Hybrid quantum-classical circuit simplification with the ZX-calculus

Agustín Borgna1,20000-0002-1688-1370 Simon
Perdrix10000-0002-1808-2409 Benoît

A. Borgna et al.

CNRS LORIA, Inria-MOCQUA, Université de Lorraine, F 54000 Nancy,
France Université Paris-Saclay, CNRS, Laboratoire Méthodes Formelles,
91405, Orsay, France École CentraleSupélec, Laboratoire Méthodes
Formelles, 91405, Orsay, France

We present a complete optimization procedure for hybrid
quantum-classical circuits with classical parity logic.

While common optimization techniques for quantum algorithms focus on rewriting
solely the pure quantum segments, there is interest in applying a global
optimization process for applications such as quantum error correction and
quantum assertions.

This work, based on the pure-quantum circuit optimization procedure by Duncan
et al., uses an extension of the formal graphical ZX-calculus called
as an intermediary representation of the hybrid circuits to allow
for granular optimizations below the quantum-gate level.

We define a translation from hybrid circuits into diagrams that admit the
graph-theoretical focused-gFlow property, needed for the final extraction back
into a circuit.

We then derive a number of gFlow-preserving optimization rules for
diagrams that reduce the size of the graph, and devise a strategy
to find optimization opportunities by rewriting the diagram guided by a Gauss
elimination process.

Then, after extracting the circuit, we present a general procedure for
detecting segments of circuit-like  diagrams which can be
implemented with classical gates in the extracted circuit.

We have implemented our optimization procedure as an extension to the
open-source python library PyZX.

§ INTRODUCTION

The description of quantum algorithms commonly involves quantum operations
interacting with classical data in its inputs, outputs, or intermediary steps
via measurements or state preparations.
Some applications such as quantum error correction [9, 2] and quantum assertions [18, 24]
explicitly introduce classical measurements and logic between quantum
In general, quantum programming languages usually allow for measurements and
classically controlled quantum operators mixed-in with unitary gates
[11, 7, 15, 22].
Furthermore, Jozsa [14] conjectured that any polynomial-time
quantum algorithm can be simulated by polylogarithmic-depth quantum computation
interleaved with polynomial-depth classical computation. As such, there is
interest in contemplating this kind of structures in circuits.

A popular alternative representation of quantum circuit is based on the
ZX-calculus [5, 6], a formal diagrammatic language which
presents a more granular representation of quantum circuits and has been
successfully used in applications such as MBQC [10],
quantum error correction [4], and quantum foundations.
Carette et al. [3] introduced an extension of the
calculus called  which allows for the representation of operations
interacting with the classical environment by adding a discarding ground
generator to the diagrams.

It is natural to look at the problem of optimizing algorithm implementations by
taking in consideration the environment in addition to the pure quantum
fragments.  However, most common optimization strategies focus solely on the
latter without contemplating the hybrid quantum-classical
structure [1, 12]. One of this
pure optimizations introduced by Duncan et al. [8] uses
the ZX-calculus to apply granular rewriting rules that ignore the boundaries of
each quantum gate. We will refer to it as the Clifford optimization
algorithm. Their rewriting steps preserve a diagram property called
gFlow admittance that is required for the final extraction of the
ZX diagrams into circuits. The ZX optimization method was latter used by
Kissinger and van de Wetering [17] in their
method to reduce the number of T-gates in quantum circuits.

In this work we define the natural extension of the pure Clifford optimization
algorithm by Duncan et al. to hybrid quantum-classical circuits using the

Our circuit optimization procedure forgets the difference between quantum and
classical wires during the simplification process, representing connections as
a single type of edge. This allows it to optimize the complete hybrid system as
an homogeneous diagram, and results in similar representations for operations
that can be done either quantumly or classically.

Generally, in a physical quantum computer, the classical operations are simpler to
implement than their quantum counterparts, and quantum simulators can exploit
the knowledge of which wires carry classical data to simplify their operation.
As such, it is beneficial to extract classical gates in the resulting circuit
where possible.

The contribution of this paper are as follows.

* We specify a translation of hybrid circuits into  diagrams in a
special graph like form that admits a gFlow, restricting the
classical segments of the input to parity circuits.

* We introduce a number of gFlow-preserving rewriting rules that interact
with the discarding generator to reduce the size of the diagrams, and devise
a strategy to find optimization opportunities using the biadjacency matrix
of the graph cut between spiders connected to ground generators and the
other nodes in the diagram.

* We define a procedure to extract  diagrams with a gFlow back
into hybrid quantum-classical circuits, including ancilla initialization and

* We define the problem of -classicalization as labelling segments
of the diagrams which can be implemented classically and present an
heuristic solution. Our method can be applied on the extracted circuits to
maximize the number of classically implemented operations.

The paper is organized in the following manner.

In Section <ref> we define the quantum circuits, present a syntactic
description of the  calculus and its equation, and give an intuition
behind the representation of hybrid quantum-classical circuits.

Section <ref> then introduces the graph-like family of diagrams and defines the focused gFlow property over the graphs.  We then define
the translation of quantum circuits into graph-like diagrams in
Section <ref>.

In Section <ref> we introduce the optimization rules and our
strategy for finding rule matches which we use to describe the complete
optimization algorithm. Then in Section <ref> we define the
extraction algorithm and finally we present our classicalization procedure in
Section <ref>.

In Section <ref> we discuss the results of testing our
procedure on randomly generated circuits.

§ HYBRID QUANTUM-CLASSICAL CIRCUITS AND THE GROUNDED ZX-CALCULUS

In pure quantum operations, a single qubit quantum state is represented as a
unitary vector in the Hilbert space $\C^2$. We use Dirac notation to talk about
such vectors and denote an arbitrary state as $\ket{\phi}$.  States can be
be described as a linear combination of vectors in a basis such as the
computational basis $\{\ket{0}, \ket{1}\}$ or the diagonal basis $\{\ket{+},
\ket{-}\}$, where $\ket{\pm} = \frac{1}{\sqrt2}(\ket{0} \pm \ket{1})$.  A third,
less commonly used basis called $Y$ is formed by the vectors $\ketR =
\frac{1}{\sqrt2}(\ket{0} + i \ket{1})$ and $\ketL = \frac{1}{\sqrt2}(\ket{0} - i
\ket{1})$. Qubit spaces can be composed using a tensor product, and we denote
$\ket{\phi\psi} = \ket{\phi} \otimes \ket{\psi}$.

Hybrid quantum-classical systems include classical data, which can be
represented in a qubit space as orthonormal basis vectors (e.g. by representing
a logical 0 as the state $\ket{0}$ and a logical 1 as $\ket{1}$), but
additionally include a trace or measurement operation, which
probabilistically projects a qubit into a vector in an orthogonal basis. The
resulting probabilistic distribution of pure states is called a mixed state, and
is better represented by a density matrix, a positive semi-definite
Hermitian operator of trace one in the $(\C^{2 \times 2})^{\otimes n}$ Hilbert
space, for an $n$-qubit system. Given a probabilistic distribution of pure
states $\{(p_i, \ket{\phi_i})\}$, their density matrix is constructed as $\sum_i
p_i \ketbra{\phi_i}{\phi_i}$, where $\bra{\phi} = \ket{\phi}^\dagger$.

Quantum circuit diagrams consist of horizontal lines carrying each the
information of one qubit, read from right to left, with some attached gates
applying unitary transformations over the qubit states. We use the universal set
of operations $\{\CNOT, \Rx[\alpha], \Z[\alpha], \H\}$ for pure-quantum
diagrams. When $\alpha$ is limited to multiples of $\frac{\pi}{4}$ this roughly
corresponds to the approximately universal Clifford+T group. Some rotation gates
have specific names, such as $\Z[] = \Z[\pi]$, $\Rx[] = \Rx[\pi]$, $\text{S} =
\Z[\frac\pi2]$, $\text{HSH} = \Rx[\frac\pi2]$, and $\text{T} = \Z[\frac\pi4]$.
We additionally include ancilla initialization and termination, and swaps.  The
representation of each mentioned gate is reproduced here.

\[ \tikzfig{circuit/cnot}
\quad
\tikzfig{circuit/hadamard}
\quad
\tikzfig{circuit/z}
\quad
\tikzfig{circuit/x}
\quad
\tikzfig{circuit/init}
\quad
\tikzfig{circuit/termination}
\quad
\tikzfig{circuit/swapQ}
\]

Hybrid circuits represent bit-carrying classical wires using doubled lines and
extend the set of gates with some classical operations such as
$\{\NOT,\XOR,\AND\}$, classical fan-out, bit swaps, measurement, qubit
preparation, and classically controlled versions of the $\Rx[\pi]$ and $\Z[\pi]$
gates. We depict them respectively as follows.

\[\scalebox{0.82}{%
\tikzfig{circuit/not}
\;
\tikzfig{circuit/xor}
\;
\tikzfig{circuit/and}
\;
\tikzfig{circuit/clone}
\;
\tikzfig{circuit/swapC}
\;
\tikzfig{circuit/meas}
\;
\tikzfig{circuit/prepare}
\;
\tikzfig{circuit/notClassic}
\;
\tikzfig{circuit/zClassic}

Circuits are inductively constructed from these generators, wire identities, and
parallel and serial composition, ensuring that only wires of the same type
connect with each other.

In this work we restrict the input to circuits with classical parity logic,
choosing not to include AND gates due to the complexity of their representation
as  diagrams, which might result in the introduction of additional
non-Clifford gates during the extraction procedure (refer to
Section <ref> for further discussion).

The ZX-calculus is a formal graphical language which provides a fine-grained
representation of quantum operations. We present a brief introduction to its
definition, including the  extension to represent classical operations.
Refer to [23] for a complete description of both

ZX diagrams representing pure-quantum linear maps are composed by
wires, spiders, and Hadamard boxes. We read the
diagrams from right to left and represent inputs and outputs as open-ended

The Hadamard box $\tikzfig{zxGnd/elem/hadamard}$ swaps the computational and
diagonal basis, mapping $\ket{0}$ to $\ket{+}$, $\ket{1}$ to $\ket{-}$ and vice

The spiders are arbitrary-degree nodes labelled with a real phase $\alpha \in
[0,2\pi)$ that come in either green or red color, named Z- and X-spiders

When $\alpha$ is a multiple $\pi$ or $\frac\pi2$, we call them Pauli
or Clifford-spiders respectively.

We refer to the set of spiders connected to outputs and inputs of
the diagram as $\outputs$ and $\inputs$ respectively, and call their members
output- and input-spiders.

A degree-2 green (resp. red) spider corresponds to applying a
[α] ([α]) operation over a qubit.  Phaseless spiders represents
nodes with phase $0$ and can be interpreted as copying the computational basis
vectors in the case of green spiders, or the diagonal basis vectors for red

\[\tikzfig{zxGnd/elem/spiderZ-2-3}\]

Spiders of the same color can be fused together, adding their phases. It is
important to note that the relative position of the nodes in ZX diagrams do
not alter their interpretation, as only the topology matters.

