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stable/0.4/.buildinfo

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+# Sphinx build info version 1
+# This file hashes the configuration used when building these files. When it is not found, a full rebuild will be done.
+config: 29bf4422ba7aa7a4a1c44bc977304a2e
+tags: 645f666f9bcd5a90fca523b33c5a78b7

stable/0.4/_sources/apidocs/array.rst.txt

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+.. _qiskit_dynamics-array:
+
+.. automodule:: qiskit_dynamics.array
+   :no-members:
+   :no-inherited-members:
+   :no-special-members:

stable/0.4/_sources/apidocs/backend.rst.txt

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+.. _qiskit_dynamics-backend:
+
+.. automodule:: qiskit_dynamics.backend
+   :no-members:
+   :no-inherited-members:
+   :no-special-members:

stable/0.4/_sources/apidocs/dispatch.rst.txt

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+.. _qiskit_dynamics-dispatch:
+
+.. automodule:: qiskit_dynamics.dispatch
+   :no-members:
+   :no-inherited-members:
+   :no-special-members:

stable/0.4/_sources/apidocs/index.rst.txt

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+.. _qiskit-dynamics:
+
+.. module:: qiskit_dynamics
+
+=============================
+Qiskit Dynamics API Reference
+=============================
+
+.. toctree::
+   :maxdepth: 2
+
+   solvers
+   backend
+   models
+   signals
+   pulse
+   perturbation
+   array
+   dispatch

stable/0.4/_sources/apidocs/models.rst.txt

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+.. _qiskit_dynamics-models:
+
+.. automodule:: qiskit_dynamics.models
+   :no-members:
+   :no-inherited-members:
+   :no-special-members:

stable/0.4/_sources/apidocs/perturbation.rst.txt

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+.. _qiskit_dynamics-perturbation:
+
+.. automodule:: qiskit_dynamics.perturbation
+   :no-members:
+   :no-inherited-members:
+   :no-special-members:

stable/0.4/_sources/apidocs/pulse.rst.txt

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+.. _qiskit_dynamics-pulse:
+
+.. automodule:: qiskit_dynamics.pulse
+   :no-members:
+   :no-inherited-members:
+   :no-special-members:

stable/0.4/_sources/apidocs/signals.rst.txt

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+.. _qiskit_dynamics-signals:
+
+.. automodule:: qiskit_dynamics.signals
+   :no-members:
+   :no-inherited-members:
+   :no-special-members:

stable/0.4/_sources/apidocs/solvers.rst.txt

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+.. _qiskit_dynamics-solvers:
+
+.. automodule:: qiskit_dynamics.solvers
+   :no-members:
+   :no-inherited-members:
+   :no-special-members:

stable/0.4/_sources/discussions/dyson_magnus.rst.txt

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+.. _perturbation review:
+
+Time-dependent perturbation theory and multi-variable series expansions review
+==============================================================================
+
+The :mod:`.perturbation` module contains functionality for
+numerically computing perturbation theory expansions used in the study of the
+dynamics of quantum systems. Following :footcite:`puzzuoli_sensitivity_2022`,
+this discussion reviews key concepts required
+to understand and utilize the module, including the Dyson series and Magnus expansion,
+their generalization to a multi-variable setting, and the notation used to represent
+multi-variable power series in terms of multisets.
+
+
+The Dyson series and Magnus expansion
+-------------------------------------
+
+The Dyson series :footcite:`dyson_radiation_1949` and Magnus expansion
+:footcite:`magnus_exponential_1954,blanes_magnus_2009`
+are time-dependent perturbation theory expansions for solutions of linear matrix differential
+equations (LMDEs). For an LMDE
+
+.. math::
+
+    \dot{U}(t) = G(t)U(t)
+
+with :math:`U(0) = I` the identity matrix, the Dyson series directly expands the solution:
+
+.. math::
+
+    U(t) &= I + \sum_{k=1}^\infty D_k(t)\textnormal{, with} \\
+    D_k(t) &= \int_0^t dt_1 \dots \int_0^{t_{k-1}} dt_k G(t_1) \dots G(t_k).
+
+The Magnus expansion alternatively seeks to construct a time-averaged generator, i.e. an operator
+:math:`\Omega(t)` for which
+
+.. math::
+
+    U(t) = \exp(\Omega(t)).
+
+The Magnus expansion provides
+a series expansion, which, under certain conditions :footcite:`blanes_magnus_2009`,
+converges to :math:`\Omega(t)`:
+
+.. math::
+
+    \Omega(t) = \sum_{k=1}^\infty \Omega_k(t),
+
+where explicit expressions for the :math:`\Omega_k(t)` are given in the literature
+:footcite:`blanes_magnus_2009`.
+
+
+Generalizing to the multi-variable case
+---------------------------------------
+
+In applications, these expansions are often used in a *multi-variable* setting, in which
+the generator :math:`G(t)` depends on several variables :math:`c_0, \dots, c_{r-1}`,
+and the Dyson series or Magnus expansion are used to understand how the evolution changes
+under perturbations to several of these parameters simultaneously. For working with
+these expansions algorithmically it is necessary to formalize
+the expression of these expansions in the multi-variable setting.
+
+Mathematically, we explicitly write the generator as a function of these variables
+:math:`G(t, c_0, \dots, c_{r-1})`, and expand :math:`G` in a
+multi-variable power series in the variables :math:`c_i`:
+
+.. math::
+
+    G(t, c_0, \dots, c_{r-1}) =
+    G_\emptyset(t) +
+    \sum_{k=1}^\infty \sum_{0 \leq i_1 \leq \dots \leq i_k \leq r-1}
+    c_{i_1} \dots c_{i_k} G_{i_1, \dots, i_k}(t).
+
+For physical applications we take the existence of such a power series for granted:
+up to constant factors, the coefficients :math:`G_{i_1, \dots, i_k}(t)` are the partial
+derivatives of :math:`G` with respect to the variables :math:`c_i`. Commonly, :math:`G`
+depends linearly on the variables, e.g. when representing couplings between quantum systems.
+
+Before defining the multi-variable Dyson series and Magnus expansions, we transform
+the generator into the *toggling frame* of :math:`G_\emptyset(t)`
+:footcite:`evans_timedependent_1967,haeberlen_1968`. Denoting
+
+.. math::
+
+    V(t) = \mathcal{T}\exp\left(\int_{t_0}^t ds G_\emptyset(s)\right),
+
+the generator :math:`G` in the toggling frame of :math:`G_\emptyset(t)`,
+the unperturbed generator, is given by:
+
+.. math::
+
+    \tilde{G}(t, c_0, \dots, c_{r-1}) &=
+    \sum_{k=1}^\infty \sum_{0 \leq i_1 \leq \dots \leq i_k \leq r-1}
+    c_{i_1} \dots c_{i_k} \tilde{G}_{i_1, \dots, i_k}(t)\textnormal{, with} \\
+    \tilde{G}_{i_1, \dots, i_k}(t) &= V^{-1}(t) G_{i_1, \dots, i_k}(t)V(t)
+
+Denoting :math:`U(t, c_0, \dots, c_{r-1})` as the solution of the LMDE with
+generator :math:`\tilde{G}`, note that
+
+.. math::
+
+    U(t, c_0, \dots, c_{r-1}) =
+    V(t)\mathcal{T}\exp\left(\int_{t_0}^t ds \tilde{G}(s, c_0, \dots, c_{r-1})\right),
+
+and hence solution for :math:`G` and :math:`\tilde{G}` are simply related by :math:`V(t)`.
+
+Using this, :footcite:`puzzuoli_sensitivity_2022` defines the multi-variable Dyson series
