[Demo] Add: Unitary synthesis with recursive KAK decompositions (#1372)

**Title:**
Unitary synthesis with recursive KAK decompositions

**Summary:**
Unitary synthesis, the process of compiling a unitary matrix into a
quantum circuit, has been a topic of study for over
25 years. Three interesting techniques for unitary synthesis, namely the
first, the (probably) most widely known, and the minimal-CNOT-count
("the best"), all use the same recursive KAK decomposition under the
hood, as we recently
elaborated [in a paper](https://arxiv.org/abs/2503.19014).

This demo builds on our demo on KAK decompositions (added in #1227) and
explains how such decompositions can be applied recursively to construct
a many-qubit circuit and how the three mentioned techniques all are
powered by the same recursion.
It defers some optimization details to the literature but provides a
brief gate count overview and showcases a simplified numerical
implementation of all three techniques.

**Relevant references:**
Quite a few, see metadata file.

**Possible Drawbacks:**
N/A

**Related GitHub Issues:**

[sc-87548]

To do

- [x] Redo circuit diagrams
- [x] Add .tex files to repo
- [x] add examples/code

----
If you are writing a demonstration, please answer these questions to
facilitate the marketing process.

* GOALS — Why are we working on this now?

- Promote `qml.liealg` features
- [promote a paper we recently did (see link above)]
- Follow up on started line of demos (#1227, #1261)


* AUDIENCE — Who is this for?
- Researchers in quantum compilation and quantum information
- Practicioners that want to learn where the complexity of their
operations comes from



* KEYWORDS — What words should be included in the marketing post?
- (Recursive) KAK decomposition
- Quantum Shannon decomposition
- Block-ZXZ decomposition
- Unitary synthesis
- Compilation


* Which of the following types of documentation is most similar to your
file?
(more details
[here](https://www.notion.so/xanaduai/Different-kinds-of-documentation-69200645fe59442991c71f9e7d8a77f8))
    
- [ ] Tutorial
- [x] Demo
- [ ] How-to

---------

Co-authored-by: cognigami <[email protected]>

Author: dwierichs

_static/demo_thumbnails/large_demo_thumbnails/thumbnail_large_unitary_synthesis_kak.png

@@ -0,0 +1,18 @@
+\documentclass[]{standalone}
+\usepackage{amsmath,quantikz,graphicx,xcolor}
+
+\begin{document}
+\begin{quantikz}
+    & & \gate{R_Y}\wire[d][3]{q} & \\
+    & \qwbundle{n-3} & |[operator]| & \\
+    & & |[operator]| & \\
+    & & |[operator]| &
+\end{quantikz}
+=
+\begin{quantikz}
+    & & \gate{R_Y(\theta_0)} & \gate{R_Y(\theta_1)} & \gate{R_Y(\theta_2)} & \ \ldots \ & \gate{R_Y(\theta_{2^{n-1}-1})} & \\
+    & \qwbundle{n-3} & \octrl{-1} & \octrl{-1} & \octrl{-1} & \ \ldots \ & \ctrl{-1} & \\
+    & & \octrl{-1} & \octrl{-1} & \ctrl{-1} & \ \ldots \ & \ctrl{-1} & \\
+    & & \octrl{-1} & \ctrl{-1} & \octrl{-1} & \ \ldots \ & \ctrl{-1} &
+\end{quantikz}
+\end{document}

_static/demo_thumbnails/opengraph_demo_thumbnails/OGthumbnail_unitary_synthesis_kak.png

@@ -0,0 +1,15 @@
+\documentclass[]{standalone}
+\usepackage{amsmath,quantikz,graphicx,xcolor}
+
+\begin{document}
+\begin{quantikz}
+    & & \gate[2]{U(2^n)} &\\
+    & \qwbundle{n-1} & &
+\end{quantikz}
+=
+\begin{quantikz}
+    & & |[operator]| & \gate{R_Y}\wire[d]{q} & |[operator]| &\\
+    & \qwbundle{n-1} & \gate{U(2^{n-1})}\wire[u]{q} & |[operator]| & \gate{U(2^{n-1})}\wire[u]{q} & 
+\end{quantikz}
+
+\end{document}