The ZX-calculus comes equipped with a complete set of formal rewrite
rules [13]. We reproduce it here ignoring scalars.

\[\scalebox{0.82}{\tikzfig{zxGnd/rule/zxRules}}\]

The  calculus [3] is an extension to the
ZX-calculus which is able to easily describe interactions with the environment.
The diagrams have a standard interpretation as completely positive linear
maps between quantum mixed states (cf. Appendix <ref> for a
formal description).

In addition to the ZX generators and rewrite rules, the calculus introduces a
ground generator ($\ground$) which represent the tracing operation, or
the discarding of information. When connected to a degree-3 green spider, this
can correspond to a measurement operation over the computational basis or a
qubit initialization from a bit.

\[
\tikzfig{circuit/meas}
\quad\sim\quad
\tikzfig{zxGnd/elem/spiderZgnd-1-1}
\quad\sim\quad
\tikzfig{circuit/prepare}
\]

We refer to the spiders attached to generators as -spiders.

Notice that we use the same kind of wire for both classical and quantum data,
since as previously discussed we can encode the latter as the former.
We will later introduce a method to differentiate between the two types of wire
by using the -spiders in Section <ref>.

extends the set of rewriting rules with the following additions.

\[
\tikzfig{zxGnd/rule/numberRule-a}
\;\overset{\mbox{\scriptsize\GndRemoveRule}}{=}\;
\tikzfig{unit-diagram}
\qquad
\tikzfig{zxGnd/rule/HRule-a}
\;\overset{\mbox{\scriptsize\GndHadamardRule}}{=}\;
\tikzfig{zxGnd/rule/HRule-b}
\qquad
\tikzfig{zxGnd/rule/MergeRule-a}
\;\overset{\mbox{\scriptsize\GndDiscardRule}}{=}\;
\tikzfig{zxGnd/rule/MergeRule-b}
\qquad
\tikzfig{zxGnd/rule/CNOTRule-a}
\;\overset{\mbox{\scriptsize\GndCNOTRule}}{=}\;
\tikzfig{zxGnd/rule/CNOTRule-b}
\]

Intuitively, the  generator discards any operation applied over a single

Multiple discards can be combined into one vio the following rule, derived from
rules , , and .

\[
\tikzfig{zxGnd/rule/DoubleRule-a}
\;\overset{\mbox{\scriptsize\GndDoubleRule}}{=}\;
\tikzfig{zxGnd/rule/DoubleRule-b}
\]

For simplicity in our diagrams, we replace solely as notation the Hadamard
boxes with “Hadamard wires” drawn in blue, as follows.

\[ \tikzfig{zxGnd/elem/hWire} \;=\; \tikzfig{zxGnd/elem/hWireDef} \]

We introduce two additional derived equations. One to erase duplicated Hadamard
wires, as proven by Duncan et al. [8], and another to
discard them, from a combination of rules  and .

\[
\tikzfig{zxGnd/rule/DoubleH-a}
\;\overset{\mbox{\scriptsize\ParHRule}}{=}\;
\tikzfig{zxGnd/rule/DoubleH-b}
\qquad\qquad
\tikzfig{simpl/gnd-rewriting/gndDiscard-a}
\;\overset{\mbox{\scriptsize\GndHDiscardRule}}{=}\;
\tikzfig{simpl/gnd-rewriting/gndDiscard-b}
\]

We utilise !-box notation [20] to represent
infinite families of diagrams with segments that can be repeated 0 or more times.
In the following sections it will be useful to use this notation for depicting
more complex diagrams. Here we present an example of its usage.

\[
\scalebox{0.85}{\tikzfig{zxGnd/bangBox}}
\;\in\;\{
\scalebox{0.85}{\tikzfig{zxGnd/bangBox-0}}
\scalebox{0.85}{\tikzfig{zxGnd/bangBox-1}}
\scalebox{0.85}{\tikzfig{zxGnd/bangBox-2}}
\dots
\}
\]

§ GRAPH-LIKE DIAGRAMS AND FOCUSED GFLOW

A ZX diagram is said to be in graph-like
form [8] when it contains only
Z-spiders connected by Hadamard wires, there are no parallel edges nor
self-loops, and no spider is connected to more than one input or output.
We define the graph-like form for  diagrams and include a weaker version
allowing a node to connect to an input, a ground, and any number of outputs
simultaneously. When defining a translation from quantum circuits into diagrams it will be simpler to initially generate weakly graph-like diagrams and
transform the final result into the strict version afterwards.

A diagram is graph-like (respectively
weakly graph-like) when:

* All spiders are Z-spiders.

* Z-spiders are only connected via Hadamard edges.

* There are no parallel Hadamard edges or self-loops.

* There is no pair of connected -spiders.

* Every input, output, or is connected to a Z-spider.

* Every Z-spider connected to a has phase 0.

* Every Z-spider is connected to
at most one input, one output, or one (at most one input and at most one ).

Every diagram is equivalent to a weakly graph-like diagram.  Indeed,
Duncan et al. [8] proved that any pure-ZX diagram is
equivalent to a graph-like one. The proof can be extended to weakly graph-like
diagrams simply by applying rule $\GndHadamardRule$ to eliminate
Hadamards connected to generators, rule $\GndDoubleRule$ to eliminate
duplicated connected to the same spider, and rule $\GndCNOTRule$ to
disconnect wires between -spiders.

There exists an algorithm to transform an arbitrary diagram into an
equivalent strictly graph-like diagram.

By adding identity spiders to the inputs and outputs. Cf.
long version on arXivAppendix <ref>.

Once a diagram is in a weakly graph-like form, all its spiders as well as all
its internal connections are of the same kind. We can refer to its underlying
structure as a simple undirected graph, marking the nodes connected to inputs
and outputs. In addition, generators or the -spiders connected to
them can be seen as outputs discarding information into the environment.  This
is known as the underlying open graph of a diagram.

An open graph is a triple $(G,S,T)$ where $G = (V,E)$ is an
undirected graph, and $S \subseteq V$ is a set of sources and $T
\subseteq V$ is a set of sinks.  For a weakly graph-like
diagram $D$, the underlying open graph $G(D)$ is the open
graph whose vertices are spiders $D$, whose edges correspond to Hadamard
edges, whose set $S$ is the subset of spiders connected to the inputs of $D$,
and whose set $T$ is the subset of spiders connected to the outputs of $D$ or
to ground generators.

The underlying open graph of a ZX diagram produced from our translation of
quantum circuits verify a graph-theoretic invariant called
focused gFlow [19]. This
structure —originally conceived for graph states in measurement based quantum
computation— gives a notion of flow of information and time on the diagram.
It will be required to guide the extraction strategy in
Section <ref>.

Given an open graph $G$, a focused gFlow $(g,\prec)$ on $G$ consists of a function
$g : \overline{T} \to 2^{\overline S}$ and a partial order $\prec$ on the vertices $V$ of $G$
such that for all $u \in \overline{T}$, $\odd_G(g(u)) \cap \overline{T} = \{u\}$
and $\forall v \in g(u), u \prec v$
where $2^{\overline S}$ is the powerset of $\overline{S}$ and
$\odd_G(A) \coloneqq \{v \in V(G) \;|\; |N(v) \cap A| \equiv 1 \text{ mod } 2\}$
is the odd neighbourhood of $A$.

§ TRANSLATION OF HYBRID QUANTUM-CLASSICAL CIRCUITS

We describe our translation from hybrid quantum-classical circuits into strictly
graph-like  diagrams by steps. First, we translate each individual gate
directly into a weakly graph-like diagram and connect them with regular wires.
We define this translation $\toZxGND{\cdot}$ by inductively translating the
gates as described in Table <ref> immediately followed
by the application of the spider fusion rule $\SpiderRule$ and
rules $\GndDoubleRule$ and $\ParHRule$ to remove
all regular wires, duplicated generators, and parallel Hadamard wires,
ensuring that the final combined diagram is in a weakly graph-like form.
An example of this translation is shown in Figure <ref>.

Notice that the translation maps both classical and quantum wires to regular
diagram edges.  We keep track of which inputs and outputs of the diagram
were connected to classical wires and introduce generators for the
operations that interact with the environment.

In Section <ref> we present a method to detect the sections of
the final circuit that can be implemented as classical operations by looking at
the classical inputs/outputs and the generators, independently of which
wires where originally classical.

\begin{gather*}%
\tikzfig{circuit/cnot}
\;\mapsto\;%
\tikzfig{circuit/cnotZX}
\qquad
\tikzfig{circuit/hadamard}
\;\mapsto\;%
\tikzfig{circuit/hadamardZX}
\qquad
\tikzfig{circuit/z}
\;\mapsto\;%
\tikzfig{circuit/zZX}
\\
\tikzfig{circuit/init}
\;\mapsto\;%
\tikzfig{circuit/initZX}
\qquad
\tikzfig{circuit/termination}
\;\mapsto\;%
\tikzfig{circuit/terminationZX}
\qquad
\tikzfig{circuit/xor}
\;\mapsto\;%
\tikzfig{circuit/xorZX}
\\
\tikzfig{circuit/not}
\;\mapsto\;%
\tikzfig{circuit/notZX}
\qquad
\tikzfig{circuit/clone}
\;\mapsto\;%
\tikzfig{circuit/cloneZX}
\qquad
\tikzfig{circuit/meas}
\;\mapsto\;%
\tikzfig{circuit/measZX}
\qquad
\tikzfig{circuit/prepare}
\;\mapsto\;%
\tikzfig{circuit/prepareZX}
\\
\tikzfig{circuit/notClassic}
\;\mapsto\;%
\tikzfig{circuit/notClassicZX}
\qquad
\tikzfig{circuit/zClassic}
\;\mapsto\;%
\tikzfig{circuit/zClassicZX}
\\
\tikzfig{circuit/wireC}
\;\mapsto\;%
\tikzfig{circuit/wireZX}
\qquad
\tikzfig{circuit/wireQ}
\;\mapsto\;%
\tikzfig{circuit/wireZX}
\qquad
\tikzfig{circuit/swapC}
\;\mapsto\;%
\tikzfig{circuit/swapZX}
\qquad
\tikzfig{circuit/swapQ}
\;\mapsto\;%
\tikzfig{circuit/swapZX}
\\
\tikzfig{circuit/serialComp}
\;\mapsto\;%
\tikzfig{circuit/serialCompZX}
\qquad\qquad
\tikzfig{circuit/parallelComp}
\;\mapsto\;%
\tikzfig{circuit/parallelCompZX}
\end{gather*}

[Translation from hybrid quantum-classical circuits into grounded ZX diagrams.]
Translation from hybrid quantum-classical circuits into diagrams.

\[
\scalebox{0.85}{\tikzfig{examples/superdenseCoding-circuit}}
\quad\mapsto
\scalebox{0.85}{\tikzfig{examples/superdenseCoding-zx}}
\]

Example translation of the superdense coding circuit into a
zx-ground diagram with labelled inputs and outputs, and subsequent
application of the spider-fusion rule.
Example translation of the superdense coding circuit into a
diagram with labelled inputs and outputs, and subsequent
application of the spider-fusion rule.

