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<p class="caption"><span class="caption-text">Contents:</span></p>
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<li class="toctree-l1"><a class="reference internal" href="01_overview.html">Krotov Python Package</a></li>
<li class="toctree-l1"><a class="reference internal" href="02_contributing.html">Contributing</a></li>
<li class="toctree-l1"><a class="reference internal" href="03_authors.html">Credits</a></li>
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<li class="toctree-l1"><a class="reference internal" href="05_history.html">History</a></li>
<li class="toctree-l1 current"><a class="current reference internal" href="#">Krotov’s Method</a><ul>
<li class="toctree-l2"><a class="reference internal" href="#functionals">Functionals</a></li>
<li class="toctree-l2"><a class="reference internal" href="#conditions-to-ensure-monotonic-convergence">Conditions to ensure monotonic convergence</a></li>
<li class="toctree-l2"><a class="reference internal" href="#first-order-update-equation">First order update equation</a></li>
<li class="toctree-l2"><a class="reference internal" href="#second-order-update-equation">Second order update equation</a></li>
<li class="toctree-l2"><a class="reference internal" href="#time-discretization">Time discretization</a></li>
<li class="toctree-l2"><a class="reference internal" href="#choice-of-a">Choice of λₐ</a></li>
<li class="toctree-l2"><a class="reference internal" href="#rotating-wave-approximation">Rotating wave approximation</a></li>
<li class="toctree-l2"><a class="reference internal" href="#optimization-in-liouville-space">Optimization in Liouville space</a></li>
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<li class="toctree-l1"><a class="reference internal" href="07_qutip_usage.html">Using Krotov with QuTiP</a></li>
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<li class="toctree-l1"><a class="reference internal" href="10_other_methods.html">Other Optimization Methods</a></li>
<li class="toctree-l1"><a class="reference internal" href="99_bibliography.html">References</a></li>
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<div class="section" id="krotovs-method">
<h1>Krotov’s Method<a class="headerlink" href="#krotovs-method" title="Permalink to this headline">¶</a></h1>
<p><em>The following overview has been adapted from Ref</em> <a class="reference internal" href="99_bibliography.html#goerzphd2015" id="id1">[GoerzPhd2015]</a>.</p>
<div class="section" id="functionals">
<h2>Functionals<a class="headerlink" href="#functionals" title="Permalink to this headline">¶</a></h2>
<p>Krotov’s method <a class="reference internal" href="99_bibliography.html#konnovarc99" id="id2">[KonnovARC99]</a>, applied to quantum control, considers one
or more quantum systems with a set of Hamiltonians
<span class="math notranslate nohighlight">\(\{\Op{H}_k(\{\epsilon_l(t)\})\}\)</span> where each Hamiltonian depends on a set
of time-continuous controls <span class="math notranslate nohighlight">\(\{\epsilon_l(t)\}\)</span>. It seeks to find control
fields that optimally steer a set of initial states <span class="math notranslate nohighlight">\(\{\ket{\phi_k}\}\)</span> in
some desired way. To this end, in each iteration <span class="math notranslate nohighlight">\((i)\)</span>, it minimizes a
functional of the form</p>
<div class="math notranslate nohighlight">
\[\begin{split}\begin{split}
J[\{\ket{\phi_k^{(i)}(t)}\}, \{\epsilon_l^{(i)}(t)\}]
&= J_T(\{\ket{\phi_k^{(i)}(T)}\}) \\
&\qquad
+ \sum_l \int_0^T g_a(\epsilon_l^{(i)}(t)) \dd t
+ \int_0^T g_b(\{\phi^{(i)}_k(t)\}) \dd t\,.
\end{split}\end{split}\]</div>
<p>where <span class="math notranslate nohighlight">\(\ket{\phi_k^{(i)}(T)}\)</span> are the time-evolved initial states
<span class="math notranslate nohighlight">\(\ket{\phi_k}\)</span> under the (guess) controls <span class="math notranslate nohighlight">\(\{\epsilon^{(i)}_l(t)\}\)</span>
of the <span class="math notranslate nohighlight">\(i\)</span>’th iteration.</p>
<p>The functional consists of three parts:</p>
<ul>
<li><p class="first">A final time functional <span class="math notranslate nohighlight">\(J_T\)</span>. This is the “main” part of the
functional, and we can usually think of <span class="math notranslate nohighlight">\(J\)</span> as being an auxiliary
functional in the optimization of <span class="math notranslate nohighlight">\(J_T\)</span>.</p>
</li>
<li><p class="first">A running cost on the control fields, <span class="math notranslate nohighlight">\(g_a\)</span>. As we will see below,
specific forms of running costs are required to obtain a closed-form update
equation. The typical form, and the only one we consider here (and that is
realized in 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</p>
<div class="math notranslate nohighlight">
\[g_a(\epsilon_l(t))
= \frac{\lambda_{a, l}}{S_l(t)} \Delta\epsilon_l^2(t)\,.\]</div>
<p>We introduce two parameters, the (inverse) Krotov “step size”
<span class="math notranslate nohighlight">\(\lambda_{a,l}\)</span> and the shape function <span class="math notranslate nohighlight">\(S_l(t)\)</span> which can be used
to influence desired properties of the optimized controls.