_static/demo_thumbnails/regular_demo_thumbnails/thumbnail_unitary_synthesis_kak.png

@@ -0,0 +1,15 @@
+\documentclass[]{standalone}
+\usepackage{amsmath,quantikz,graphicx,xcolor}
+
+\begin{document}
+\begin{quantikz}
+    & \ghost{R_Z} & |[operator]| &\\
+    & \qwbundle{n-1} & \gate{U(2^{n-1})}\wire[u]{q} &
+\end{quantikz}
+=
+\begin{quantikz}
+    & & & \gate{R_Z}\wire[d]{q} & &\\
+    & \qwbundle{n-1} & \gate{U(2^{n-1})} & |[operator]| & \gate{U(2^{n-1})} & 
+\end{quantikz}
+
+\end{document}

_static/demonstration_assets/unitary_synthesis_kak/AIII_generic_CSG.pdf

@@ -0,0 +1,47 @@
+\documentclass[]{standalone}
+\usepackage{amsmath,quantikz,graphicx,xcolor}
+
+\begin{document}
+\begin{quantikz}
+    & \gate[2]{G} & \\
+    &&
+\end{quantikz}
+=
+\begin{quantikz}
+    & \gate[2]{K_2} & \gate[2]{A} & \gate[2]{K_1} & \\
+    &&&&
+\end{quantikz}
+% Iterate the above:
+
+% \begin{align*}
+% \begin{quantikz}
+%     & \ctrl[open]{1} & \ctrl[open]{1} & \ctrl{1} & \ctrl{1} & \\
+%     & \ctrl[open]{1} & \ctrl{1} & \ctrl[open]{1} & \ctrl{1} & \\
+%     & \gate{A} & \gate{B} & \gate{C} & \gate{D} &
+% \end{quantikz}
+% &=
+% \begin{quantikz}
+%     & \ctrl[open]{1} & \ctrl{1} & \\
+%     & \gate[2]{A\oplus B} & \gate[2]{C\oplus D} & \\
+%     & & &
+% \end{quantikz}\\
+% &=
+% \begin{quantikz}
+%     & & \ctrl{1} & \\
+%     & \gate[2]{A\oplus B} & \gate[2]{A^\dagger C\oplus B^\dagger D} & \\
+%     & & &
+% \end{quantikz}\\
+% &=
+% \begin{quantikz}
+%     & & & \ctrl{1} & \ctrl{1} & \\
+%     & \ctrl[open]{1} & \ctrl{1} & \ctrl[open]{1} & \ctrl{1} & \\
+%     & \gate{A} & \gate{B} & \gate{A^\dagger C} & \gate{B^\dagger D} &
+% \end{quantikz}\\
+% &=
+% \begin{quantikz}
+%     & & & \ctrl{2} & \ctrl{1} & \\
+%     & & \ctrl{1} & & \ctrl{1} & \\
+%     & \gate{A} & \gate{A^\dagger B} & \gate{A^\dagger C} & \gate{C^\dagger A B^\dagger D} &
+% \end{quantikz}
+% \end{align*}
+\end{document}

_static/demonstration_assets/unitary_synthesis_kak/AIII_generic_CSG.png

@@ -0,0 +1,17 @@
+\documentclass[]{standalone}
+\usepackage{amsmath,quantikz,graphicx,xcolor}
+
+
+\begin{document}
+% \begin{align*}
+\begin{quantikz}
+    & & \gate[3]{U(2^n)} &
+    \midstick[3,brackets=none]{=}
+    & & & &[-0.5cm] \gate{R_Z}\wire[d][2]{q} &[-0.5cm] & & &[-0.5cm] \gate{R_X}\wire[d][2]{q} &[-0.5cm] & & &[-0.5cm] \gate{R_Z}\wire[d][2]{q} &[-0.5cm] & & \\
+    & \qwbundle{n-3} & &
+    & \qwbundle{n-3} & \gate[2]{U(2^{n-1})} & \gate{H} & |[operator]| & \gate{H} & \gate[2]{U(2^{n-1})} & \gate{H} & |[operator]| & \gate{H} & \gate[2]{U(2^{n-1})} & \gate{H} & |[operator]| & \gate{H} & \gate[2]{U(2^{n-1})} & \\
+    & \qwbundle{2} & &
+    & \qwbundle{2} & & \gate{E^\dagger} & |[operator]| & \gate{E} & & \gate{E^\dagger} & |[operator]| & \gate{E} & & \gate{E^\dagger} & |[operator]| & \gate{E} & & 
+\end{quantikz}
+
+\end{document}