The diagram resulting from the translation $\toZxGND{\cdot}$ is weakly

By induction on the circuit construction. Cf.
long version on arXivAppendix <ref>.

After the translation, we can apply Lemma <ref> to
obtain a strictly graph-like diagram.
This step essentially separates the generators from the inputs and
outputs, allowing the optimization procedure to move them around and let them
interact with other parts of the diagram.

If $C$ is a hybrid quantum-classical circuit and $D$ is the graph-like
diagram obtained from the translation $\toZxGND{C}$ and
Lemma <ref>, then $G(D)$ admits
a focused gFlow.

By induction on $C$. Cf.
long version on arXivAppendix <ref>.

§ GROUNDED ZX OPTIMIZATION

Our simplification strategy for  diagrams is based on eliminating nodes
from the diagram by systematically applying a number of rewriting rules while
preserving the existence of a focused gFlow. In this section we introduce the
new rules, define a strategy to maximize their effectiveness, and finally use
it together with the pure-ZX optimization to define our algorithm.

§.§ Basic simplification rules

Duncan et al. [8] presented the following gFlow-preserving
local complementation and  pivoting rules for the ZX calculus
in their optimization procedure. These rules effectively reduce the size of the
diagram by at least one node on each application by eliminating internal
proper-Clifford spiders and Pauli spider pairs respectively.

\[
\hfill \scalebox{0.82}{\tikzfig{simpl/lc-simp}} \hfill
\]

\[
\scalebox{0.82}{\tikzfig{simpl/pivot-a}}
\quad\overset{\PivotRule}{=}\quad
\scalebox{0.82}{\tikzfig{simpl/pivot-b}}
\]

These rules can be applied directly in  diagrams when the target spiders
are not connected to a generator. For the cases where some of the target
spiders are -spiders, we introduce the following altered rules.
Their derivation can be found in
the long version on arXivAppendix <ref>.

\[
\scalebox{0.75}{\tikzfig{simpl/gnd-rewriting/gndComplement-a}}
\!\!\!\overset{\LocalCompGndRule}{=}
\scalebox{0.75}{\tikzfig{simpl/gnd-rewriting/gndComplement-b}}
\quad
\scalebox{0.75}{\tikzfig{simpl/gnd-rewriting/gndPivot-a}}
\overset{\PivotGndRule}{=}
\scalebox{0.75}{\tikzfig{simpl/gnd-rewriting/gndPivot-b}}
\]

\[
\scalebox{0.82}{\tikzfig{simpl/gnd-rewriting/gndDeletion-a}}
\quad\overset{\PauliPivotGndRule}{=}\quad
\scalebox{0.82}{\tikzfig{simpl/gnd-rewriting/gndDeletion-b}}
\]

Notice that both rules $\LocalCompGndRule$ and $\PivotGndRule$ do not decrease
the number of spiders in the diagram. As such, we will focus on rule
$\PauliPivotGndRule$ for our optimization.

If  is applied with a non-spider connected to a
boundary, the rule produces a -spider connected to an input or output
thus needing to add an identity operation as described in
Lemma <ref> to preserve the graph-like property.
Since in this case we add additional nodes to the graph, we will only apply rule
on a boundary spider if it can be followed by another
node-removing rule.

Additionally, we will use rules $\GndHDiscardRule$ and $\GndRemoveRule$ directly
to remove nodes in the diagram when there are -spiders with degree 1 or
0 in the graph, respectively.

If the non-spider in the lhs of the discarding rule   is not connected to an output or input, then applying the rule over a
graph-like diagram $D$ preserves the existence of a focused gFlow.

If the non-spider in the lhs is not connected to an input or output
of the diagram, then applying the rule does not break the graph-like
property of $D$. The preservation of the gFlow follows from -spiders
being sinks of the underlying open graph.

Rules , , and  preserve the
existence of a focused gFlow.

Notice that that rules , ,
and  are compositions of gFlow-preserving rules.

§.§ Ground-cut simplification

The previously introduced rewriting rules require a simplification strategy to
apply them. A simple solution is to try to find a match for each rule
and apply them iteratively until no more matches are available. We describe a
strategy that can find additional rule matches by operating on the biadjacency
matrix between the -spiders and the non-spiders.

The ground-cut of a graph-like diagram $D$
is the cut resulting from splitting the and non-spiders in $G(D)$.

Since the diagram is graph-like, there are no internal wires in the partition.

Given a diagram $D$, we denote $M_D$ the biadjacency matrix of its
ground-cut, where rows correspond to -spiders and columns correspond to

We can apply all elementary row operations on the matrix by rewriting the
diagram. The addition operation between the rows corresponding to the
-spider $u$ and the -spider $v$ can be implemented via the
following rule, the derivation of which can be found in
Appendix <ref>.

\[
\scalebox{0.85}{\tikzfig{simpl/gnd-row-sum-0}}
\quad\overset{\RowSumGndRule}{=}\quad
\scalebox{0.85}{\tikzfig{simpl/gnd-row-sum-4}}
\]

Using the elementary row operations we can apply Gaussian elimination on the
ground-cut biadjacency matrix of a graph-like  diagram, generating in the
process an equivalent diagram whose ground-cut biadjacency matrix is in
reduced echelon form.

Any row in the ground-cut biadjacency matrix left without non-zero elements
after applying Gaussian elimination corresponds to an isolated
-spiders in the diagram that can be eliminated by rule .

If the reduced row echelon form of the biadjacency matrix contains a row with
exactly one non-zero elements, then that element corresponds to an isolated
-spider and non-spider pair in the diagram and therefore we can
apply rule  to remove the non-spider.

§.§ The Algorithm

Based on the previous strategy, we define a terminating procedure which turns
any graph-like diagram into an equivalent simplified diagram
that cannot be further reduced.

A graph-like diagram is in simplified-form if it does not contain any
of the following, up to single-qubit unitaries on the inputs and outputs.

* Interior proper Clifford spiders.

* Adjacent pairs of interior Pauli spiders.

* Interior Pauli spiders adjacent to boundary spiders.

* Interior Pauli spiders adjacent to -spiders.

* Degree-1 -spiders not connected to input or output spiders.

* Connected components not containing inputs nor outputs.

An example of diagrams satisfying and not satisfying this property is shown
in Figure <ref>

\[\tikzfig{examples/simplified-not}\]

Diagram not in simplified-form

\[\tikzfig{examples/simplified}\]

Diagram in simplified-form

Example of simplified and non-simplified diagrams.

We define an optimization algorithm that produces diagrams in simplified-form by
piggybacking on the pure optimization procedure. This optimization applies the
local complementations  and pivoting  rules until
there are no interior proper Clifford spiders or adjacent pairs of non-interior Pauli spiders.

After the initial pure simplification, we continue our optimization as follows.

Repeat until no rule matches, removing wires between -spiders and
parallel Hadamard connections after each step:

* Run Gaussian elimination on the ground-cut of the diagram
as described in Section <ref>.

Remove the grounds corresponding to null rows with rule .

If any row of the biadjacency matrix has a single non-zero element,
corresponding to a -spider connected to a spider $v$, then:

* If $v$ is not a boundary spider, apply rule .

* If $v$ is a boundary spider and $v$ is adjacent to a Pauli spider,
apply rule  immediately followed by
the procedure from Lemma <ref>
to make the diagram graph-like again.
Then delete the Pauli neighbour using rule ,
to ensure that the step removes at least one node.

* Apply Pauli spider elimination rule
until there are no Pauli spiders connected to ground spiders.

* Remove any connected component of the graph without inputs or outputs.

Notice that each cycle the loop reduces the number of nodes in the graph, so
this is a terminating procedure. Additionally, since each applied rule preserves
the existence of a gFlow the final diagram admits a gFlow. An example run of
the algorithm is shown in Figure <ref>.

\(
\tikzfig{examples/optimizationB-0}
\;\to\;
\tikzfig{examples/optimizationB-1}
\;\to\;
\tikzfig{examples/optimizationB-2}
\;\to\;
\tikzfig{examples/optimizationB-3}
\)

Example of a diagram optimization applying a ground-cut
simplification, a discard rule, and a Pauli elimination.

§ CIRCUIT EXTRACTION

Here we describe a general circuit extraction procedure for graph-like
diagrams admitting a focused gFlow into hybrid quantum-classical
circuits, by modifying the procedure for pure diagrams from the Clifford
optimization. We present the pseudocode in
Algorithm <ref> and an example of an execution in
Appendix <ref>.

The algorithm progresses through the diagram from right-to-left, maintaining a
set of spiders $\frontier$, called the frontier, which represents the
unextracted spiders connected to the extracted segment. Each frontier spider is
assigned an output qubit line $\fQubit(v)$. This set is initially populated by
the nodes connected to outputs of the diagram. The strategy is to proceed
backwards by steps, adding unextracted spiders into the frontier and deleting
some of them to extract operations on the output circuit, in back-to-front

To find candidate spiders to add to the frontier we apply Gaussian elimination
on the biadjacency matrix of the frontier and non-frontier spiders, similarly to
the optimization method described in Section <ref>.
The gFlow property of the graph ensures that we can always progress by
extracting a node.  It suffices to look at the set of non-frontier vertices
maximal in the order and notice that, after the Gauss elimination, either we can
choose a -spider from the set, or a non-spider that has a single
connection to the frontier. A careful implementation of the biadjacency matrix
row and column ordering can reduce the number of -spider extractions when
no non-candidates are available. We require the following proposition
to apply the row additions on the graph
(Duncan et al. [8], Proposition 7.1).

For any diagram $D$, the following equation holds:

\[
\tikzfig{extraction-v2/row-sum-0}
\quad=\quad
\tikzfig{extraction-v2/row-sum-1}
\]

where $M$ describes the biadjacency matrix of the relevant vertices, and
$M'$ is the matrix produced by adding row 2 to row 1 in $M$.
Furthermore, if the diagram on the LHS has a focused gFlow, then so does the

In our pseudocode, the call to CleanFrontier ensures that $\frontier$
only contains phaseless spiders without internal wires. Notice that it preserves
the gFlow since it only modifies edges between sink nodes, and removes spiders
with no other connections.

After the while loop terminates, all outputs of the circuit will have been
extracted. If there are inputs left unextracted, and since the diagram had a
gFlow, we can discard them directly via measurement operations.

Finally, we add any necessary swap operations to map the inputs to the
correct lines, and insert qubit initializations and measurements at
inputs and outputs marked as classical. In Section <ref>
we detail a method to better detect the internal parts of the circuit that can
be implemented classically.