<span class="math notranslate nohighlight">\(\Delta\epsilon_l(t)\)</span> is the update of the control in a single iteration of
the optimization algorithm. It is best to think of this running cost as a
technical requirement, and not to assign physical meaning to it. Note that as
the optimization converges, <span class="math notranslate nohighlight">\(\Delta \epsilon_l(t) \rightarrow 0\)</span>, so that the
minimization of <span class="math notranslate nohighlight">\(J\)</span> is equivalent to the minimization of <span class="math notranslate nohighlight">\(J_T\)</span>
(for <span class="math notranslate nohighlight">\(g_b \equiv 0\)</span>).</p>
</li>
<li><p class="first">An optional state-dependent running cost, <span class="math notranslate nohighlight">\(g_b\)</span>, e.g., to penalize
population in a subspace. This is rarely used, as there are other methods to
achieve the same effect, like placing artificially high dissipation on a
“forbidden” subspace.</p>
</li>
</ul>
<p>The most commonly used functionals (cf. <a class="reference internal" href="API/krotov.functionals.html#module-krotov.functionals" title="krotov.functionals"><code class="xref py py-mod docutils literal notranslate"><span class="pre">krotov.functionals</span></code></a>) optimize for
a set of initial states <span class="math notranslate nohighlight">\(\{\ket{\phi_k}\}\)</span> to evolve to a set of target
states <span class="math notranslate nohighlight">\(\{\ket{\phi_k^\tgt}\}\)</span>. The functionals can then be expressed in
terms of the complex overlaps of the final-time states with the target states
under the given control. Thus,</p>
<div class="math notranslate nohighlight" id="equation-tauk">
<span class="eqno">(1)<a class="headerlink" href="#equation-tauk" title="Permalink to this equation">¶</a></span>\[ \tau_k = \Braket{\phi_k(T)}{\phi_k^\tgt}\]</div>
<p>in Hilbert space, or</p>
<div class="math notranslate nohighlight">
\[\tau_k
= \langle\!\langle \Op{\rho}_k(T) \vert \Op{\rho}^{\tgt} \rangle\!\rangle
\equiv \tr\left[\Op{\rho}^{\dagger}_k(T) \Op{\rho}_k^{\tgt} \right]\]</div>
<p>in Liouville space.</p>
<p>The following functionals <span class="math notranslate nohighlight">\(J_T\)</span> can be formed from these complex
overlaps, taking into account that any optimization functional <span class="math notranslate nohighlight">\(J_T\)</span> must
be real. They differ by the way they treat the phases <span class="math notranslate nohighlight">\(\varphi_k\)</span> in the
physical optimization goal <span class="math notranslate nohighlight">\(\ket{\phi_k(T)} \overset{!}{=}
e^{i\varphi_k}\ket{\phi_k^{\tgt}}\)</span> <a class="reference internal" href="99_bibliography.html#palaopra2003" id="id3">[PalaoPRA2003]</a>:</p>
<ul>
<li><p class="first">Optimize for simultaneous state-to-state transitions, with completely
arbitrary phases <span class="math notranslate nohighlight">\(\varphi_k\)</span>,</p>
<div class="math notranslate nohighlight" id="equation-jtss">
<span class="eqno">(2)<a class="headerlink" href="#equation-jtss" title="Permalink to this equation">¶</a></span>\[J_{T,\text{ss}} = 1- \frac{1}{N} \sum_{k=1}^{N} \Abs{\tau_k}^2\,,\]</div>
<p>cf. <a class="reference internal" href="API/krotov.functionals.html#krotov.functionals.J_T_ss" title="krotov.functionals.J_T_ss"><code class="xref py py-func docutils literal notranslate"><span class="pre">J_T_ss()</span></code></a>.</p>
</li>
<li><p class="first">Optimize for simultaneous state-to-state transitions, with an arbitrary
<em>global</em> phase, i.e., <span class="math notranslate nohighlight">\(\varphi_k = \varphi_{\text{global}}\)</span> for all
<span class="math notranslate nohighlight">\(k\)</span> with arbitrary <span class="math notranslate nohighlight">\(\varphi_{\text{global}}\)</span>,</p>
<div class="math notranslate nohighlight" id="equation-jtsm">
<span class="eqno">(3)<a class="headerlink" href="#equation-jtsm" title="Permalink to this equation">¶</a></span>\[J_{T,\text{sm}} = 1- \frac{1}{N^2} \Abs{\sum_{k=1}^{N} \tau_k}^2
= 1- \frac{1}{N^2} \sum_{k=1}^{N} \sum_{k'=1}^{N} \tau_{k'}^* \tau_{k}\,,\]</div>
<p>cf. <a class="reference internal" href="API/krotov.functionals.html#krotov.functionals.J_T_sm" title="krotov.functionals.J_T_sm"><code class="xref py py-func docutils literal notranslate"><span class="pre">J_T_sm()</span></code></a>.</p>
</li>
