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<div class="section" id="using-krotov-with-qutip">
<span id="id1"></span><h1>Using Krotov with QuTiP<a class="headerlink" href="#using-krotov-with-qutip" title="Permalink to this headline">¶</a></h1>
<p>The <a class="reference internal" href="API/krotov.html#module-krotov" title="krotov"><code class="xref py py-mod docutils literal notranslate"><span class="pre">krotov</span></code></a> package is designed around <a class="reference external" href="http://qutip.org">QuTiP</a>, a very powerful “Quantum
Toolbox” in Python. This means that all operators and states are expressed as
<a class="reference external" href="http://qutip.org/docs/latest/apidoc/classes.html#qutip.Qobj" title="(in QuTiP: Quantum Toolbox in Python v4.4)"><code class="xref py py-class docutils literal notranslate"><span class="pre">qutip.Qobj</span></code></a> quantum objects. The <a class="reference internal" href="API/krotov.optimize.html#krotov.optimize.optimize_pulses" title="krotov.optimize.optimize_pulses"><code class="xref py py-func docutils literal notranslate"><span class="pre">optimize_pulses()</span></code></a> interface
for Krotov’s optimization method is closely linked to the interface of QuTiP’s
central <a class="reference external" href="http://qutip.org/docs/latest/apidoc/functions.html#qutip.mesolve.mesolve" title="(in QuTiP: Quantum Toolbox in Python v4.4)"><code class="xref py py-func docutils literal notranslate"><span class="pre">mesolve()</span></code></a> routine for simulating the system
dynamics of a closed or open quantum system. In particular, when setting up an
optimization, the (time-dependent) system Hamiltonian should be represented by
a nested list. This is, a Hamiltonian of the form <span class="math notranslate nohighlight">\(\Op{H} = \Op{H}_0 +
\epsilon(t) \Op{H}_1\)</span> is represented as <code class="docutils literal notranslate"><span class="pre">H</span> <span class="pre">=</span> <span class="pre">[H0,</span> <span class="pre">[H1,</span> <span class="pre">eps]]</span></code> where <code class="docutils literal notranslate"><span class="pre">H0</span></code>
and <code class="docutils literal notranslate"><span class="pre">H1</span></code> are <a class="reference external" href="http://qutip.org/docs/latest/apidoc/classes.html#qutip.Qobj" title="(in QuTiP: Quantum Toolbox in Python v4.4)"><code class="xref py py-class docutils literal notranslate"><span class="pre">Qobj</span></code></a> operators, and <code class="docutils literal notranslate"><span class="pre">eps</span></code> is a function with
signature <code class="docutils literal notranslate"><span class="pre">eps(t,</span> <span class="pre">args)</span></code>, or an array of control values with the length of the
time grid (<cite>tlist</cite> parameter in <a class="reference external" href="http://qutip.org/docs/latest/apidoc/functions.html#qutip.mesolve.mesolve" title="(in QuTiP: Quantum Toolbox in Python v4.4)"><code class="xref py py-func docutils literal notranslate"><span class="pre">mesolve()</span></code></a>). The operator
can depend on multiple controls, resulting in expressions of the form <code class="docutils literal notranslate"><span class="pre">H</span> <span class="pre">=</span>
<span class="pre">[H0,</span> <span class="pre">[H1,</span> <span class="pre">eps1],</span> <span class="pre">[H2,</span> <span class="pre">eps2],</span> <span class="pre">...]</span></code>.</p>
<p>The central routine provided by the <a class="reference internal" href="API/krotov.html#module-krotov" title="krotov"><code class="xref py py-mod docutils literal notranslate"><span class="pre">krotov</span></code></a> package is
<a class="reference internal" href="API/krotov.optimize.html#krotov.optimize.optimize_pulses" title="krotov.optimize.optimize_pulses"><code class="xref py py-func docutils literal notranslate"><span class="pre">optimize_pulses()</span></code></a>. It takes as input a list of objectives, each of which
is an instance of <a class="reference internal" href="API/krotov.objectives.html#krotov.objectives.Objective" title="krotov.objectives.Objective"><code class="xref py py-class docutils literal notranslate"><span class="pre">Objective</span></code></a>. Each objective has an
<a class="reference internal" href="API/krotov.objectives.html#krotov.objectives.Objective.initial_state" title="krotov.objectives.Objective.initial_state"><code class="xref py py-attr docutils literal notranslate"><span class="pre">initial_state</span></code></a>, which is a <a class="reference external" href="http://qutip.org/docs/latest/apidoc/classes.html#qutip.Qobj" title="(in QuTiP: Quantum Toolbox in Python v4.4)"><code class="xref py py-class docutils literal notranslate"><span class="pre">qutip.Qobj</span></code></a> representing
a Hilbert space state or density matrix, a <a class="reference internal" href="API/krotov.objectives.html#krotov.objectives.Objective.target" title="krotov.objectives.Objective.target"><code class="xref py py-attr docutils literal notranslate"><span class="pre">target</span></code></a> (usually
the target state that the <a class="reference internal" href="API/krotov.objectives.html#krotov.objectives.Objective.initial_state" title="krotov.objectives.Objective.initial_state"><code class="xref py py-attr docutils literal notranslate"><span class="pre">initial_state</span></code></a> should evolve into
when the objective is fulfilled), and a Hamiltonian <a class="reference internal" href="API/krotov.objectives.html#krotov.objectives.Objective.H" title="krotov.objectives.Objective.H"><code class="xref py py-attr docutils literal notranslate"><span class="pre">H</span></code></a> in
the nested-list format described above. For dissipative dynamics,
<a class="reference internal" href="API/krotov.objectives.html#krotov.objectives.Objective.H" title="krotov.objectives.Objective.H"><code class="xref py py-attr docutils literal notranslate"><span class="pre">H</span></code></a> should be a Liouvillian, which can be obtained from the
Hamiltonian and a set of Lindblad operators via
<a class="reference internal" href="API/krotov.objectives.html#krotov.objectives.liouvillian" title="krotov.objectives.liouvillian"><code class="xref py py-func docutils literal notranslate"><span class="pre">krotov.objectives.liouvillian()</span></code></a>. The Liouvillian again is in nested list
format to express time-dependencies. Alternatively, each objective could also
directly include a list <a class="reference internal" href="API/krotov.objectives.html#krotov.objectives.Objective.c_ops" title="krotov.objectives.Objective.c_ops"><code class="xref py py-attr docutils literal notranslate"><span class="pre">c_ops</span></code></a> of collapse (Lindblad)
operators , where each collapse operator is a <a class="reference external" href="http://qutip.org/docs/latest/apidoc/classes.html#qutip.Qobj" title="(in QuTiP: Quantum Toolbox in Python v4.4)"><code class="xref py py-class docutils literal notranslate"><span class="pre">Qobj</span></code></a> operator.