In any case, each step of the while loop in
Algorithm <ref> line <ref>,
preserves the gFlow of the diagram, and we can show that it terminates
in at most $|V|$ steps: Indeed, if there are no non-frontier spiders,
then a call to CleanFrontier will remove all nodes from the
frontier. Moreover, each step of the while loop in
line <ref> moves one non-frontier spider to

Circuit extraction
[1]
$\frontier : Set\langle Node\rangle \gets \outputs$,
$\fQubit : Map\langle Node, int\rangle \gets \emptyset$
$v \in \frontier$
$\fQubit(v) \gets$ Output connected to $v$
CleanFrontier$(\frontier, \fQubit)$
$\frontier \neq \emptyset$
Run Gauss elimination on the frontier biadjacency matrix $M$
(Proposition <ref>)

a row of $M$ has a single non-zero element
Let $u$ and $v$ be the corresponding non-frontier and frontier node
$\fQubit(u) \gets \fQubit(v)$
Remove $v$ from the diagram and add $u$ to $\frontier$
$v \gets$ Arbitrary -spider in the neighbourhood of $\frontier$
$\fQubit(v) \gets$ New qubit line id
Extract a classical bit termination on $\fQubit(v)$ and
add $u$ to $\frontier$
CleanFrontier$(\frontier, \fQubit)$
Unextracted $v \in I$
$\fQubit(v) \gets$ Input connected to $v$
Extract a measurement gate
and a classical bit termination on $\fQubit(v)$
Assign the corresponding input to $\fQubit(v)$

CleanFrontier$\frontier, \fQubit$
$v \in \frontier$
$v$ is a -spider
Remove the , extract a measurement on $\fQubit(v)$
$v$ has a phase $\alpha \neq 0$
Set $\alpha = 0$, extract a [α] gate on $\fQubit(v)$
$u\in\frontier, v \sim u$
Remove the wire, extract a  gate on $\fQubit(v)$, $\fQubit(u)$
$v$ is not connected to any other node
Remove $v$
$v \in \inputs$
assign the input to qubit $\fQubit$

extract a $\ket{+}$ qubit initialization on $\fQubit(v)$

§ CIRCUIT CLASSICALIZATION

The extraction procedure described in Section <ref> produces
correct circuits that are almost completely composed by quantum gates and wires,
without any classical operation.
In this section we describe the general problem of detecting parts of the
circuits that can be realized as classical operations, and introduce an
efficient heuristic solution based on a local-search.

Notice, however, that while we aim to recognize all classically realizable
operations in the circuit the characteristics of each quantum computer may
dictate the final choice between quantum and classical operators by taking into
account the costs of exchanging data between both realms.

Given a diagram, we decorate its wires using the set of labels $\labels =
\{\Q, \X, \Y, \Z, \bot\}$. The label $\Z$ means that this particular wire can be
replaced by a classical wire (possibly precomposed with a standard basis
measurement and postcomposed by a qubit initialisation in the standard basis
depending on whether the connected wires are also classical or not), and similarly for
$\X$ and $\Y$ by adapting the basis of measurement/initialisation to the diagonal
or $Y$ basis. $\Q$ means that the wire is a quantum wire, and finally $\bot$
means that the wire can be removed by precomposing with a and postcomposing
with a maximally mixed state.  The set of labels form a partial order, $\Q \ge
\X,\Y,\Z \ge \bot$.

A labelling $L$ of a diagram $D$ is a map from its edges into a pair of labels.
The two labels, drawn at each end of the wire, indicate the origin of the
constraint. Intuitively, $\tikzfig{classical/id-za}$ means that the wire is
produced in such a way that guarantees that the qubit carries classical
information encoded in the computational basis, whereas
$\tikzfig{classical/id-az}$ means that the wire can be replaced by a classical
wire because some process will force this qubit to be in that basis —for
instance, it is going to be measured in the standard basis and thus one can
already measure this qubit in the standard basis and use a classical wire—.
We define a partial order between labellings of a diagram as the natural lift
from the partial order of the labels.

Each label corresponds to a density matrix subspace of $\C^{2\times2}$,
representing all possible mixed states allowed by that particular kind of wire.

\[
\begin{array}{ll}
\interpretLbl{\Q} = \C^{2\times2}
\interpretLbl{\Z} =%
\{\alpha \ketbra{0}{0} + \beta \ketbra{0}{0} \mid%
\alpha,\beta\in\R_{\geq0}, \alpha+\beta=1\}
\\
\interpretLbl{\bot} = \{\frac12 \ketbra00 + \frac12 \ketbra11\}
\quad
\interpretLbl{\X} =%
\{\alpha \ketbra{+}{+} + \beta \ketbra{-}{-} \mid%
\alpha,\beta\in\R_{\geq0}, \alpha+\beta=1\}
\\
\interpretLbl{\Y} =%
\{\alpha \ketbra{\circlearrowright}{\circlearrowright} + \beta
\ketbra{\circlearrowleft}{\circlearrowleft} \mid%
\alpha,\beta\in\R_{\geq0}, \alpha+\beta=1\}
\end{array}%
\]

Notice that the greatest common ancestor $\interpretLbl{A \sqcup B}$ corresponds
to the intersection of the sets.

Intuitively, a labelling is valid if we can cut any wire in the diagram and,
after forcing a valid state in the inputs and outputs, we get a valid state in
the cut terminals. That is, we rearrange the diagram to transform all outputs
into inputs and connect the cut terminals as outputs, as shown on the right.
Then, applying an arbitrary input
\(
\rho \in (\bigotimes_i^n \interpretLbl{A_i})
\otimes(\bigotimes_j^m \interpretLbl{D_j})
\)
to the diagrams produces a result in $\interpretLbl{E}\otimes\interpretLbl{F}$.

Notice that if $A$ is a valid label for a wire then any $B \geq A$ is also
valid, and in particular  is always a valid label. We can then omit
unnecessary labels in the diagrams, marking them implicitly as .

Given a diagram $D$ with marked classical inputs and outputs,
we define the classicalization problem as finding a minimal valid labeling
where the inputs and outputs are labelled as  or  accordingly.

§.§ Local-search algorithm

We present a local-search labelling procedure for  diagrams with explicit
Hadamard gates —replacing the Hadamard wires— and only green spiders, that
produces locally minimal labellings by propagating the labels over individual

A diagram resulting from the circuit extraction in Section <ref>
can be transformed to have only green spiders by applying the color-change rule
. This restriction is purely for simplicity in our definition, as
the equivalent functions can be defined easily for red spiders.

We introduce a number of operations over the labels.
First, a binary function representing the result of combining two wires via a
phaseless green spider, $\tJoin:\labels\times\labels\to\labels$.

\[
\begin{array}{lllll}
\Z\tJoin A=Z \quad & \X\tJoin A=A \quad & \Y\tJoin\Y=\X \quad & \Q\tJoin\Y = \Q \quad & \bot\tJoin\Y = \bot \\
A\tJoin\Z=Z & A\tJoin\X=A & \Y\tJoin\Q=\Q & \Q\tJoin\Q=\Q & \bot\tJoin\Q=\Z \\
& & \Y\tJoin\bot=\bot & \Q\tJoin\bot=\Z & \bot\tJoin\bot=\bot \\
\end{array}
\]

Notice that $(\labels, \tJoin)$ is a commutative monoid with  as neutral
We also define a “Z rotation" operation for $\alpha\in[0,2\pi)$,
$\rot: \labels \to \labels$.

\[
\begin{array}{llllll}
\multicolumn{2}{l}{\rot(\Z) = \Z \qquad\qquad} &
\multicolumn{2}{l}{\rot(\Q) = \Q \qquad\qquad} &
\multicolumn{2}{l}{\rot(\bot) = \bot} \\
\multicolumn{3}{l}{%
\rot(\X) =
\begin{cases}
\X & \text{if $\alpha \in \{0,\pi\}$} \\
\Y & \text{if $\alpha \in \{\frac12\pi,\frac34\pi\}$} \\
\Q & \text{otherwise}
\end{cases}
\qquad
} &
\multicolumn{3}{l}{%
\rot(\Y) =
\begin{cases}
\Y & \text{if $\alpha \in \{0,\pi\}$} \\
\X & \text{if $\alpha \in \{\frac12\pi,\frac34\pi\}$} \\
\Q & \text{otherwise}
\end{cases}
\end{array}
\]

This corresponds to the identity if $\alpha\in\{0,\pi\}$, and in general
\(
\rot(A) \tJoin b \geq \rot(A \tJoin B)
\).

Finally, we define a function $\hadamLbl$ representing the application of the
Hadamard operation over a label, $\hadamLbl: \labels \to \labels$.

\[
\hadamLbl(\Q) = \Q
\qquad
\hadamLbl(\X) = \Z
\qquad
\hadamLbl(\Z) = \X
\qquad
\hadamLbl(\Y) = \Y
\qquad
\hadamLbl(\bot) = \bot
\]

Our classical detection procedure starts by labelling any classical input or
output with a $\Z$ label, and any with a $\bot$ label, and the rest of
the diagram wires with $\Q$.

It then proceeds by propagating the labels using the
following rules:

\[
\tikzfig{classical/had-rule0}
\;\;\overset{\mbox{\scriptsize\ClassHadamRule}}{\to}\;\;
\tikzfig{classical/had-rule1}
\]

\[
\tikzfig{classical/Zn-p-rule0}
\;\;\overset{\mbox{\scriptsize\ClassZRule}}{\to}\;\;
\tikzfig{classical/Zn-p-rule1}
\]

For any labels $A,B,C,D,E,F\in\labels$.

We apply these rules until there are no more labels to change. Since each time we
replace labels with lesser ones in the order, the procedure terminates.
Finally, we can interpret wires with a classical label in any direction as
classically realisable.

We show an example of a labeled diagram in
Figure <ref>.

\[
\tikzfig{examples/classicalization-start-circ}
\;\sim\;
\tikzfig{examples/classicalization-labelled}
\;\sim\;
\tikzfig{examples/classicalization-labelled-circ}
\]

Example of a local-search classicalization.
labels are omitted.

The local-search labelling algorithm produces a valid labeling according to
the standard interpretation of the  calculus.

By proving that both rules  and  produce valid
labellings, cf.
long version on arXivAppendix <ref>.

The local-search labelling does not always reach a globally minimum labeling.
Consider for example the following input diagram.

\[
\tikzfig{classical/cnot-cnot-}
\]

Here, although the diagram is equivalent to the identity and the outputs could
theoretically be labelled as $\Z$, the entangled wires in the middle do not
allow for local propagation of the classical labels past the double controlled
Z gates.

§ IMPLEMENTATION

We have implemented each of the algorithms presented in this work as an
extension to the open source Python library
PyZX [16] by modifying its implementation of ZX
diagrams to admit  primitives. A repository with the code is
available at <http://github.com/aborgna/pyzx/tree/zxgnd>.

We additionally implemented a naïve  extension of the pure Clifford
optimization for comparison purposes, which doesn't use any of our rewriting rules.