_static/demonstration_assets/unitary_synthesis_kak/AIII_generic_CSG.tex

@@ -0,0 +1,15 @@
+\documentclass[]{standalone}
+\usepackage{amsmath,quantikz,graphicx,xcolor}
+
+\begin{document}
+\begin{quantikz}
+    & & \gate[2]{U(2^n)} &\\
+    & \qwbundle{n-1} & &
+\end{quantikz}
+=
+\begin{quantikz}
+    & & & \gate{R_Z}\wire[d]{q} & & \gate{R_Y}\wire[d]{q} & & \gate{R_Z}\wire[d]{q} & &\\
+    & \qwbundle{n-1} & \gate{U(2^{n-1})} & |[operator]| & \gate{U(2^{n-1})} & |[operator]| & \gate{U(2^{n-1})} & |[operator]| & \gate{U(2^{n-1})} & 
+\end{quantikz}
+
+\end{document}

_static/demonstration_assets/unitary_synthesis_kak/AIII_generic_complete.pdf

@@ -0,0 +1,85 @@
+\documentclass[preview]{standalone}
+\usepackage{amsmath,quantikz,graphicx,xcolor,amssymb,amsfonts}
+
+\definecolor{yellow}{HTML}{E2A300}
+\definecolor{blue}{HTML}{4D53C8}
+\definecolor{green}{HTML}{52BD7C}
+\definecolor{red}{HTML}{D7333B}
+
+\begin{document}
+\begin{align*}
+\begin{quantikz}
+    \ \ldots \  & &
+    \gate[style={dashed}]{\phantom{U}}\wire[d][2]{q} & &
+    \gate[style={dashed}]{\phantom{U}}\wire[d][2]{q} & & \\
+    \ \ldots \ & \gate[2]{SU(4)} &
+    |[operator]| & \gate[2]{SU(4)} &
+    |[operator]| & \gate[2]{SU(4)} & \\
+    \ \ldots \ & &
+    |[operator]| & &
+    |[operator]| & & 
+\end{quantikz}
+&=
+\begin{quantikz}
+    \ \ldots \  & &
+    \gate[style={dashed}]{\phantom{U}}\wire[d][2]{q} & &
+    \gate[style={dashed}]{\phantom{U}}\wire[d][2]{q} &[-0.9cm] & [0.2cm] & & & & & & \\
+    \ \ldots \ & \gate[2]{SU(4)} &
+    |[operator]| & \gate[2]{SU(4)} &
+    |[operator]| & \gategroup[2,steps=2,style={inner sep=0pt, rounded corners, color=red}]{} & \gate[2]{\Delta} & \gate[style={dotted}]{\phantom{U}} & \ctrl{1} & \gate{R_Y} & \ctrl{1} & \gate[style={dotted}]{\phantom{U}} & \\
+    \ \ldots \ & &
+    |[operator]| & &
+    |[operator]| & & & \gate[style={dotted}]{\phantom{U}} & \targ{} & \gate{R_Y} & \targ{} & \gate[style={dotted}]{\phantom{U}} & 
+\end{quantikz}\\
+&=
+\begin{quantikz}
+    \ \ldots \  & &
+    \gate[style={dashed}]{\phantom{U}}\wire[d][2]{q} & & & 
+    \gate[style={dashed}]{\phantom{U}}\wire[d][2]{q} & & & & & & \\
+    \ \ldots \ & \gate[2]{SU(4)} &
+    |[operator]| & \gate[2]{SU(4)}\gategroup[2,steps=2,style={inner sep=0pt, rounded corners, color=blue}]{} &
+    \gate[2]{\Delta} & |[operator]| & \gate[style={dotted}]{\phantom{U}} & \ctrl{1} & \gate{R_Y} & \ctrl{1} & \gate[style={dotted}]{\phantom{U}} & \\
+    \ \ldots \ & &
+    |[operator]| & &
+    & |[operator]| & \gate[style={dotted}]{\phantom{U}} & \targ{} & \gate{R_Y} & \targ{} & \gate[style={dotted}]{\phantom{U}} & 
+\end{quantikz}\\
+&=
+\begin{quantikz}
+    \ \ldots \  & &
+    \gate[style={dashed}]{\phantom{U}}\wire[d][2]{q} &[-0.9cm] & [0.2cm] & & & & & & 
+    \gate[style={dashed}]{\phantom{U}}\wire[d][2]{q} & & & & & & \\