<li><p class="first">Optimize for simultaneous state-to-state transitions, with a global phase of zero, i.e.,
<span class="math notranslate nohighlight">\(\varphi_k = 0\)</span> for all <span class="math notranslate nohighlight">\(k\)</span>,</p>
<div class="math notranslate nohighlight" id="equation-jtre">
<span class="eqno">(4)<a class="headerlink" href="#equation-jtre" title="Permalink to this equation">¶</a></span>\[J_{T,\text{re}} = 1-\frac{1}{N} \Re \left[\, \sum_{k=1}^{N} \tau_k \,\right]\,,\]</div>
<p>cf. <a class="reference internal" href="API/krotov.functionals.html#krotov.functionals.J_T_re" title="krotov.functionals.J_T_re"><code class="xref py py-func docutils literal notranslate"><span class="pre">J_T_re()</span></code></a>.</p>
</li>
</ul>
</div>
<div class="section" id="conditions-to-ensure-monotonic-convergence">
<h2>Conditions to ensure monotonic convergence<a class="headerlink" href="#conditions-to-ensure-monotonic-convergence" title="Permalink to this headline">¶</a></h2>
<p>Krotov’s method is based on a rigorously examination of the conditions for
constructing updated fields <span class="math notranslate nohighlight">\(\epsilon_l^{(i+1)}(t)\)</span> such that
<span class="math notranslate nohighlight">\(J(\{\ket{\phi_k^{(i+1)}(t)}\}, \{\epsilon_l^{(i+1)}\}) \leq
J(\{\ket{\phi_k^{(i)}(t)}\}, \{\epsilon_l^{(i)}\})\)</span> is mathematically
guaranteed. The main difficulty is disentangling the
interdependence of the states and the field. Krotov tackles
this by introducing an auxiliary functional <span class="math notranslate nohighlight">\(L[\{\ket{\phi_k^{(i)}(t)}\},
\{\epsilon_l^{(i)}(t)\}, \Phi]\)</span> that is equivalent to
<span class="math notranslate nohighlight">\(J[\{\ket{\phi_k^{(i)}(t)}\}, \{\epsilon_l^{(i)}(t)\}]\)</span>, but includes an
arbitrary scalar potential <span class="math notranslate nohighlight">\(\Phi\)</span>. The freedom in this scalar potential is then
used to formulate a condition to ensure monotonic convergence,</p>
<div class="math notranslate nohighlight" id="equation-krotov-proto-update">
<span class="eqno">(5)<a class="headerlink" href="#equation-krotov-proto-update" title="Permalink to this equation">¶</a></span>\[\begin{split} \left.\frac{\partial g_a}{\partial \epsilon}\right\vert_{\epsilon^{(i+1)}(t)}
= 2 \Im
\sum_{k=1}^{N}
\Bigg\langle
\chi_k^{(i)}(t)
\Bigg\vert
\Bigg(
\left.\frac{\partial \Op{H}}{\partial \epsilon}\right\vert_{{\scriptsize \begin{matrix}\phi^{(i+1)}(t)\\\epsilon^{(i+1)}(t)\end{matrix}}}
\Bigg)
\Bigg\vert
\phi_k^{(i+1)}(t)
\Bigg\rangle\,,\end{split}\]</div>
<p>assuming the equation of motion for the forward propagation of
<span class="math notranslate nohighlight">\(\ket{\phi_k}\)</span> under the optimized controls to be written as</p>
<div class="math notranslate nohighlight" id="equation-fw-eqm">
<span class="eqno">(6)<a class="headerlink" href="#equation-fw-eqm" title="Permalink to this equation">¶</a></span>\[\frac{\partial}{\partial t} \Ket{\phi_k^{(i+1)}(t)}
= -\frac{\mathrm{i}}{\hbar} \Op{H}^{(i+1)} \Ket{\phi_k^{(i+1)}(t)}\,.\]</div>
<p>The co-states <span class="math notranslate nohighlight">\(\Ket{\chi_k^{(i)}(t)}\)</span> are propagated backwards under the
guess controls of iteration <span class="math notranslate nohighlight">\((i)\)</span>, i.e., the optimized controls from the previous
iteration, as</p>
<div class="math notranslate nohighlight" id="equation-bw-eqm">
<span class="eqno">(7)<a class="headerlink" href="#equation-bw-eqm" title="Permalink to this equation">¶</a></span>\[\frac{\partial}{\partial t} \Ket{\chi_k^{(i)}(t)}
= -\frac{\mathrm{i}}{\hbar} \Op{H}^{\dagger\,(i)} \Ket{\chi_k^{(i)}(t)}
+ \left.\frac{\partial g_b}{\partial \Bra{\phi_k}}\right\vert_{\phi^{(i)}(t)}\,,\]</div>
<p>with the boundary condition</p>
<div class="math notranslate nohighlight" id="equation-chi-boundary">
<span class="eqno">(8)<a class="headerlink" href="#equation-chi-boundary" title="Permalink to this equation">¶</a></span>\[\Ket{\chi_k^{(i)}(T)}
= - \left.\frac{\partial J_T}{\partial \Bra{\phi_k}}\right\vert_{\phi^{(i)}(T)}\,.\]</div>
<p>Note that the backward propagation uses the adjoint Hamiltonian, which becomes
relevant for non-Hermitian Hamiltonians or dissipative dynamics in Liouville
space. In Hilbert space, and without any state-dependent constraints
(<span class="math notranslate nohighlight">\(g_b \equiv 0\)</span>), this is still the standard Schrödinger equation running