When applied to pure quantum circuits, our algorithm does not perform additional
optimizations after the Clifford procedure and therefore achieves the same
benchmark results recorded by Duncan et al. on the circuit set described by Amy et
al. [1].

We tested the procedure over two classes of randomly generated circuits, and
measured the size of the resulting diagram as the number of spiders left after
the optimization. This metric correlates with the size of the final circuit,
although the algorithmic noise caused by the arbitrary choices in the extraction
procedure may result in some cases in bigger extracted circuit after a reduction

The first test generates Clifford+T circuits with measurements by applying randomly
chosen gates from the set {, S, HSH, T, Meas} over a fixed number of
qubits, where Meas are measurement gates on a qubit immediately followed by a
qubit initialization.  We fix the probability of choosing a , S, or HSH
gate to $0.2$ each and vary the probabilities for T and Meas in the remaining
$0.4$. These circuits present a general worst case, where there is no
additional classical structure to exploit during the hybrid circuit

The second type of generated operations are classical parity-logic circuits.
These consist on a number of classical inputs, fixed at 10, where we apply
randomly chosen operations from the set {NOT, XOR, Fanout} with probabilities
0.3, 0.3, and 0.4 respectively.

In Figure <ref> we compare the results of our optimization using
the Clifford optimization as baseline.

Figure <ref> shows the reduction of diagram size when
running the algorithm on randomly generated Clifford+T circuits with
measurement. We vary the probability of generating a measurement gate between
$0$ and $0.2$ while correspondingly changing the probability of generating a
T-gate between $0.4$ and $0.2$, and show the results for different combinations
of qubit and gate quantities.

We remark that the optimization produces noticeably smaller diagrams once enough
generators start interacting with each other. There is a critical
threshold of measurement gate probability, specially visible in the cases with 8
qubits and 1024 gates, where with high probability the outputs of the diagram become
disconnected from the inputs due to the interactions. This results in
our algorithm optimizing the circuits to produce a constant result while
discarding their input.

Figure <ref> shows the comparison of diagram size between
our procedure and the Clifford optimization when run over classical parity

The optimization produces consistently smaller diagrams, generally
achieving the theoretical minimal number of generators, equal to the
number of inputs.

We further remark that in all of the tested cases the classicalization procedure
was able to detect that all the extracted operations on the optimized
parity-logic circuit were classically realisable.

Diagram size reduction on Clifford+T circuits with

Diagram size reduction on parity-logic circuits.

Benchmark results on randomly generated diagrams.

The runtime of our algorithm implementation is polynomial in the size of the
circuit. As with the Clifford optimization, the cost of our optimization and
extraction processes is dominated by the Gauss elimination steps.
For the ground-node rewriting rules, our unoptimized implementation is roughly
$\bigo(n^2 * k^2)$ in the worst case with $k$ being the number of measurement gates and $n$ the number of gates,
but in practice it behaves cubically on the number of gates due to the sparseness of the diagrams.
The implementation was not developed with a focus on the runtime cost, and some possible optimizations may reduce this bound.

§ DISCUSSION AND FUTURE WORK

We introduced an optimization procedure for optimizing hybrid classical circuits
inspired by previous work on pure circuit optimization using the ZX calculus.
The process is composed by a translation step, the optimization of the diagrams,
an extraction back into circuits and finally a detection of
classically-realisable operations.

Our translation operation produces diagrams which admit a focused gFlow, a
property that we maintain during the optimization and require during the

For our optimization step we defined a series of rewrite strategies to reduce
the size of the diagrams, and introduced a strategy to find additional
optimization opportunities by applying Gaussian elimination on the biadjacency
matrix of the ground-cut of the diagram. Our extraction procedure initially
generates circuits without classical operations. Hence, we introduced a
classicalization heuristic for arbitrary circuits that is able to replace
quantum operations by their classical equivalent, where possible.

Kissinger and van de Wetering [17] defined a
procedure based on the Clifford optimization to reduce the T-gate count in
quantum circuits by defining new structures in the graphs called phase
gadgets and operating over their phases. Their work can be easily extended to
, where the generators act as an absorbing element for the
gadgets phases. However, rules such as  prove to be strictly
more powerful than applying the pure phase gadget rules over -gadgets.

In general, the phase-gadget optimization affects an independent section of the
structure of the diagram compared to ours, and can be applied  with it.

During our definition of the optimization process we decided to restrict the
input circuits to parity classical logic, excluding  and  gates. This
does not raise from an inherent limitation of the system but from a practical
standpoint.  The  calculus is able to represent  operations in what
equates to the Clifford+T decomposition of the Toffoli gate, introducing
multiple T-gates and  gates to the circuit [21]. The
multiple spiders would be dispersed around the diagram during the optimization
step, potentially breaking the pattern formed by the  gate and replacing it
with multiple quantum operations. This can produce the unexpected result of
introducing expensive quantum operations in an originally pure classical

A possible next step for this work would be to use an alternative diagrammatic
representation better adapted to represent arbitrary boolean circuit such as the
ZH calculus.

The authors would like to thank Kostia Chardonnet and Renaud Vilmart for their
suggestions on the classicalization problem, and John van de Wetering for his
help with the pyzx library.

This work was supported in part by the French National Research Agency (ANR)
under the research project SoftQPRO ANR-17-CE25-0009-02, and by the DGE of the
French Ministry of Industry under the research project PIA-GDN/QuantEx
P163746-484124, and by the project STIC-AmSud project Qapla' 21-SITC-10.

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[1]
Amy, M., Maslov, D., Mosca, M.: Polynomial-time T-depth optimization of
Clifford+T circuits via matroid partitioning. IEEE T. Comp. Aid. D.
33(10),  1476–1489 (2014).

[2]
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lattice surgery. Quantum  4,  218 (2020).

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Carette, T., Jeandel, E., et al.: Completeness of
graphical languages for mixed states quantum mechanics. In: Proc. ICALP. pp.
108:1–108:15 (2019).

[4]
Chancellor, N., Kissinger, et al.: Graphical
structures for design and verification of quantum error correction (2016),

[5]
Coecke, B., Duncan, R.: Interacting quantum observables. In: Proc. ICALP. pp.
298–310 (2008).

[6]
Coeke, B., Kissinger, A.: Picturing Quantum Processes.

[7]
Cross, A.W., Bishop, L.S., et al.: Open quantum assembly
language (2017).

[8]
Duncan, R., Kissinger, et al.: Graph-theoretic
simplification of quantum circuits with the zx-calculus. Quantum  4, 279 (2020).

[9]
Duncan, R., Lucas, M.: Verifying the Steane code with Quantomatic. In:
Proc. QPL. pp. 33–49 (2013).

[10]
Duncan, R., Perdrix, S.: Rewriting measurement-based quantum computations with
generalised flow. In: Proc. ICALP. pp. 285–296 (2010).

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Green, A.S., Lumsdaine, et al.: Quipper: a
scalable quantum programming language. In: Proc. PLDI. pp. 333–342 (2013).

[12]
Heyfron, L., Campbell, E.T.: An efficient quantum compiler that reduces
T count. (2018)

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Jeandel, E., Perdrix, S., Vilmart, R.: Completeness of the ZX-calculus.
LMCS, 16(2), pp.11-1 (2020).

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Jozsa, R.: An introduction to measurement based quantum
computation. NATO Science Series, III, vol. 199, pp. 137–158 (2005).

[15]
Khammassi, N., Guerreschi, et al.: cqasm v1.0: Towards a common quantum assembly language (2018).

[16]
Kissinger, A., van de Wetering, J.: PyZX: Large scale automated
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#  Theory and phenomenology of the three-gluon vertex

J. Papavassiliou  Department of Theoretical Physics and IFIC, University of
Valencia and CSIC, E-46100, Valencia, Spain  A. C. Aguilar  University of
Campinas - UNICAMP, Institute of Physics “Gleb Wataghin”, 13083-859 Campinas,
São Paulo, Brazil  M. N. Ferreira  University of Campinas - UNICAMP, Institute
of Physics “Gleb Wataghin”, 13083-859 Campinas, São Paulo, Brazil

###### Abstract

The three-gluon vertex is a fundamental ingredient of the intricate QCD
dynamics, being inextricably connected to key nonperturbative phenomena, such
as the emergence of a mass scale in the gauge sector of the theory. In this
presentation we review the main theoretical properties of the three-gluon
vertex in the Landau gauge, obtained from the fruitful synergy between
functional methods and lattice simulations. We pay particular attention to the
manifestation and origin of the infrared suppression of its main form factors
and the associated zero crossing. In addition, we discuss certain
characteristic phenomenological applications that require this special vertex
as input.

##  I Introduction

The three-gluon vertex plays a pivotal role in the structure and dynamics of
Yang-Mills theories, reflecting their non-Abelian nature in the form of the
gauge boson self-interaction that it induces Marciano and Pagels (1978); Ball
and Chiu (1980); Davydychev  _et al._ (1996). In fact, the most preeminent
perturbative property of these theories, namely asymptotic freedom Gross and
Wilczek (1973); Politzer (1973), is intimately linked to the action of this
vertex.

In recent years, the QCD community has been gradually unveiling the rich
infrared facets of this vertex, which are instrumental to a wide array of
nonperturbative phenomena; for a representative set of references, see Alkofer
_et al._ (2005); Huber  _et al._ (2012); Pelaez  _et al._ (2013); Aguilar  _et
al._ (2014); Blum  _et al._ (2014); Eichmann  _et al._ (2014); Mitter  _et
al._ (2015); Williams  _et al._ (2016); Blum  _et al._ (2015); Cyrol  _et al._
(2016); Corell  _et al._ (2018); Boucaud  _et al._ (2017); Huber (2020);
Aguilar  _et al._ (2019a, 2020a, b); Parrinello (1994); Alles  _et al._
(1997); Parrinello  _et al._ (1998); Boucaud  _et al._ (1998); Cucchieri  _et
al._ (2006, 2008); Athenodorou  _et al._ (2016); Duarte  _et al._ (2016);
Boucaud  _et al._ (2017); Vujinovic and Mendes (2019). Several of these works
have underscored the subtle interplay of the three-gluon vertex with the two-
point sector of the theory, and in particular the mass-generating patterns
associated with the gluon and ghost propagators Cornwall (1982); Alkofer  _et
al._ (2009, 2010); Aguilar  _et al._ (2008); Huber  _et al._ (2012); Pelaez
_et al._ (2013); Aguilar  _et al._ (2014); Blum  _et al._ (2014, 2015);
Eichmann  _et al._ (2014); Vujinovic  _et al._ (2014); Mitter  _et al._
(2015); Williams  _et al._ (2016); Cyrol  _et al._ (2016). As a result, the
three-gluon vertex provides an outstanding testing ground for a variety of
physical ideas and field-theoretic mechanisms Aguilar  _et al._ (2021a);
Eichmann  _et al._ (2021); Eichmann and Pawlowski (2021); Meyers and Swanson
(2013); Souza  _et al._ (2020); Huber  _et al._ (2021). In this presentation
we provide a synopsis of some of the most important findings of this
exploration.