+    \ \ldots \ & \gate[2]{SU(4)} &
+    |[operator]| &\gategroup[2,steps=2,style={inner sep=0pt, rounded corners, color=red}]{} & \gate[2]{\Delta} & \gate[style={dotted}]{\phantom{U}} & \ctrl{1} & \gate{R_Y} & \ctrl{1} & \gate[style={dotted}]{\phantom{U}}  & |[operator]| & \gate[style={dotted}]{\phantom{U}} & \ctrl{1} & \gate{R_Y} & \ctrl{1} & \gate[style={dotted}]{\phantom{U}} & \\
+    \ \ldots \ & &
+    |[operator]| & & & \gate[style={dotted}]{\phantom{U}} & \targ{} & \gate{R_Y} & \targ{} & \gate[style={dotted}]{\phantom{U}}
+    & |[operator]| & \gate[style={dotted}]{\phantom{U}} & \targ{} & \gate{R_Y} & \targ{} & \gate[style={dotted}]{\phantom{U}} & 
+\end{quantikz}\\
+% &=   
+% \begin{quantikz}
+%     & & & \gate{R_Z}\wire[d][2]{q} & & \gate{R_X}\wire[d][2]{q} & & \gate{R_Z}\wire[d][2]{q} & &\\
+%     & & \gate[2]{U(2^{n-1})} & |[operator]| & \gate[2]{U(2^{n-1})} & |[operator]| & \gate[2]{U(2^{n-1})} & |[operator]| & \gate[2]{U(2^{n-1})} & \\
+%     & \qwbundle{n-2} & & |[operator]| & & |[operator]| & & |[operator]| & & 
+% \end{quantikz}\\
+% &=\begin{quantikz}
+%     & & & \gate{R_Z}\wire[d][2]{q} & \gate{H} & \gategroup[3,steps=3,style={inner
+% sep=3pt, rounded corners,color=green}]{} & \gate{R_Z}\wire[d][2]{q} & & \gate{H} & \gate{R_Z}\wire[d][2]{q} & &\\
+%     & & \gate[2]{U(2^{n-1})} & |[operator]| & & \gate[2]{U(2^{n-1})} & |[operator]| & \gate[2]{U(2^{n-1})} & & |[operator]| & \gate[2]{U(2^{n-1})} & \\
+%     & \qwbundle{n-2} &  & |[operator]| & &  & |[operator]| &  & & |[operator]| &  & 
+% \end{quantikz}\\
+% &=\begin{quantikz}
+%     & & & \gate{R_Z}\wire[d][2]{q} & \targ{}\gategroup[2,steps=3,style={inner
+% sep=0pt, rounded corners,color=yellow}]{} & \gate{H} & \ctrl{0}\gategroup[3,steps=3,style={inner
+% sep=3pt, rounded corners,color=blue}]{} & |[operator]| & \ctrl{0}\gategroup[2,steps=3,style={inner
+% sep=0pt, rounded corners,color=yellow}]{} & \gate{H} & \targ{} & \gate{R_Z}\wire[d][2]{q} & &\\
+%     & & \gate[2]{U(2^{n-1})} & |[operator]| & \ctrl{-1} & &\ctrl{-1} & \gate[2]{U(2^{n-1})}\wire[u]{q} & \ctrl{-1} & & \ctrl{-1}& |[operator]| & \gate[2]{U(2^{n-1})} & \\
+%     & \qwbundle{n-2} &  & |[operator]| & & & & & & & & |[operator]| &  & 
+% \end{quantikz}\\
+% &=\begin{quantikz}
+%     & & & \gate{R_Z}\wire[d][2]{q}\gategroup[3,steps=2,style={inner
+% sep=0pt, rounded corners,color=red}]{} & \targ{} & & \gate{R_X}\wire[d][2]{q} & & \targ{}\gategroup[3,steps=2,style={inner
+% sep=0pt, rounded corners,color=red}]{} & \gate{R_Z}\wire[d][2]{q} & &\\
+%     & & \gate[2]{U(2^{n-1})} & |[operator]| & \ctrl{-1} & \gate[2]{U(2^{n-1})} & |[operator]| & \gate[2]{U(2^{n-1})} & \ctrl{-1} & |[operator]| & \gate[2]{U(2^{n-1})} & \\
+%     & \qwbundle{n-2} &  & |[operator]| & & & |[operator]| & & & |[operator]| &  & 
+% \end{quantikz}
+\end{align*}
+\end{document}