backwards in time (<span class="math notranslate nohighlight">\(\dd t \rightarrow -\dd t\)</span>). A state-dependent
constraint introduces an inhomogeneity in Eq. <a class="reference internal" href="#equation-bw-eqm">(7)</a>. The equations in
Liouville space follow an analogous structure, with <span class="math notranslate nohighlight">\(\Op{H} \rightarrow i
\Liouville\)</span>, see <a class="reference internal" href="API/krotov.mu.html#module-krotov.mu" title="krotov.mu"><code class="xref py py-mod docutils literal notranslate"><span class="pre">krotov.mu</span></code></a> for details. For details on the derivation of
the above equations, see Ref. <a class="reference internal" href="99_bibliography.html#reichjcp12" id="id4">[ReichJCP12]</a>. Here, and in the following,
we have dropped the index <span class="math notranslate nohighlight">\(l\)</span> of the controls and the associated
<span class="math notranslate nohighlight">\(\lambda_{a,l}\)</span> and <span class="math notranslate nohighlight">\(S_l(t)\)</span>; all equations are valid for each
individual control.</p>
</div>
<div class="section" id="first-order-update-equation">
<h2>First order update equation<a class="headerlink" href="#first-order-update-equation" title="Permalink to this headline">¶</a></h2>
<p>In order to obtain an explicit equation for <span class="math notranslate nohighlight">\(\epsilon^{(i+1)}(t)\)</span> – the
optimized pulse in iteration <span class="math notranslate nohighlight">\((i)\)</span> – a running cost
<span class="math notranslate nohighlight">\(g_a(\epsilon^{(i+1)}(t))\)</span> must be specified. It usually takes the form</p>
<div class="math notranslate nohighlight">
\[g_a(\epsilon^{(i+1)}(t))
= \frac{\lambda_a}{S(t)} (\epsilon^{(i+1)}(t) - \epsilon^{\text{ref}}(t))^2\,,\]</div>
<p>with a scaling parameter <span class="math notranslate nohighlight">\(\lambda_a\)</span> and a shape function <span class="math notranslate nohighlight">\(S(t) \in
[0,1]\)</span>. When <span class="math notranslate nohighlight">\(\epsilon^{\text{ref}}(t)\)</span> is set to the guess pulse
<span class="math notranslate nohighlight">\(\epsilon^{(i)}(t)\)</span> of the iteration <span class="math notranslate nohighlight">\((i)\)</span> – the optimized pulse
from the previous iteration – this yields</p>
<div class="math notranslate nohighlight">
\[g_a(\epsilon^{(i+1)}(t))
= \frac{\lambda_a}{S(t)} \Delta\epsilon^2(t)\,,
\quad
\Delta\epsilon(t) \equiv \epsilon^{(i+1)}(t) - \epsilon^{(i)}(t)\,.\]</div>
<p>Thus, we obtain the first-order Krotov update equation as <a class="reference internal" href="99_bibliography.html#palaopra2003" id="id5">[PalaoPRA2003]</a><a class="reference internal" href="99_bibliography.html#sklarzpra2002" id="id6">[SklarzPRA2002]</a>,</p>
<div class="math notranslate nohighlight" id="equation-krotov-first-order-update">
<span class="eqno">(9)<a class="headerlink" href="#equation-krotov-first-order-update" title="Permalink to this equation">¶</a></span>\[\begin{split}\Delta\epsilon(t)
=
\frac{S(t)}{\lambda_a} \Im \left[
\sum_{k=1}^{N}
\Bigg\langle
\chi_k^{(i)}(t)
\Bigg\vert
\Bigg(
\left.\frac{\partial \Op{H}}{\partial \epsilon}\right\vert_{{\scriptsize \begin{matrix}\phi^{(i+1)}(t)\\\epsilon^{(i+1)}(t)\end{matrix}}}
\Bigg)
\Bigg\vert
\phi_k^{(i+1)}(t)
\Bigg\rangle
\right]\,.\end{split}\]</div>
<p>If <span class="math notranslate nohighlight">\(S(t) \in [0,1]\)</span> is chosen as a function that smoothly goes to zero at
<span class="math notranslate nohighlight">\(t=0\)</span> and <span class="math notranslate nohighlight">\(t=T\)</span>, then the update will be suppressed near the edges
of the optimization time interval. Thus, a smooth switch-on and switch-off can
be maintained. The scaling factor <span class="math notranslate nohighlight">\(\lambda_a\)</span> controls the overall
magnitude of the pulse update thereby taking the role of an (inverse) “step
size”. Values that are too large will change <span class="math notranslate nohighlight">\(\epsilon^{(i)}(t)\)</span> by
only a small amount in every iteration, causing slow convergence. Values that
are too small will cause sharp spikes in the optimized control and numerical
instabilities (including a loss of monotonic convergence).</p>
<p>We have assumed that the Hamiltonian is linear in the controls. If this is not