The outline of this contribution is as follows. In Sec. II we introduce the
notation and comment on the general properties of the three-gluon vertex, give
one of its standard tensorial decompositions, and report the Slavnov-Taylor
identity (STI) that is satisfies. In Sec. III we discuss the three main
nonperturbative approaches used in the scrutiny of the three-gluon vertex,
namely functional methods, lattice simulations and STI-based constructions.
Next, in Sec. IV we analyse in some detail one of the most exceptional
nonperturbative features of the three-gluon vertex, namely the suppression of
its predominant form factors for Euclidean momenta comparable to the
fundamental QCD scale, and the associated logarithmic infrared divergence at
the origin. In Sec. V we discuss two phenomenological applications of the
three-gluon vertex, namely (a) the effective charge obtained from it, and (b)
its impact on the computation of the mass of the pseudoscalar glueball.
Finally, in Sec. VI we summarize our conclusions.

##  II General properties

We work in the Landau gauge, where the gluon propagator,
$\Delta^{ab}_{\mu\nu}\left(q\right)=-i\delta^{ab}\Delta_{\mu\nu}(q)$, is fully
transverse, i.e.,

$\Delta_{\mu\nu}\left(q\right)=P_{\mu\nu}(q)\Delta(q^{2})\,,\quad\Delta(q^{2})=\mathcal{Z}(q^{2})/{q^{2}}\,,$
(1)

where $P_{\mu\nu}(q):=g_{\mu\nu}-{q_{\mu}q_{\nu}}/{q^{2}}\ $ is the usual
transverse projector, and $\Delta(q^{2})$ the scalar component of the gluon
propagator. In addition, we have defined the gluon dressing function, denoted
by $\mathcal{Z}(q^{2})$.

It is also convenient to introduce the ghost propagator,
$D^{ab}(q^{2})=i\delta^{ab}D(q^{2})$, related to its dressing function,
$F(q^{2})$, by

$D(q^{2})=F(q^{2})/{q^{2}}\,.$ (2)

The full three-gluon vertex will be denoted by
${\rm{I}}\\!\Gamma^{abc}_{\alpha\mu\nu}(q,r,p)=gf^{abc}{\rm{I}}\\!\Gamma^{\alpha\mu\nu}(q,r,p)$,
and is represented in Fig. 1, with $q+p+r=0$, and $g$ the gauge coupling.

Figure 1: The diagrammatic representations of the three-gluon vertex,
${\rm{I}}\\!\Gamma_{\alpha\mu\nu}^{abc}(q,r,p)$, and the ghost-gluon
scattering kernel, $H_{\nu\alpha}(q,r,p)$, with the respective conventions of
momenta and indices.

It is convenient to decompose $\Gamma_{\alpha\mu\nu}(q,r,p)$ into two distinct
pieces  Ball and Chiu (1980); Davydychev  _et al._ (1996); Aguilar  _et al._
(2019b),

${\rm{I}}\\!\Gamma^{\alpha\mu\nu}(q,r,p)=\Gamma_{\\!{\scriptscriptstyle
L}}^{\alpha\mu\nu}(q,r,p)+\Gamma_{\\!{\scriptscriptstyle
T}}^{\alpha\mu\nu}(q,r,p)\,,$ (3)

where $\Gamma_{\\!{\scriptscriptstyle L}}^{\alpha\mu\nu}(q,r,p)$ and
$\Gamma_{\\!{\scriptscriptstyle T}}^{\alpha\mu\nu}(q,r,p)$ are the
“longitudinal” and “transverse” parts of the three-gluon vertex, respectively.
While the former saturates the corresponding STIs [see Eq. (8)], the latter is
automatically conserved when contracted by $q^{\alpha}$, $r^{\mu}$, and
$p^{\nu}$, i.e.,  $q^{\alpha}\Gamma_{\\!{\scriptscriptstyle
T}}^{\alpha\mu\nu}=r^{\mu}\Gamma_{\\!{\scriptscriptstyle
T}}^{\alpha\mu\nu}=p^{\nu}\Gamma_{\\!{\scriptscriptstyle
T}}^{\alpha\mu\nu}=0$.

The tensorial decompositions of $\Gamma_{\\!{\scriptscriptstyle
L}}^{\alpha\mu\nu}(q,r,p)$ and $\Gamma_{\\!{\scriptscriptstyle
T}}^{\alpha\mu\nu}(q,r,p)$ reads

$\displaystyle\Gamma_{\\!{\scriptscriptstyle L}}^{\alpha\mu\nu}(q,r,p)$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{10}X_{i}(q,r,p)\ell_{i}^{\alpha\mu\nu}\,,$
$\displaystyle\Gamma_{\\!{\scriptscriptstyle T}}^{\alpha\mu\nu}(q,r,p)$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{4}Y_{i}(q,r,p)t_{i}^{\alpha\mu\nu}\,,$ (4)

where the explicit expressions of the basis elements $\ell_{i}^{\alpha\mu\nu}$
and $t_{i}^{\alpha\mu\nu}$ are given in Eqs. (3.4) and (3.6) of Aguilar  _et
al._ (2019a), respectively.

Another familiar quantity introduced in the studies of the three-gluon vertex
is the transversally projected vertex,
$\overline{\Gamma}_{\alpha\mu\nu}(q,r,p)$, defined as Eichmann  _et al._
(2014); Huber (2020)

$\\!\\!\overline{{\rm{I}}\\!\Gamma}_{\alpha\mu\nu}(q,r,p)\\!=\\!{\rm{I}}\\!\Gamma^{\alpha\\!^{\prime}\mu\\!^{\prime}\nu\\!^{\prime}}\\!(q,r,p)P_{\alpha^{\prime}\\!\alpha}(q)P_{\mu^{\prime}\\!\mu}(r)P_{\nu^{\prime}\\!\nu}(p)\,.$
(5)

In addition, we define the tree-level counterpart of Eq. (5),

$\overline{\Gamma}_{\alpha\mu\nu}(q,r,p)=\Gamma_{\\!0}^{\alpha\\!^{\prime}\mu\\!^{\prime}\nu\\!^{\prime}}\\!(q,r,p)P_{\alpha^{\prime}\\!\alpha}(q)P_{\mu^{\prime}\\!\mu}(r)P_{\nu^{\prime}\\!\nu}(p),$
(6)

where
$\Gamma_{\\!0}^{\alpha\\!^{\prime}\mu\\!^{\prime}\nu\\!^{\prime}}\\!(q,r,p)$
is the standard tree-level expression, given by

$\Gamma_{\\!0}^{\alpha\mu\nu}=(q-r)^{\nu}g^{\alpha\mu}+(r-p)^{\alpha}g^{\mu\nu}+(p-q)^{\mu}g^{\alpha\nu}\,;$
(7)

it may be obtained from Eq. (3) by setting $X_{1}=X_{4}=X_{7}=1$, and zero for
all other form-factors.

The STI satisfied by ${\rm{I}}\\!\Gamma_{\alpha\mu\nu}(q,r,p)$ reads

$p^{\nu}{\rm{I}}\\!\Gamma_{\alpha\mu\nu}(q,r,p)=F(p^{2})[{\cal
T}_{\mu\alpha}(r,p,q)-{\cal T}_{\alpha\mu}(q,p,r)]\,,$ (8)

with

${\cal
T}_{\mu\alpha}(r,p,q):=\Delta^{-1}(r^{2})P_{\mu}^{\sigma}(r)H_{\sigma\alpha}(r,p,q)\,,$
(9)

where $H_{\nu\mu}(q,p,q)$ denotes the ghost-gluon scattering kernel,
represented diagrammatically in the panel (b) of Fig. 1. Its tensorial
decomposition is given by Ball and Chiu (1980); Davydychev  _et al._ (1996);
Aguilar  _et al._ (2019c)

$\displaystyle H_{\nu\mu}(q,p,r)$ $\displaystyle=$ $\displaystyle
g_{\nu\mu}A_{1}+q_{\mu}q_{\nu}A_{2}+r_{\mu}r_{\nu}A_{3}$ (10) $\displaystyle+$
$\displaystyle q_{\mu}r_{\nu}A_{4}+r_{\mu}q_{\nu}A_{5}\,,$

where we use the compact notation $A_{i}:=A_{i}(q,p,r)$.

##  III  Nonperturbative methods

The rich kinematic structure of the three-gluon vertex makes its
nonperturbative study particularly challenging. There are three main
frameworks for dealing with this problem: (i) Functional methods, such as the
Schwinger-Dyson equations (SDEs) Schleifenbaum  _et al._ (2005); Huber and von
Smekal (2013); Aguilar  _et al._ (2013); Huber  _et al._ (2012); Blum  _et
al._ (2014); Eichmann  _et al._ (2014); Williams  _et al._ (2016); Binosi  _et
al._ (2017) and the functional renormalization group Corell  _et al._ (2018);
Cyrol  _et al._ (2018a, 2016); (ii) large-volume lattice simulations
Parrinello  _et al._ (1998); Boucaud  _et al._ (1998); Cucchieri  _et al._
(2006, 2008); Athenodorou  _et al._ (2016); Duarte  _et al._ (2016); Boucaud
_et al._ (2017); Vujinovic and Mendes (2019); Aguilar  _et al._ (2020a,
2021b); and (iii) STI-based reconstructions of the longitudinal part,
$\Gamma_{\\!{\scriptscriptstyle L}}^{\alpha\mu\nu}(q,r,p)$, in the spirit of
the “gauge-technique” Salam (1963); Salam and Delbourgo (1964); Delbourgo and
West (1977a, b).

(i) _ Functional methods_: The diagrammatic representation of the SDE that
governs the evolution of the three-gluon vertex is shown in Fig. 2. The self-
consistent treatment of this equation is particularly complicated, and entails
its coupling to additional related equations, such as the SDEs of the gluon
and ghost propagators. In practice, this task is considerably simplified by
using as inputs the lattice results for $\Delta(q^{2})$ and $F(q^{2})$.

Figure 2:  The SDE of the three-gluon vertex. The white (gray) circles
(ellipses) indicate fully dressed propagators (kernels), while the dots
indicate the omitted terms.

(ii) _ Lattice simulations_: In this case the three-gluon vertex is accessed
through the functional averaging of the quantity
$\langle\widetilde{A}^{a}_{\alpha}(q)\widetilde{A}^{b}_{\mu}(r)\widetilde{A}^{c}_{\nu}(p)\rangle$,
where $\widetilde{A}^{a}_{\alpha}(q)$ denotes the SU(3) gauge field.
Specifically, the connected three-point function, ${\cal
G}_{\alpha\mu\nu}(q,r,p)$, defined as

${\cal
G}_{\alpha\mu\nu}(q,r,p)=g\overline{{\rm{I}}\\!\Gamma}_{\alpha\mu\nu}(q,r,p)\Delta(q^{2})\Delta(r^{2})\Delta(p^{2})\
,$ (11)

is given by
$\langle\widetilde{A}^{a}_{\alpha}(q)\widetilde{A}^{b}_{\mu}(r)\widetilde{A}^{c}_{\nu}(p)\rangle=f^{abc}{\cal
G}_{\alpha\mu\nu}(q,r,p)$.
$\overline{{\rm{I}}\\!\Gamma}_{\alpha\mu\nu}(q,r,p)$ is finally obtained after
an appropriate amputation of the gluon propagators.