+for the generator :math:`\tilde{G}(t, c_0, \dots, c_{r-1})` as:
+
+.. math::
+
+    U(t, c_0, \dots, c_{r-1}) = I +
+    \sum_{k=1}^\infty \sum_{0 \leq i_1 \leq \dots \leq i_k \leq r-1}
+    c_{i_1} \dots c_{i_k} \mathcal{D}_{i_1, \dots, i_k}(t),
+
+where the :math:`\mathcal{D}_{i_1, \dots, i_k}(t)` are defined implicitly by the above
+equation, and are called the *multi-variable Dyson series terms*. Similarly the
+multi-variable Magnus expansion for :math:`\tilde{G}` is given as:
+
+.. math::
+
+    \Omega(t, c_0, \dots, c_{r-1}) =
+    \sum_{k=1}^\infty \sum_{0 \leq i_1 \leq \dots \leq i_k \leq r-1}
+    c_{i_1} \dots c_{i_k} \mathcal{O}_{i_1, \dots, i_k}(t),
+
+with the :math:`\mathcal{O}_{i_1, \dots, i_k}(t)` again defined implicitly, and called the
+*multi-variable Magnus expansion terms*.
+
+
+Computing multi-variable Dyson series and Magnus expansion terms
+----------------------------------------------------------------
+
+Given a power series decomposition of the generator as above,
+the function :func:`.solve_lmde_perturbation` computes,
+in the toggling frame of the unperturbed generator, either multi-variable
+Dyson series or Magnus expansion terms via the algorithms in
+:footcite:`puzzuoli_sensitivity_2022`. It can also be used to compute Dyson-like terms via
+the algorithm in :footcite:`haas_engineering_2019`. In the presentation here and elsewhere,
+the expansions are phrased as infinite series, but of course in practice truncated
+versions must be specified and computed.
+
+Utilizing this function, and working with the other objects in the module, requires
+understanding the notation and data structures used to represent power series.
+
+.. _multiset power series:
+
+Multiset power series notation
+------------------------------
+
+Following :footcite:`puzzuoli_sensitivity_2022`, the :mod:`.perturbation`
+module utilizes a *multiset* notation to more compactly represent and work with power series.
+
+Consider the power series expansion above for the generator :math:`G(t, c_0, \dots, c_{r-1})`.
+Structurally, each term in the power series is labelled by the number of times each
+variable :math:`c_0, \dots, c_{r-1}` appears in the product :math:`c_{i_1} \dots c_{i_k}`.
+Equivalently, each term may be indexed by the number of times each variable label
+:math:`0, \dots, r-1` appears. The data structure used to represent these labels in this
+module is that of a *multiset*, i.e. a "set with repeated entries". Denoting multisets
+with round brackets, e.g. :math:`I = (i_1, \dots, i_k)`, we define
+
+.. math::
+
+    c_I = c_{i_1} \times \dots \times c_{i_k}.
+
+and similarly denote :math:`G_I = G_{i_1, \dots, i_k}`. This notation is chosen due to
+the simple relationship between algebraic operations and multiset operations. E.g.,
+for two multisets :math:`I, J`, it holds that:
+
+.. math::
+
+    c_{I + J} = c_I \times c_J,
+
+where :math:`I + J` denotes the multiset whose object counts are the sum of both :math:`I` and
+:math:`J`.
+
+Some example usages of this notation are:
+
+    - :math:`c_{(0, 1)} = c_0 c_1`,
+    - :math:`c_{(1, 1)} = c_1^2`, and
+    - :math:`c_{(1, 2, 2, 3)} = c_1 c_2^2 c_3`.
+
+Finally, we denote the set of multisets of size $k$ with elements in :math:`\{0, \dots, r-1\}`
+as :math:`\mathcal{I}_k(r)`. Combining everything, the power series for :math:`G` may be
+rewritten as:
+
+.. math::
+
+    G(t, c_0, \dots, c_{r-1}) = G_\emptyset(t)
+    + \sum_{k=1}^\infty \sum_{I \in \mathcal{I}_k(r)} c_I G_I(t).
+
+Similarly, the multi-variable Dyson series is written as:
+
+.. math::
+
+    U(t, c_0, \dots, c_{r-1}) =
+            I + \sum_{k=1}^\infty \sum_{I \in \mathcal{I}_k(r)} c_I \mathcal{D}_I(t),
+
+and the multi-variable Magnus expansion as:
+
+.. math::
+
+    \Omega(t, c_0, \dots, c_{r-1}) =
+            \sum_{k=1}^\infty \sum_{I \in \mathcal{I}_k(r)} c_I \mathcal{O}_I(t).
+
+In the module, multisets are represented using the ``Multiset`` object in the
+`multiset package <https://pypi.org/project/multiset/>`_. Arguments to functions
+which must specify a multiset or a list of multisets accept either ``Multiset`` instances
+directly, or a valid argument to the constructor to ``Multiset``, with the restriction that
+the multiset entries must be non-negative integers.
+
+
+
+
+.. footbibliography::