_static/demonstration_assets/unitary_synthesis_kak/AIII_generic_complete.png

@@ -0,0 +1,17 @@
+\documentclass[]{standalone}
+\usepackage{amsmath,quantikz,graphicx,xcolor}
+
+
+\begin{document}
+% \begin{align*}
+\begin{quantikz}
+    & & \gate[2]{U(2^n)} &\\
+    & \qwbundle{n-1} & &
+\end{quantikz}
+=
+\begin{quantikz}
+    & & & \gate{R_Z}\wire[d]{q} & & \gate{R_X}\wire[d]{q} & & \gate{R_Z}\wire[d]{q} & &\\
+    & \qwbundle{n-1} & \gate{U(2^{n-1})} & |[operator]| & \gate{U(2^{n-1})} & |[operator]| & \gate{U(2^{n-1})} & |[operator]| & \gate{U(2^{n-1})} & 
+\end{quantikz}
+
+\end{document}

_static/demonstration_assets/unitary_synthesis_kak/AIII_generic_complete.tex

@@ -0,0 +1,44 @@
+\documentclass[]{article}
+\usepackage{amsmath,quantikz,graphicx,xcolor,amssymb,amsfonts}
+
+\definecolor{yellow}{HTML}{E2A300}
+\definecolor{blue}{HTML}{4D53C8}
+\definecolor{green}{HTML}{52BD7C}
+\definecolor{red}{HTML}{D7333B}
+
+\begin{document}
+\begin{align*}
+\hspace{-5cm}\begin{quantikz}
+    & & \gate[3]{U(2^n)} &\\
+    & & &\\
+    & \qwbundle{n-2} & &
+\end{quantikz}
+&=
+\begin{quantikz}
+    & & & \gate{R_Z}\wire[d][2]{q} & & \gate{R_X}\wire[d][2]{q} & & \gate{R_Z}\wire[d][2]{q} & &\\
+    & & \gate[2]{U(2^{n-1})} & |[operator]| & \gate[2]{U(2^{n-1})} & |[operator]| & \gate[2]{U(2^{n-1})} & |[operator]| & \gate[2]{U(2^{n-1})} & \\
+    & \qwbundle{n-2} & & |[operator]| & & |[operator]| & & |[operator]| & & 
+\end{quantikz}\\
+&=\begin{quantikz}
+    & & & \gate{R_Z}\wire[d][2]{q} & \gate{H} & \gategroup[3,steps=3,style={inner
+sep=3pt, rounded corners,color=green}]{} & \gate{R_Z}\wire[d][2]{q} & & \gate{H} & \gate{R_Z}\wire[d][2]{q} & &\\
+    & & \gate[2]{U(2^{n-1})} & |[operator]| & & \gate[2]{U(2^{n-1})} & |[operator]| & \gate[2]{U(2^{n-1})} & & |[operator]| & \gate[2]{U(2^{n-1})} & \\
+    & \qwbundle{n-2} &  & |[operator]| & &  & |[operator]| &  & & |[operator]| &  & 
+\end{quantikz}\\
+&=\begin{quantikz}
+    & & & \gate{R_Z}\wire[d][2]{q} & \targ{}\gategroup[2,steps=3,style={inner
+sep=0pt, rounded corners,color=yellow}]{} & \gate{H} & \ctrl{0}\gategroup[3,steps=3,style={inner
+sep=3pt, rounded corners,color=blue}]{} & |[operator]| & \ctrl{0}\gategroup[2,steps=3,style={inner
+sep=0pt, rounded corners,color=yellow}]{} & \gate{H} & \targ{} & \gate{R_Z}\wire[d][2]{q} & &\\
+    & & \gate[2]{U(2^{n-1})} & |[operator]| & \ctrl{-1} & &\ctrl{-1} & \gate[2]{U(2^{n-1})}\wire[u]{q} & \ctrl{-1} & & \ctrl{-1}& |[operator]| & \gate[2]{U(2^{n-1})} & \\
+    & \qwbundle{n-2} &  & |[operator]| & & & & & & & & |[operator]| &  & 
+\end{quantikz}\\
+&=\begin{quantikz}