The typical structure of lattice “observables” is

$L(q,p,r)=\frac{W^{\alpha\mu\nu}(q,r,p)\overline{{\rm{I}}\\!\Gamma}_{\alpha\mu\nu}(q,r,p)}{\rule[0.0pt]{0.0pt}{12.80365pt}W_{\alpha\mu\nu}(q,r,p)W^{\alpha\mu\nu}(q,r,p)}\rule[0.0pt]{0.0pt}{14.22636pt}\;,$
(12)

where the $W^{\alpha\mu\nu}(q,r,p)$ are appropriately chosen projectors
Athenodorou  _et al._ (2016); Duarte  _et al._ (2016); Boucaud  _et al._
(2017). In what follows we will focus our attention on two special kinematic
limits involving a single momentum variable.

(a) _Soft limit_ , corresponding to the kinematic choice

$q\to 0\,,\quad p=-r\,,\quad\theta:=\widehat{pr}=\pi\,,$ (13)

obtained by setting $W^{\alpha\mu\nu}(q,r,p)\to 2r^{\alpha}P^{\mu\nu}(r)$,
namely

$L_{sg}(r^{2})=\frac{\Gamma_{0}^{\alpha\mu\nu}(q,r,p)\overline{{\rm{I}}\\!\Gamma}_{\alpha\mu\nu}(q,r,p)}{\rule[0.0pt]{0.0pt}{12.80365pt}\;{\Gamma_{0}^{\alpha\mu\nu}(q,r,p)\overline{\Gamma}_{\alpha\mu\nu}(q,r,p)}}\rule[0.0pt]{0.0pt}{14.22636pt}\Bigg{|}_{\begin{subarray}{c}\\!\\!q\to
0\\\ p\to-r\end{subarray}}\,.$ (14)

(b) _Totally symmetric_ limit,

$q^{2}=p^{2}=r^{2}:=s^{2}\,,\quad\theta:=\widehat{qr}=\widehat{qp}=\widehat{rp}=2\pi/3\,;$
(15)

the corresponding $W^{\alpha\mu\nu}(q,r,p)$ and the expression for
$\mathit{L}_{{sym}}(s^{2})$ may be found in Eqs. (2.18) and (2.19) of Aguilar
_et al._ (2020a).

(iii) _STI_ : As was first shown in Ball and Chiu (1980), the STI of Eq. (8),
together with its cyclic permutation, determines the form factors
$X_{i}(q,r,p)$ in terms of the kinetic part of the gluon propagator, to be
denoted by $J(q^{2})$, the ghost dressing function, and three form factors of
the ghost-gluon kernel.

Figure 3: A representative case of the three-gluon form factor
$X_{1}(r^{2},p^{2},\theta)$ (left panel) and for
$|p||r|X_{3}(r^{2},p^{2},\theta)$ (right panel) for a fixed value of the
angle, $\theta:=\widehat{pr}=\pi$.

Specifically,

$\displaystyle
X_{1}=\frac{1}{4}[(q^{2}-r^{2})(b_{rpq}+b_{pqr}-b_{qpr}-b_{prq})]\,,$
$\displaystyle+\\!2(a_{pqr}\\!+\\!a_{prq})\\!+\\!p^{2}(b_{qrp}\\!+\\!b_{rqp})\\!+\\!2(\,q\cdot
p\,d_{prq}\\!+\\!\,r\cdot p\,d_{pqr})]\,,$ $\displaystyle
X_{2}=\frac{1}{4}[2(a_{prq}-a_{pqr})-(q^{2}-r^{2})(b_{qrp}+b_{rqp})$
$\displaystyle+2(\,q\cdot p\,d_{prq}-\,r\cdot
p\,d_{pqr})+p^{2}(b_{prq}-b_{pqr}+b_{qpr}-b_{rpq})]\,,$ $\displaystyle
X_{3}=\frac{1}{q^{2}-r^{2}}[a_{rpq}-a_{qpr}+r\cdot p\,d_{qpr}-q\cdot
p\,d_{rpq}]\,,$ $\displaystyle
X_{10}=-\frac{1}{2}[b_{qrp}+b_{rpq}+b_{pqr}-b_{qpr}-b_{rqp}-b_{prq}],$ (16)

where we introduced the following compact notation

$\displaystyle a_{qrp}:=$ $\displaystyle F(r)J(p)A_{1}(p,r,q)\,,$
$\displaystyle b_{qrp}:=$ $\displaystyle F(r)J(p)A_{3}(p,r,q)\,,$
$\displaystyle d_{qrp}:=$ $\displaystyle
F(r)J(p)[A_{4}(p,r,q)-A_{3}(p,r,q)]\,.$ (17)

Due to the Bose symmetry of the three-gluon vertex, the remaining six $X_{i}$
may be computed by permuting the arguments appropriately (see Eq. (3.8) of
Aguilar  _et al._ (2019a)).

It is important to emphasize that in the original work of Ball and Chiu (1980)
the kinetic term of the gluon propagator was defined as
$\Delta^{-1}(q^{2})=q^{2}J(q^{2})$, while in the nonperturbative
generalization presented in Aguilar  _et al._ (2019a) we have
$\Delta^{-1}(q^{2})=q^{2}J(q^{2})+m^{2}(q^{2})$, where $m^{2}(q^{2})$ is the
running gluon mass Cornwall (1982); Aguilar  _et al._ (2008); Binosi and
Papavassiliou (2009); Cornwall (1982); Aguilar  _et al._ (2008); Roberts
(2020).

Two representative results for the form factors $X_{1}(r^{2},p^{2},\theta$)
and $|q||r|X_{3}(r^{2},p^{2},\theta$), obtained with Eq. (16), are shown on
Fig. 3. In this figure we present both form factors as a function of the two
momenta $p^{2}$ and $r^{2}$ when the angle between these two momenta is fixed
at the value $\theta=\pi$.

Clearly, one can see that $X_{1}(r^{2},p^{2},\theta)$ has a completely
nontrivial structure, which persists for general values of the angle.
Evidently, the most striking feature of this result is the reduction of the
size of $X_{1}(r^{2},p^{2},\theta)$ with respect to its tree-level value
(unity); this effect is known in the literature as “infrared suppression”
Aguilar  _et al._ (2014); Athenodorou  _et al._ (2016); Boucaud  _et al._
(2017); Blum  _et al._ (2015); Corell  _et al._ (2018); Huber (2020); Aguilar
_et al._ (2019a).

Let us also point out that the projection of the three gluon vertex in the
totally symmetric limit, defined in Eq. (15), can be written as Aguilar  _et
al._ (2019a)

$\displaystyle\mathit{L}_{{sym}}(s^{2})$ $\displaystyle=$ $\displaystyle
X_{1}(s^{2})-\frac{s^{2}}{2}X_{3}(s^{2})$ (18) $\displaystyle+$
$\displaystyle\frac{s^{4}}{4}Y_{1}(s^{2})-\frac{s^{2}}{2}Y_{4}(s^{2})\,.$

Figure 4:  The projection of the three-gluon vertex in the soft gluon,
$\mathit{L}_{{sg}}(r^{2})$ (left) and in the symmetric,
$\mathit{L}_{{sym}}(s^{2})$, (right) kinematic configurations obtained from
lattice QCD of Athenodorou  _et al._ (2016); Boucaud  _et al._ (2017); Aguilar
_et al._ (2021b) (solid circles) and from the SDE-STIs approach (magenta
continuous curve) Aguilar  _et al._ (2021b).

Figure 5:  Left panel: The unquenched $\mathit{L}_{{sg}}(r^{2})$ (left) and
$\mathit{L}_{{sym}}(s^{2})$ (right) obtained from lattice QCD with $N_{f}=2+1$
dynamical quarks Aguilar  _et al._ (2020a), and from the SDE-STI approach
(magenta continuous curve).

On other hand, for the case of the soft limit configuration of Eq. (13), the
expression for $\mathit{L}_{{sg}}(r^{2})$ is given by

$\mathit{L}_{{sg}}(r^{2})=X_{1}(r^{2},r^{2},\pi)-r^{2}X_{3}(r^{2},r^{2},\pi)\,\,;$
(19)

note that the result is free of transverse form factors $Y_{i}$.

Notice that the soft gluon kinematic limit of $X_{1}$ and $X_{3}$ corresponds
to the curves that lie on the diagonal “slice” of the 3D plots of Fig. 3 where
$p^{2}=r^{2}$. In the left panel of Fig. 4 we show a comparison of the
$\mathit{L}_{{sg}}(r^{2})$ computed using the SDE-STI approach (magenta
continuous curve) and a combination of the lattice data of Athenodorou  _et
al._ (2016); Boucaud  _et al._ (2017); Aguilar  _et al._ (2021b) (solid
circles), for the case of quenched QCD. It is clear that both methods
corroborate the infrared suppression of the three-gluon vertex. In the right
panel we show the results for $\mathit{L}_{{sym}}(s^{2})$, obtained when we
set $Y_{i}=0$ in Eq. (18). Once again the coincidence with the lattice data is
rather notable, and the presence of the steep decline in the infrared is
visible in both approaches. In addition, the same pattern (suppression and
zero crossing) persists qualitatively unaltered when $N_{f}=2+1$ dynamical
quarks are added Aguilar  _et al._ (2020a), as can be clearly seen in Fig. 5.

##  IV Infrared suppression

One of the most remarkable nonperturbative features of the three-gluon vertex
in the Landau gauge is its infrared suppression, as established clearly in the
results of the previous section. Thus, form factors such as $X_{1}$, $X_{4}$,
and $X_{7}$, which, due to renormalization, acquire their tree level value
(unity) at $4.3$ GeV, reduce their size to half at around $1$ GeV. This
tendency culminates with a characteristic reversal of the sign, known as “zero
crossing” Aguilar  _et al._ (2014); Eichmann  _et al._ (2014); Blum  _et al._
(2014); Vujinovic  _et al._ (2014); Athenodorou  _et al._ (2016); Boucaud  _et
al._ (2017), followed by a logarithmic divergence of the corresponding form
factor at the origin.

This type of behavior is in sharp contradistinction to what happens with the
other vertices of the theory that have been explored so far, such as the
quark-gluon or the ghost-gluon vertex. Indeed, as one can see in Fig. 6, the
analogous form factors display a clear enhancement for the same range of
intermediate and infrared momenta.