the case, <span class="math notranslate nohighlight">\(\epsilon^{(i+1)}(t)\)</span> will still show up on the right hand side
of Eq. <a class="reference internal" href="#equation-krotov-first-order-update">(9)</a>. In order to remove the implicit nature
of Eq. <a class="reference internal" href="#equation-krotov-first-order-update">(9)</a>, we approximate
<span class="math notranslate nohighlight">\(\epsilon^{(i+1)}(t) \approx \epsilon^{(i)}(t)\)</span> on the right hand side,
in other words, we assume <span class="math notranslate nohighlight">\(\Abs{\Delta \epsilon(t)} \ll
\Abs{\epsilon(t)}\)</span>. This can be ensured by choosing a sufficiently large value
for <span class="math notranslate nohighlight">\(\lambda_a\)</span>.</p>
<p>The functional <span class="math notranslate nohighlight">\(J_T\)</span> enters the update equation only implicitly in the
boundary condition for the backward propagated co-state,
Eq. <a class="reference internal" href="#equation-chi-boundary">(8)</a>. For example, the standard functionals defined in
Eq. <a class="reference internal" href="#equation-jtsm">(3)</a> and Eq. <a class="reference internal" href="#equation-jtre">(4)</a> yield</p>
<div class="math notranslate nohighlight">
\[\begin{split}\begin{aligned}
- \left.\frac{\partial J_{T,\text{sm}}}{\partial \Bra{\phi_k}}\right\vert_{\phi_k^{(i)}(T)}
&= \left( \frac{1}{N^2} \sum_{l=1}^N \tau_l \right) \Ket{\phi_k^\tgt}\,,
\\
- \left.\frac{\partial J_{T,\text{re}}}{\partial \Bra{\phi_k}}\right\vert_{\phi_k^{(i)}(T)}
&= \frac{1}{2N} \Ket{\phi_k^\tgt}\,,
\end{aligned}\end{split}\]</div>
<p>cf. <a class="reference internal" href="API/krotov.functionals.html#krotov.functionals.chis_sm" title="krotov.functionals.chis_sm"><code class="xref py py-func docutils literal notranslate"><span class="pre">chis_sm()</span></code></a>, <a class="reference internal" href="API/krotov.functionals.html#krotov.functionals.chis_re" title="krotov.functionals.chis_re"><code class="xref py py-func docutils literal notranslate"><span class="pre">chis_re()</span></code></a>.</p>
</div>
<div class="section" id="second-order-update-equation">
<h2>Second order update equation<a class="headerlink" href="#second-order-update-equation" title="Permalink to this headline">¶</a></h2>
<p>The condition <a class="reference internal" href="#equation-krotov-proto-update">(5)</a> and the update
Eq. <a class="reference internal" href="#equation-krotov-first-order-update">(9)</a> are based on a first-order expansion of the
auxiliary potential <span class="math notranslate nohighlight">\(\Phi\)</span> with respect to the states, see
Ref. <a class="reference internal" href="99_bibliography.html#reichjcp12" id="id7">[ReichJCP12]</a> for details. This is sufficient in most cases, in
particular if the equation of motion is linear (<span class="math notranslate nohighlight">\(\Op{H}\)</span> does not depend
on the states <span class="math notranslate nohighlight">\(\ket{\phi_k(t)}\)</span>), the functional <span class="math notranslate nohighlight">\(J_T\)</span> is convex,
and no state-dependent constraints are used (<span class="math notranslate nohighlight">\(g_b\equiv 0\)</span>). Even for
some types of state-dependent constraints, the first-order expansion is
sufficient, specifically for keeping the population in an allowed
subspace <a class="reference internal" href="99_bibliography.html#palaopra2008" id="id8">[PalaoPRA2008]</a>.</p>
<p>When these conditions are not fulfilled, it is still possible to derive conditions
to ensure monotonic convergence via an expansion of <span class="math notranslate nohighlight">\(\Phi\)</span> to second order
in the states, resulting in a second term in Eq. <a class="reference internal" href="#equation-krotov-proto-update">(5)</a>,</p>
<div class="math notranslate nohighlight" id="equation-krotov-proto-update2">
<span class="eqno">(10)<a class="headerlink" href="#equation-krotov-proto-update2" title="Permalink to this equation">¶</a></span>\[\begin{split}\begin{split}
\left.\frac{\partial g_a}{\partial \epsilon}\right\vert_{\epsilon^{(i+1)}(t)}
& =
2 \Im \left[
\sum_{k=1}^{N}
\Bigg\langle
\chi_k^{(i)}(t)
\Bigg\vert
\Bigg(
\left.\frac{\partial \Op{H}}{\partial \epsilon}\right\vert_{{\scriptsize \begin{matrix}\phi^{(i+1)}(t)\\\epsilon^{(i+1)}(t)\end{matrix}}}
\Bigg)
\Bigg\vert
\phi_k^{(i+1)}(t)
\Bigg\rangle
\right. \\ & \qquad \left.
