`h_{pqrs}` tensor.
numpy array giving the $h_{pqrs}$ tensor.
spin_basis (bool): True if the two-body terms are passed in spin
orbital basis. False if already in spatial orbital basis.
Returns:
one_body_correction (ndarray): an N x N array of floats giving
coefficients of the :math:`a^\dagger_p a_q` terms that come out.
coefficients of the $a^\dagger_p a_q$ terms that come out.
chemist_two_body_coefficients (ndarray): an N x N x N x N numpy array
giving the :math:`g_{pqrs}` tensor in chemist notation.
giving the $g_{pqrs}$ tensor in chemist notation.
Raises:
TypeError: Input must be two-body number conserving
Expand Down Expand Up @@ -80,14 +80,14 @@ def low_rank_two_body_decomposition(two_body_coefficients,
r"""Convert two-body operator into sum of squared one-body operators.
As in arXiv:1808.02625, this function decomposes
:math:`\sum_{pqrs} h_{pqrs} a^\dagger_p a^\dagger_q a_r a_s` as
:math:`\sum_{l} \lambda_l (\sum_{pq} g_{lpq} a^\dagger_p a_q)^2`
$\sum_{pqrs} h_{pqrs} a^\dagger_p a^\dagger_q a_r a_s$ as
$\sum_{l} \lambda_l (\sum_{pq} g_{lpq} a^\dagger_p a_q)^2$
l is truncated to take max value L so that
:math:`\sum_{l=0}^{L-1} (\sum_{pq} |g_{lpq}|)^2 |\lambda_l| < x`
$\sum_{l=0}^{L-1} (\sum_{pq} |g_{lpq}|)^2 |\lambda_l| < x$
Args:
two_body_coefficients (ndarray): an N x N x N x N
numpy array giving the :math:`h_{pqrs}` tensor.
numpy array giving the $h_{pqrs}$ tensor.
This tensor must be 8-fold symmetric (real integrals).
truncation_threshold (optional Float): the value of x, above.
final_rank (optional int): if provided, this specifies the value of
Expand All @@ -97,13 +97,13 @@ def low_rank_two_body_decomposition(two_body_coefficients,
Returns:
eigenvalues (ndarray of floats): length L array
giving the :math:`\lambda_l`.
giving the $\lambda_l$.
one_body_squares (ndarray of floats): L x N x N array of floats
corresponding to the value of :math:`g_{pql}`.
corresponding to the value of $g_{pql}$.
one_body_correction (ndarray): One-body correction terms that result
from reordering to chemist ordering, in spin-orbital basis.
truncation_value (float): after truncation, this is the value
:math:`\sum_{l=0}^{L-1} (\sum_{pq} |g_{lpq}|)^2 |\lambda_l| < x`
$\sum_{l=0}^{L-1} (\sum_{pq} |g_{lpq}|)^2 |\lambda_l| < x$
Raises:
TypeError: Invalid two-body coefficient tensor specification.
Expand Down Expand Up @@ -163,22 +163,22 @@ def prepare_one_body_squared_evolution(one_body_matrix, spin_basis=True):
r"""Get Givens angles and DiagonalHamiltonian to simulate squared one-body.
The goal here will be to prepare to simulate evolution under
:math:`(\sum_{pq} h_{pq} a^\dagger_p a_q)^2` by decomposing as
:math:`R e^{-i \sum_{pq} V_{pq} n_p n_q} R^\dagger' where
:math:`R` is a basis transformation matrix.
$(\sum_{pq} h_{pq} a^\dagger_p a_q)^2$ by decomposing as
$R e^{-i \sum_{pq} V_{pq} n_p n_q} R^\dagger$ where
$R$ is a basis transformation matrix.
TODO: Add option for truncation based on one-body eigenvalues.
Args:
one_body_matrix (ndarray of floats): an N by N array storing the
coefficients of a one-body operator to be squared. For instance,
in the above the elements of this matrix are :math:`h_{pq}`.
in the above the elements of this matrix are $h_{pq}$.
spin_basis (bool): Whether the matrix is passed in the
spin orbital basis.
Returns:
density_density_matrix(ndarray of floats) an N by N array storing
the diagonal two-body coefficeints :math:`V_{pq}` above.
the diagonal two-body coefficeints $V_{pq}$ above.
basis_transformation_matrix (ndarray of floats) an N by N array
storing the values of the basis transformation.
Expand Down
30 changes: 15 additions & 15 deletions src/openfermion/circuits/primitives/bogoliubov_transform.py
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Original file line number Diff line number Diff line change
Expand Up @@ -28,41 +28,41 @@ def bogoliubov_transform(
r"""Perform a Bogoliubov transformation.
This circuit performs the transformation to a basis determined by a new set
of fermionic ladder operators. It performs the unitary :math:`U` such that
.. math::
of fermionic ladder operators. It performs the unitary $U$ such that
$$
U a^\dagger_p U^{-1} = b^\dagger_p
$$
where the :math:`a^\dagger_p` are the original creation operators and the
:math:`b^\dagger_p` are the new creation operators. The new creation
where the $a^\dagger_p$ are the original creation operators and the
$b^\dagger_p$ are the new creation operators. The new creation
operators are linear combinations of the original ladder operators with
coefficients given by the matrix `transformation_matrix`, which will be
referred to as :math:`W` in the following.
referred to as $W$ in the following.
If :math:`W` is an `N \times N` matrix, then the :math:`b^\dagger_p` are
If $W$ is an $N \times N$ matrix, then the $b^\dagger_p$ are
given by
.. math::
$$
b^\dagger_p = \sum_{q=1}^N W_{pq} a^\dagger_q.
$$
If :math:`W` is an `N \times 2N` matrix, then the :math:`b^\dagger_p` are
If $W$ is an $N \times 2N$ matrix, then the $b^\dagger_p$ are
given by
.. math::
$$
b^\dagger_p = \sum_{q=1}^N W_{pq} a^\dagger_q
+ \sum_{q=N+1}^{2N} W_{pq} a_q.
$$
This algorithm assumes the Jordan-Wigner Transform.
Args:
qubits: The qubits to which to apply the circuit.
transformation_matrix: The matrix :math:`W` holding the coefficients
transformation_matrix: The matrix $W$ holding the coefficients
that describe the new creation operators in terms of the original
ladder operators. Its shape should be either :math:`NxN` or
:math:`Nx(2N)`, where :math:`N` is the number of qubits.
ladder operators. Its shape should be either $NxN$ or
$Nx(2N)$, where $N$ is the number of qubits.
initial_state: Optionally specifies a computational basis state
to assume that the qubits start in. This assumption enables
optimizations that result in a circuit with fewer gates.
Expand Down
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