+    & & & \gate{R_Z}\wire[d][2]{q}\gategroup[3,steps=2,style={inner
+sep=0pt, rounded corners,color=red}]{} & \targ{} & & \gate{R_X}\wire[d][2]{q} & & \targ{}\gategroup[3,steps=2,style={inner
+sep=0pt, rounded corners,color=red}]{} & \gate{R_Z}\wire[d][2]{q} & &\\
+    & & \gate[2]{U(2^{n-1})} & |[operator]| & \ctrl{-1} & \gate[2]{U(2^{n-1})} & |[operator]| & \gate[2]{U(2^{n-1})} & \ctrl{-1} & |[operator]| & \gate[2]{U(2^{n-1})} & \\
+    & \qwbundle{n-2} &  & |[operator]| & & & |[operator]| & & & |[operator]| &  & 
+\end{quantikz}
+\end{align*}
+\end{document}

_static/demonstration_assets/unitary_synthesis_kak/A_generic_complete.pdf

@@ -0,0 +1,42 @@
+\documentclass[]{article}
+\usepackage{amsmath,quantikz,graphicx,xcolor,amssymb,amsfonts}
+
+\begin{document}
+\begin{align*}
+\begin{quantikz}
+    & & \gate[2]{U(2^n)} &\\
+    & \qwbundle{n-1} & &
+\end{quantikz}
+&=
+\begin{quantikz}
+    & & |[operator]| & \gate{R_X(\vec{\theta})}\wire[d]{q} & |[operator]| &\\
+    & \qwbundle{n-1} & \gate{W_0,W_1}\wire[u]{q} & |[operator]| & \gate{V_0,V_1}\wire[u]{q} & 
+\end{quantikz}\\
+&=
+\begin{quantikz}
+    & & |[operator]| & \gate{H} & \gate{R_Z(\vec{\theta})}\wire[d]{q} & \gate{H} & |[operator]| &\\
+    & \qwbundle{n-1} & \gate{W_0,W_1}\wire[u]{q} & & |[operator]| & & \gate{V_0,V_1}\wire[u]{q} & 
+\end{quantikz}\\
+&=
+\begin{quantikz}
+    & & |[operator]| & \gate{H} & & \gate{R_Z(\vec{\theta})}\wire[d]{q} & \gate{H} & |[operator]| &\\
+    & \qwbundle{n-1} & \gate{W_0,W_1}\wire[u]{q} & \gate{W_0^\dagger} & \gate{W_0} & |[operator]| & & \gate{V_0,V_1}\wire[u]{q} & 
+\end{quantikz}\\
+&=
+\begin{quantikz}
+    & & |[operator]| & \gate{H} & \gate{R_Z(\vec{\theta})}\wire[d]{q} & & \gate{H} & |[operator]| &\\
+    & \qwbundle{n-1} & \gate{\mathbb{I},W_0^\dagger W_1}\wire[u]{q} & \gate{W_0} & |[operator]| & \gate{\Delta(-\vec{\theta})} & \gate{\Delta(\vec{\theta})} & \gate{V_0,V_1}\wire[u]{q} & 
+\end{quantikz}\\
+&=
+\begin{quantikz}
+    & & \ctrl{1} & \gate{H} & \ctrl{1} & & & \gate{H} & |[operator]| &\\
+    & \qwbundle{n-1} & \gate{W_0^\dagger W_1}\wire[u]{q} & \gate{W_0} & \gate{\Delta(-2\vec{\theta})} & \gate{W_0^\dagger} & \gate{W_0} & \gate{\Delta(\vec{\theta})} & \gate{V_0,V_1}\wire[u]{q} & 
+\end{quantikz}\\
+&=
+\begin{quantikz}
+    & & \ctrl{1} & \gate{H} & \ctrl{1} & \gate{H} & |[operator]| &\\
+    & \qwbundle{n-1} & \gate{W_0^\dagger W_1}\wire[u]{q} & & \gate{W_0^\dagger \Delta(-2\vec{\theta}) W_0} & & \gate{V_0 \Delta(\vec{\theta})W_0,V_1\Delta(\vec{\theta})W_0}\wire[u]{q} & 
+\end{quantikz}
+\end{align*}
+
+\end{document}