stable/0.4/_sources/discussions/index.rst.txt

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+####################
+Dynamics Discussions
+####################
+
+Discussions about topics and concepts related to the package.
+
+.. toctree::
+   :maxdepth: 1
+
+   dyson_magnus.rst
+
+
+.. Hiding - Indices and tables
+   :ref:`genindex`
+   :ref:`modindex`
+   :ref:`search`

stable/0.4/_sources/index.rst.txt

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+#############################
+Qiskit Dynamics documentation
+#############################
+
+Qiskit Dynamics is an open-source project for building, transforming, and solving
+models of quantum systems in Qiskit.
+
+The goal of Qiskit Dynamics is to provide access to different numerical
+methods, and to automate common processes typically performed by hand,
+e.g. entering rotating frames, or doing the rotating wave approximation.
+
+Qiskit Dynamics can be configured to use either
+`NumPy <https://github.com/numpy/numpy>`_ or `JAX <https://github.com/google/jax>`_
+as the backend for array operations. `NumPy <https://github.com/numpy/numpy>`_ is the default,
+and `JAX <https://github.com/google/jax>`_ is an optional dependency, which enables
+just-in-time compilation, automatic differentiation, and GPU execution of Qiskit Dynamics code.
+
+.. warning::
+
+   This package is still in the early stages of development and it is very likely
+   that there will be breaking API changes in future releases.
+   If you encounter any bugs please open an issue on
+   `Github <https://github.com/Qiskit-Extensions/qiskit-dynamics/issues>`_
+
+
+.. toctree::
+  :maxdepth: 1
+
+  Tutorials <tutorials/index>
+  User Guide <userguide/index>
+  API References <apidocs/index>
+  Discussions <discussions/index>
+  Release Notes <release_notes>
+
+.. Hiding - Indices and tables
+   :ref:`genindex`
+   :ref:`modindex`
+   :ref:`search`

stable/0.4/_sources/release_notes.rst.txt

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+.. release-notes:: Release Notes

stable/0.4/_sources/stubs/qiskit_dynamics.array.Array.rst.txt

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+Array
+=====
+
+.. currentmodule:: qiskit_dynamics.array
+
+.. autoclass:: Array
+   :no-members:
+   :show-inheritance:
+   :no-inherited-members:
+   :no-special-members:
+
+   .. rubric:: Methods
+
+
+   .. automethod:: Array.available_backends
+   .. automethod:: Array.default_backend
+   .. automethod:: Array.set_default_backend
+
+
+
+   .. rubric:: Attributes
+
+   .. autoattribute:: backend
+   .. autoattribute:: data

stable/0.4/_sources/stubs/qiskit_dynamics.array.wrap.rst.txt

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+qiskit\_dynamics.array.wrap
+===========================
+
+.. currentmodule:: qiskit_dynamics.array
+
+.. autofunction:: wrap