Figure 6:  The behavior of the form factors, $L_{1}^{sg}(q^{2})$,
$B_{1}^{sg}(q^{2})$, and $\mathit{L}_{{sg}}(q^{2})$ associated with the
classical tensor structures of the quark-gluon (green dotted), ghost-gluon
(blue dashed) and three-gluon (magenta continuous) vertices, respectively, in
the soft gluon limit.

From the theoretical point of view, this particular feature of the three-gluon
vertex hinges on the subtle interplay between dynamical effects originating
from the two-point sector of the theory Roberts and Williams (1994); Alkofer
and von Smekal (2001); Fischer (2006); Cloet and Roberts (2014); Aguilar  _et
al._ (2016); Binosi  _et al._ (2015). This may be understood at the level of
the one-loop dressed version of the SDE in Fig. 2, which is shown in Fig. 7.
The crucial theoretical ingredient is that, whereas the gluon acquires
dynamically an effective mass, the ghost remains massless even
nonperturbatively. As a result, the loops of the three-gluon vertex containing
gluons (such as the ($d_{1}$) in Fig. 7) give rise to “protected” logarithms,
because the effective gluon mass $m$ acts as an infrared regulator. Instead,
loops containing ghosts (such as the ($d_{2}$) in Fig. 7) produce
“unprotected” logarithms, which diverge at the origin Aguilar  _et al._
(2014).

In the simplified kinematic circumstances where only a single representative
momentum $q^{2}$ is considered, a basic model describing qualitatively the
resulting dynamics is given by

$L(q^{2})=b_{0}+b_{\rm
gl}\ln\left(\frac{q^{2}+m^{2}}{\Lambda^{2}}\right)+b_{\rm
gh}\ln\left(\frac{q^{2}}{\Lambda^{2}}\right)\,,$ (20)

where $L(q^{2})$ denotes the particular combination of form factors, such
that, at tree-level, $L_{0}(q^{2})=1$, and $b_{0}$, $b_{\rm gl}$, and $b_{\rm
gh}$ are positive constants.

It is clear that, as $q\to 0$, the term with the unprotected logarithm will
dominate over the others, forcing $L(q^{2})$ to reverse its sign (zero
crossing), and finally diverge, $L(0)\to-\infty$. Because, in practice,
$b_{\rm gl}$ is about one order of magnitude larger than $b_{\rm gh}$, the
point where the unprotected logarithm overtakes the protected one is rather
deep in the infrared, and the location of the zero-crossing is at about $120$
MeV. Thus, in the intermediate region of momenta, which is typically relevant
for the onset of nonperturbative dynamics, we have $L(q^{2})<1$; this effect
is known as the infrared suppression of the three-gluon vertex.

Figure 7:  The SDE of the three-gluon vertex at the one-loop dressed level.
The white (red and blue) circles indicate fully dressed propagators
(vertices). The diagrams $(d_{1})$ and $(d_{2})$ are the gluon and the ghost
triangle contributions entering in the skeleton expansion of three-gluon
vertex.

##  V  Phenomenology

In this section we discuss two representative phenomenological applications,
where the infrared suppression of the corresponding form factors plays a
crucial role.

###  V.1  Effective couplings

A typical quantity employed in a variety of phenomenological applications is
the effective charge, defined as a special renormalization-group invariant
combination of propagators and vertex form factors. In the case of the three-
gluon vertex in the soft-gluon limit, the corresponding charge, to be denoted
by ${\alpha}_{\rm{3g}}(q^{2})$, is defined as Alkofer  _et al._ (2005);
Fischer (2006); Athenodorou  _et al._ (2016); Fu  _et al._ (2020); Aguilar
_et al._ (2021c)

${\alpha}_{\rm{3g}}(q^{2})={\alpha}_{s}(\mu^{2})\mathit{L}_{{sg}}^{2}(q^{2})\mathcal{Z}^{3}(q^{2})\,,$
(21)

with $\mathcal{Z}(q^{2})$ defined in Eq. (1).

It is natural to expect that the infrared suppression of
$\mathit{L}_{{sg}}^{2}(q^{2})$ will affect the shape and size of
${\alpha}_{\rm{3g}}(q^{2})$. In order to meaningfully quantify this
suppression, we compare ${\alpha}_{\rm{3g}}(q^{2})$ with the corresponding
quantity defined from the ghost-gluon vertex, to be denoted
${\alpha}_{\rm{cg}}(q^{2})$, namely (see, e.g., Alkofer  _et al._ (2005);
Fischer (2006); Aguilar  _et al._ (2009))

${\alpha}_{\rm{cg}}(q^{2})={\alpha}_{s}(\mu^{2})B_{1\,sg}^{2}(q^{2})F^{2}(q^{2})\mathcal{Z}(q^{2})\,,$
(22)

where $B_{1\,sg}(q^{2})$ is the ghost-gluon form factor introduced in Fig. 6.

Figure 8:  The comparison of the effective couplings,
${\alpha}_{\rm{cg}}(q^{2})$ (blue dashed line) and ${\alpha}_{\rm{3g}}(q^{2})$
(magenta continuous curve).

It is important to mention that both effective couplings are computed in the
same renormalization scheme, namely the Taylor scheme Boucaud  _et al._ (2009,
2011); von Smekal  _et al._ (2009) where we have fixed that
${\alpha}_{s}(\mu)=0.244$, at $\mu=4.3$ GeV (for more details see Aguilar  _et
al._ (2021c)).

The comparison of the two effective charges is displayed in Fig. 8. One
clearly sees that, as the momentum $q$ decreases, ${\alpha}_{\rm{3g}}(q^{2})$
(magenta continuous) becomes considerably smaller than
${\alpha}_{\rm{cg}}(q^{2})$ (blue dashed line). The suppression of
${\alpha}_{\rm{3g}}(q^{2})$, located in the region below $2$ GeV is consistent
with previous finding Huber and von Smekal (2013); Blum  _et al._ (2014);
Williams (2015); Cyrol  _et al._ (2016, 2018a); Aguilar  _et al._ (2020b), and
its origin is exclusively associated with the suppression of the
$\mathit{L}_{{sg}}(q^{2})$.

###  V.2  Pseudoscalar glueball

The dynamical generation of a mass gap in pure-gauge QCD is intimately
connected with the attendant appearance of glueball bound-states Cornwall
(1982). The rich glueball spectrum, and related fundamental properties, has
been obtained by means of detailed lattice simulations, see e.g., Morningstar
and Peardon (1999); Bali  _et al._ (1993); McNeile (2009); Chen  _et al._
(2006); Athenodorou and Teper (2020). Evidently, these results serve as
valuable benchmarks in the ongoing effort of continuum bound-state methods to
reach an intuitive understanding of the underlying dynamics Dudal  _et al._
(2011); Meyers and Swanson (2013); Souza  _et al._ (2020); Huber  _et al._
(2021).

In this context, the $J^{PC}=0^{-+}$ glueball represents the simplest case,
because the pertinent Bethe-Salpeter equation possesses a single dynamical
kernel, which essentially describes the four-gluon scattering process. The
lowest-order contribution of this kernel is shown in Fig. 9; evidently, the
three-gluon vertex constitutes one of its central ingredients Souza  _et al._
(2020).

Moreover, the corresponding amplitude involves only one scalar function,
namely

$\chi_{\mu\nu}(k_{+},k_{-})=\epsilon_{\mu\nu\alpha\beta}k^{\alpha}P^{\beta}\mathcal{F}(k;P)\,,$
(23)

simplifying considerably the treatment of this problem.

Figure 9: The diagrammatic representation of the Bethe-Salpeter equation for
the pseudoscalar glueball with total momentum $P$, and
$\ell_{\pm}:=P/2\pm\ell$, for $\ell=k,q$ .

It turns out that the infrared suppression of the three-gluon vertex, and the
overall attenuation of the interaction strength that it induces is
instrumental for the formation of the pseudoscalar glueball state, with a mass
compatible with that obtained from the lattice Souza  _et al._ (2020).

Let us finally mention that the need for a considerable suppression has also
been established in studies of hybrid states by means of Faddeev equations Xu
_et al._ (2019).

##  VI Conclusions

In this presentation we have reviewed some of the most characteristic
nonpertubative features of the three-gluon vertex, unraveled by the ongoing
synergy of a multitude of techniques and approaches, such as functional
methods, lattice simulations, and STI-based constructions.

We have focused on the interplay between the dynamics of the three-gluon
vertex and the Landau-gauge two-point sector of the theory. In particular, as
has been argued in Sec. IV, the characteristic infrared suppression displayed
by the main form factors of the three-gluon vertex is tightly interlocked with
the mass generating pattern established in the gauge sector of QCD.

There is an additional key aspect of the three-gluon vertex, which is worth
mentioning, albeit in passing. In particular, the three-gluon vertex develops
longitudinally coupled bound state massless poles, which trigger the well-
known Schwinger mechanism Schwinger (1962a, b), endowing the gluons with a
dynamical mass scale Aguilar  _et al._ (2008); Binosi and Papavassiliou (2009,
2009). Due to their special kinematic properties, these poles decouple from
the transversally projected vertex $\overline{\Gamma}_{\alpha\mu\nu}(q,r,p)$
[see Eq. (5)], which enters in the lattice quantities defined according to Eq.
(12). Consequently, these dynamically produced poles do not induce divergences
in the results displayed in Fig. 4 and Fig. 5. Nonetheless, as has been
recently demonstrated in Aguilar  _et al._ (2021a), the massless poles leave
smoking-gun signals of their presence, by inducing finite displacements to the
non-Abelian Ward identity satisfied by the pole-free part of the three-gluon
vertex. Quite interestingly, this displacement is identical to the Bethe-
Salpeter amplitude that controls the dynamical formation of the massless poles
Aguilar  _et al._ (2012); Ibañez and Papavassiliou (2013); Aguilar  _et al._
(2018); Binosi and Papavassiliou (2018), thus establishing a powerful
constraint on the entire mass generating mechanism put forth in a series of
works (see Aguilar  _et al._ (2021a) and references therein).

It would be clearly important to continue the research activity surrounding
the three-gluon vertex in the future. In this context, a major challenge for
functional methods is the extension of the results for this vertex from space-
like to time-like momenta. Such information will be particularly important,
both from the theoretical as well as the phenomenological point of view. The
methods and techniques developed in Cyrol  _et al._ (2018b); Horak  _et al._
(2020, 2021a, 2021b) may be decisive for making progress with this demanding
endeavor.

## Acknowledgments

We thank the organizers of the 19th International Conference on Hadron
Spectroscopy and Structure (Hadron 21-virtual) for the kind invitation. The
work of J. P. is supported by the Spanish AEI-MICINN grant
PID2020-113334GB-I00/AEI/10.13039/501100011033, and the grant
Prometeo/2019/087 of the Generalitat Valenciana. A. C. A. is supported by the
CNPq grants 307854/2019-1 and 464898/2014-5 (INCT-FNA). A. C. A. and M. N. F.
also acknowledge financial support from the FAPESP projects 2017/05685-2 and
2020/12795-1, respectively.

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