+ \frac{1}{2} \sigma(t)
\Bigg\langle
\Delta\phi_k(t)
\Bigg\vert
\Bigg(
\left.\frac{\partial \Op{H}}{\partial \epsilon}\right\vert_{{\scriptsize \begin{matrix}\phi^{(i+1)}(t)\\\epsilon^{(i+1)}(t)\end{matrix}}}
\Bigg)
\Bigg\vert
\phi_k^{(i+1)}(t)
\Bigg\rangle
\right]\,,
\end{split}\end{split}\]</div>
<p>with</p>
<div class="math notranslate nohighlight">
\[\ket{\Delta \phi_k(t)} \equiv \ket{\phi_k^{(i+1)}(t)} - \ket{\phi_k^{(i)}(t)}\,.\]</div>
<p>In Eq. <a class="reference internal" href="#equation-krotov-proto-update2">(10)</a>, <span class="math notranslate nohighlight">\(\sigma(t)\)</span> is a scalar function that
must be properly chosen to ensure monotonic convergence.</p>
<p>As shown in Ref. <a class="reference internal" href="99_bibliography.html#reichjcp12" id="id9">[ReichJCP12]</a>, it is possible to numerically approximate
<span class="math notranslate nohighlight">\(\sigma(t)\)</span>. In Refs <a class="reference internal" href="99_bibliography.html#wattspra2015" id="id10">[WattsPRA2015]</a><a class="reference internal" href="99_bibliography.html#goerzpra2015" id="id11">[GoerzPRA2015]</a>, non-convex
final-time functionals that depend higher than quadratically on the states are
considered, for a standard equation of motion given by a linear Schrödinger
equation. In this case,</p>
<div class="math notranslate nohighlight">
\[\sigma(t) \equiv -\max\left(\varepsilon_A,2A+\varepsilon_A\right)\,,
\label{eq:sigma_A}\]</div>
<p>where <span class="math notranslate nohighlight">\(\varepsilon_A\)</span> is a small non-negative number that can be used to
enforce strict inequality in the second order optimality condition. The optimal
value for <span class="math notranslate nohighlight">\(A\)</span> in each iteration can be approximated numerically
as <a class="reference internal" href="99_bibliography.html#reichjcp12" id="id12">[ReichJCP12]</a></p>
<div class="math notranslate nohighlight">
\[A =
\frac{\sum_{k=1}^{N} 2 \Re\left[
\langle \chi_k(T) \vert \Delta\phi_k(T) \rangle
\right]
+ \Delta J_T}
{\sum_{k=1}^{N} \Abs{\Delta\phi_k(T)}^2}
\,,\]</div>
<p>cf. <a class="reference internal" href="API/krotov.second_order.html#krotov.second_order.numerical_estimate_A" title="krotov.second_order.numerical_estimate_A"><code class="xref py py-func docutils literal notranslate"><span class="pre">krotov.second_order.numerical_estimate_A()</span></code></a>, with</p>
<div class="math notranslate nohighlight">
\[\Delta J_T \equiv J_T(\{\phi_k^{(i+1)}(T)\}) -J_T(\{\phi_k^{(i)}(T)\})\,.\]</div>
<p>See the <a class="reference internal" href="notebooks/07_example_PE.html"><span class="std std-ref">Optimization towards a Perfect Entangler</span></a> for an example.</p>
<div class="admonition note">
<p class="first admonition-title">Note</p>
<p class="last">Even when the second order update equation is mathematically required to
guarantee monotonic convergence, very often an optimization with the
first-order update equation <a class="reference internal" href="#equation-krotov-first-order-update">(9)</a> will give
converging results. Since the second order update requires
more numerical resources (calculation and storage of the states
<span class="math notranslate nohighlight">\(\ket{\Delta\phi_k(t)}\)</span>), you should always try the optimization with
the first-order update equation first.</p>
</div>
</div>
<div class="section" id="time-discretization">
<h2>Time discretization<a class="headerlink" href="#time-discretization" title="Permalink to this headline">¶</a></h2>
<div class="figure" id="id14">
<span id="figkrotovscheme"></span><a class="reference internal image-reference" href="_images/krotovscheme.svg"><img alt="Sequential update scheme in Krotov’s method on a time grid." src="_images/krotovscheme.svg" width="100%" /></a>
<p class="caption"><span class="caption-number">Fig. 1 </span><span class="caption-text">Sequential update scheme in Krotov’s method on a time grid.</span></p>
</div>
<p>The derivation of Krotov’s method assumes time-continuous control fields. In
this case, it mathematically guarantees monotonic convergence. However, for
practical numerical applications, we have to consider controls on a discrete
time grid with <span class="math notranslate nohighlight">\(nt\)</span> points running from <span class="math notranslate nohighlight">\(t=0\)</span> to <span class="math notranslate nohighlight">\(t=T\)</span>, with
a time step <span class="math notranslate nohighlight">\(\dd t\)</span> . The states are defined on the points of the time
grid, while the controls are assumed to be constant on the intervals of the
time grid. See the notebook <a class="reference external" href="https://nbviewer.jupyter.org/gist/goerz/21e46ea7b45c9514e460007de14419bd/Krotov_time_discretization.ipynb#">Time Discretization in Quantum Optimal Control</a>
for details. This discretization yields the numerical scheme shown in
<a class="reference internal" href="#figkrotovscheme"><span class="std std-numref">Fig. 1</span></a>. The scheme proceeds as follows:</p>