_static/demonstration_assets/unitary_synthesis_kak/A_generic_complete.png

@@ -0,0 +1,12 @@
+\documentclass[]{standalone}
+\usepackage{amsmath,quantikz,graphicx,xcolor}
+
+\begin{document}
+\begin{quantikz}
+    & \gate[2]{E} & \midstick[2,brackets=none]{=} 
+    & \gate{R_Z(\tfrac{\pi}{2})} & \targ{} & & \\
+    & &
+    & \gate{R_X(-\tfrac{\pi}{2})} & \ctrl{-1} & \gate{R_Z(-\tfrac{\pi}{2})} &
+\end{quantikz}
+
+\end{document}

_static/demonstration_assets/unitary_synthesis_kak/A_generic_complete.tex

@@ -0,0 +1,32 @@
+\documentclass[]{standalone}
+\usepackage{amsmath,quantikz,graphicx,xcolor}
+
+\begin{document}
+% \begin{quantikz} % Intermediate decompositions
+%     & |[operator]| & \midstick[3,brackets=none]{=} & & \ctrl{2} & & \ctrl{2} & \\
+%     & |[operator]| & & |[operator]| & & |[operator]| & & \\
+%     & \gate{R_P}\wire[u][2]{q} & & \gate{R_P}\wire[u][1]{q} & \targ{} & \gate{R_P}\wire[u][1]{q} & \targ{} &
+% \end{quantikz}
+
+% \begin{quantikz}
+%     & |[operator]| & \midstick[3,brackets=none]{=} 
+%     & & \ctrl{2} & & \ctrl{2} & \midstick[3,brackets=none]{=}
+%     & & & & \ctrl{2} & & & & \ctrl{2} & \\
+%     & |[operator]| &
+%     & |[operator]| & & |[operator]| & & 
+%     & & \ctrl{1} & & & & \ctrl{1} & & & \\
+%     & \gate{R_P}\wire[u][2]{q} &
+%     & \gate{R_P}\wire[u][1]{q} & \targ{} & \gate{R_P}\wire[u][1]{q} & \targ{} &
+%     & \gate{R_P} & \targ{} & \gate{R_P} & \targ{} & \gate{R_P} & \targ{} & \gate{R_P} & \targ{} &
+% \end{quantikz}
+
+\begin{quantikz}
+    & |[operator]| & \midstick[3,brackets=none]{=} 
+    & & & & \ctrl{2} & & & & \ctrl{2} & \\
+    & |[operator]| &
+    & & \ctrl{1} & & & & \ctrl{1} & & & \\
+    & \gate{R_P}\wire[u][2]{q} &
+    & \gate{R_P} & \targ{} & \gate{R_P} & \targ{} & \gate{R_P} & \targ{} & \gate{R_P} & \targ{} &
+\end{quantikz}
+
+\end{document}