However, this only makes sense if the time propagation routine takes the
collapse operators into account explicitly, such as in the Monte-Carlo
<a class="reference external" href="http://qutip.org/docs/latest/apidoc/functions.html#qutip.mcsolve.mcsolve" title="(in QuTiP: Quantum Toolbox in Python v4.4)"><code class="xref py py-func docutils literal notranslate"><span class="pre">mcsolve()</span></code></a>. Otherwise, the use of
<a class="reference internal" href="API/krotov.objectives.html#krotov.objectives.Objective.c_ops" title="krotov.objectives.Objective.c_ops"><code class="xref py py-attr docutils literal notranslate"><span class="pre">c_ops</span></code></a> is strongly discouraged.</p>
<p>In order to simulate the dynamics of the guess control, you can use
<a class="reference internal" href="API/krotov.objectives.html#krotov.objectives.Objective.mesolve" title="krotov.objectives.Objective.mesolve"><code class="xref py py-meth docutils literal notranslate"><span class="pre">Objective.mesolve()</span></code></a>, which delegates to <a class="reference external" href="http://qutip.org/docs/latest/apidoc/functions.html#qutip.mesolve.mesolve" title="(in QuTiP: Quantum Toolbox in Python v4.4)"><code class="xref py py-func docutils literal notranslate"><span class="pre">qutip.mesolve.mesolve()</span></code></a>.
There is also a related method <a class="reference internal" href="API/krotov.objectives.html#krotov.objectives.Objective.propagate" title="krotov.objectives.Objective.propagate"><code class="xref py py-meth docutils literal notranslate"><span class="pre">Objective.propagate()</span></code></a> that uses a
different sampling of the control values, see <a class="reference internal" href="API/krotov.propagators.html#module-krotov.propagators" title="krotov.propagators"><code class="xref py py-mod docutils literal notranslate"><span class="pre">krotov.propagators</span></code></a>.</p>
<p>The optimization routine will automatically extract all controls that it can
find in the objectives, and iteratively calculate updates to all controls in
order to meet all <cite>objectives</cite> simultaneously. The result of the optimization
will be in the returned <a class="reference internal" href="API/krotov.result.html#krotov.result.Result" title="krotov.result.Result"><code class="xref py py-class docutils literal notranslate"><span class="pre">Result</span></code></a> object, with a list of the optimized
controls in <a class="reference internal" href="API/krotov.result.html#krotov.result.Result.optimized_controls" title="krotov.result.Result.optimized_controls"><code class="xref py py-attr docutils literal notranslate"><span class="pre">optimized_controls</span></code></a>.
The <a class="reference internal" href="API/krotov.result.html#krotov.result.Result.optimized_objectives" title="krotov.result.Result.optimized_objectives"><code class="xref py py-attr docutils literal notranslate"><span class="pre">optimized_objectives</span></code></a> property contains a copy of the
objectives with the <a class="reference internal" href="API/krotov.result.html#krotov.result.Result.optimized_controls" title="krotov.result.Result.optimized_controls"><code class="xref py py-attr docutils literal notranslate"><span class="pre">optimized_controls</span></code></a> plugged into the
Hamiltonian or Liouvillian and/or collapse operators. The dynamics under the
optimized controls can then again be simulated through
<a class="reference internal" href="API/krotov.objectives.html#krotov.objectives.Objective.mesolve" title="krotov.objectives.Objective.mesolve"><code class="xref py py-meth docutils literal notranslate"><span class="pre">Objective.mesolve()</span></code></a>.</p>
<p>While the guess controls that are in the <cite>objectives</cite> on input may be
functions, or an array of control values on the time grid, the output
<a class="reference internal" href="API/krotov.result.html#krotov.result.Result.optimized_controls" title="krotov.result.Result.optimized_controls"><code class="xref py py-attr docutils literal notranslate"><span class="pre">optimized_controls</span></code></a> will always be an array of control values.</p>
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