<ol class="arabic">
<li><p class="first">Construct the states <span class="math notranslate nohighlight">\(\ket{\chi_k(T)}\)</span> according to
Eq. <a class="reference internal" href="#equation-chi-boundary">(8)</a>. This may depend on the states forward-propagated
under the optimized pulse from the previous iteration, that is, the guess
pulse in the current iteration.</p>
</li>
<li><p class="first">Perform a backward-propagation using Eq. <a class="reference internal" href="#equation-bw-eqm">(7)</a> as the equation of
motion over the entire time grid. The resulting state at each point in the
time grid must be stored in memory.</p>
</li>
<li><p class="first">Starting from the known initial state <span class="math notranslate nohighlight">\(\ket{\phi_k(t=0)}\)</span>, calculate
the pulse update for the first time step according to
Eq. <a class="reference internal" href="#equation-krotov-first-order-update">(9)</a>, with <span class="math notranslate nohighlight">\(t=\dd t/2\)</span> on the left-hand
side (representing the first <em>interval</em> in the time grid, on which the
control pulse is defined), and <span class="math notranslate nohighlight">\(t=0\)</span> on the right-hand side
(representing the first <em>point</em> on the time grid). This approximation of
<span class="math notranslate nohighlight">\(t \approx t + \dd t /2\)</span> is what constitutes the “time discretization”
mathematically, and what resolves the seeming contradiction in the
time-continuous Eq. <a class="reference internal" href="#equation-krotov-first-order-update">(9)</a>, i.e., that the
calculation of <span class="math notranslate nohighlight">\(\epsilon^{(i+1)}(t)\)</span> requires knowledge of the states
<span class="math notranslate nohighlight">\(\ket{\phi_k^{(i+1)}(t)}\)</span> propagated under
<span class="math notranslate nohighlight">\(\epsilon^{(i+1)}(t)\)</span>.</p>
</li>
<li><p class="first">Use the updated control field for the first interval to propagate
<span class="math notranslate nohighlight">\(\ket{\phi_k(t=0)} \rightarrow \ket{\phi_k(t=\dd t)}\)</span> for a single
time step, with Eq. <a class="reference internal" href="#equation-fw-eqm">(6)</a> as the equation of motion. The updates then
proceed sequentially, until the final forward-propagated state
<span class="math notranslate nohighlight">\(\ket{\phi_k(T)}\)</span> is reached.</p>
<p>For numerical stability, it is useful to define the normalized states</p>
<div class="math notranslate nohighlight">
\[\ket{\phi_k^{\text{bw}}(T)} = \frac{1}{\Norm{\ket{\chi_k}}} \ket{\chi_{k}(T)}\]</div>
<p>for use in the backward propagation, and then later multiply again
with <span class="math notranslate nohighlight">\(\Norm{\ket{\chi_k}}\)</span> when calculating the pulse update.</p>
</li>
</ol>
<p>Note that for multiple objectives the scheme can run in parallel and each
objective contributes a term to the update. Summation of these terms yields the
sum in Eq. <a class="reference internal" href="#equation-krotov-first-order-update">(9)</a>. See <a class="reference internal" href="API/krotov.parallelization.html#module-krotov.parallelization" title="krotov.parallelization"><code class="xref py py-mod docutils literal notranslate"><span class="pre">krotov.parallelization</span></code></a> for
details. For a second-order update, the forward propagated states from step 4,
both for the current iteration and the previous iteration, must be stored in
memory over the entire time grid.</p>
</div>
<div class="section" id="choice-of-a">
<h2>Choice of λₐ<a class="headerlink" href="#choice-of-a" title="Permalink to this headline">¶</a></h2>
<p>The monotonic convergence of Krotov’s method is only guaranteed in the
continuous limit; a coarse time step must be compensated by larger values of
the inverse step size <span class="math notranslate nohighlight">\(\lambda_a\)</span>, slowing down convergence. Generally,
choosing <span class="math notranslate nohighlight">\(\lambda_a\)</span> too small will lead to numerical instabilities and
unphysical features in the optimized pulse. A lower limit for <span class="math notranslate nohighlight">\(\lambda_a\)</span>
can be determined from the requirement that the change
<span class="math notranslate nohighlight">\(\Delta\epsilon(t)\)</span> should be at most of the same order of magnitude as
the guess pulse <span class="math notranslate nohighlight">\(\epsilon^{(i)}(t)\)</span> for that iteration. The
Cauchy-Schwarz inequality applied to the update equation yields</p>
<div class="math notranslate nohighlight">
\[\Norm{\Delta \epsilon(t)}_{\infty}
\le
\frac{\Norm{S(t)}}{\lambda_a}
\sum_{k} \Norm{\ket{\chi_k (t)}}_{\infty} \Norm{\ket{\phi_k (t)}}_{\infty}
\Norm{\frac{\partial \Op{H}}{\partial \epsilon}}_{\infty}
\stackrel{!}{\le}
\Norm{\epsilon^{(i)}(t)}_{\infty}\,,\]</div>
<p>where <span class="math notranslate nohighlight">\(\norm{\partial \Op{H}/\partial \epsilon}_{\infty}\)</span> denotes the
supremum norm (with respect to time) of the operator norms of the operators
<span class="math notranslate nohighlight">\(\partial \Op{H}/\partial \epsilon\)</span> obtained at time <span class="math notranslate nohighlight">\(t\)</span>. Since
<span class="math notranslate nohighlight">\(S(t) \in [0,1]\)</span> and <span class="math notranslate nohighlight">\(\ket{\phi_k}\)</span> is normalized, the condition
for <span class="math notranslate nohighlight">\(\lambda_a\)</span> becomes</p>
<div class="math notranslate nohighlight">
\[\lambda_a \ge
\frac{1}{\Norm{\epsilon^{(i)}(t)}_{\infty}}
\left[ \sum_{k} \Norm{\ket{\chi_k(t)}}_{\infty} \right]
\Norm{\frac{\partial \Op{H}}{\partial \epsilon}}_{\infty}\,.\]</div>
<p>From a practical point of view, the best strategy is to start the
optimization with a comparatively large value of <span class="math notranslate nohighlight">\(\lambda_a\)</span>, and
after a few iterations lower <span class="math notranslate nohighlight">\(\lambda_a\)</span> as far as possible
without introducing numerical instabilities. The value of
<span class="math notranslate nohighlight">\(\lambda_a\)</span> may be adjusted dynamically with respect to the rate of
convergence. Generally, the optimal choice of <span class="math notranslate nohighlight">\(\lambda_a\)</span> requires
some trial and error.</p>
</div>
<div class="section" id="rotating-wave-approximation">
<h2>Rotating wave approximation<a class="headerlink" href="#rotating-wave-approximation" title="Permalink to this headline">¶</a></h2>
<p>When using the rotating wave approximation (RWA), it is important to remember
that the target states are usually defined in the lab frame, not in the
rotating frame. This is relevant for the construction of
<span class="math notranslate nohighlight">\(\ket{\chi_k(T)}\)</span>. When doing a simple optimization, such as a
state-to-state or a gate optimization, the easiest approach is to transform
the target states to the rotating frame before calculating
<span class="math notranslate nohighlight">\(\ket{\chi_k(T)}\)</span>. This is both straightforward and numerically
efficient.</p>
<p>Another solution would be to transform the result of the forward propagation
<span class="math notranslate nohighlight">\(\ket{\phi_k(T)}\)</span> from the rotating frame to the lab frame, then
constructing <span class="math notranslate nohighlight">\(\ket{\chi_k(T)}\)</span>, and finally to transform
<span class="math notranslate nohighlight">\(\ket{\chi_k(T)}\)</span> back to the rotating frame, before starting the
backward propagation.</p>
<p>When the RWA is used the control fields are
complex-valued. In this case the Krotov update equation is valid for
both the real and the imaginary part independently. The most straightforward
implementation of the method is for real controls only, requiring that any
complex control Hamiltonian is rewritten as two independent control
Hamiltonians, one for the real part and one for the imaginary part of the
control field. For example,</p>
<div class="math notranslate nohighlight">
\[\epsilon^*(t) \Op{a} + \epsilon(t) \Op{a}^\dagger
= \epsilon_{\text{re}}(t) (\Op{a} + \Op{a}^\dagger) + \epsilon_{\text{im}}(t) (i \Op{a}^\dagger - i \Op{a})\]</div>
<p>with two independent control fields <span class="math notranslate nohighlight">\(\epsilon_{\text{re}}(t)= \Re[\epsilon(t)]\)</span> and
<span class="math notranslate nohighlight">\(\epsilon_{\text{im}}(t) = \Im[\epsilon(t)]\)</span>.</p>
<p>See the <a class="reference internal" href="notebooks/02_example_lambda_system_rwa_complex_pulse.html"><span class="std std-ref">Optimization of a state-to-state transfer in a lambda system with RWA</span></a> for an
example.</p>
</div>
<div class="section" id="optimization-in-liouville-space">
<h2>Optimization in Liouville space<a class="headerlink" href="#optimization-in-liouville-space" title="Permalink to this headline">¶</a></h2>
<p>The control equations have been written in the notation of Hilbert space.
However, they are equally valid for a gate optimization in Liouville space, by
replacing Hilbert space states with density matrices, <span class="math notranslate nohighlight">\(\Op{H}\)</span> with
<span class="math notranslate nohighlight">\(i \Liouville\)</span> (cf. <a class="reference internal" href="API/krotov.mu.html#module-krotov.mu" title="krotov.mu"><code class="xref py py-mod docutils literal notranslate"><span class="pre">krotov.mu</span></code></a>), and inner products with
Hilbert-Schmidt products, <span class="math notranslate nohighlight">\(\langle \cdot \vert \cdot \rangle \rightarrow
\langle\!\langle \cdot \vert \cdot \rangle\!\rangle\)</span>, cf., e.g.,
Ref. <a class="reference internal" href="99_bibliography.html#goerznjp2014" id="id13">[GoerzNJP2014]</a>.</p>
<p>See the <a class="reference internal" href="notebooks/04_example_dissipative_qubit_reset.html"><span class="std std-ref">Optimization of Dissipative Qubit Reset</span></a> for an
example.</p>
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