Variational quantum eigensolver for causal loop Feynman diagrams and directed acyclic graphs Giuseppe Clemente1Arianna Crippa1Karl Jansen1Selomit Ram ırez-Uribe2 3 4 Andr es

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Variational quantum eigensolver for causal loop Feynman diagrams and directed acyclic

graphs

Giuseppe Clemente,

1, ∗

Arianna Crippa,

1, †

Karl Jansen,

1, ‡

Selomit Ram´ırez-Uribe,

2, 3, 4, §

Andr´es

E. Renter´ıa-Olivo,

2, ¶

Germ´an Rodrigo,

2, ∗∗

German F. R. Sborlini,

5, 6, ††

and Luiz Vale Silva

2, ‡‡

Deutsches Elektronen-Synchrotron DESY, Platanenallee 6, 15738 Zeuthen, Germany.

Instituto de F´ısica Corpuscular, Universitat de Val`encia – Consejo Superior de Investigaciones Cient´ıﬁcas,

Parc Cient´ıﬁc, E-46980 Paterna, Valencia, Spain.

Facultad de Ciencias F´ısico-Matem´aticas, Universidad Aut´onoma de Sinaloa,

Ciudad Universitaria, CP 80000 Culiac´an, Mexico.

Facultad de Ciencias de la Tierra y el Espacio, Universidad Aut´onoma de Sinaloa,

Ciudad Universitaria, CP 80000 Culiac´an, Mexico.

Departamento de F´ısica Fundamental e IUFFyM,

Universidad de Salamanca, 37008 Salamanca, Spain.

Escuela de Ciencias, Ingenier´ıa y Dise˜no, Universidad Europea de Valencia,

Paseo de la Alameda 7, 46010 Valencia, Spain.

We present a variational quantum eigensolver (VQE) algorithm for the eﬃcient bootstrapping

of the causal representation of multiloop Feynman diagrams in the Loop-Tree Duality (LTD) or,

equivalently, the selection of acyclic conﬁgurations in directed graphs. A loop Hamiltonian based on

the adjacency matrix describing a multiloop topology, and whose diﬀerent energy levels correspond

to the number of cycles, is minimized by VQE to identify the causal or acyclic conﬁgurations. The

algorithm has been adapted to select multiple degenerated minima and thus achieves higher detection

rates. A performance comparison with a Grover’s based algorithm is discussed in detail. The VQE

approach requires, in general, fewer qubits and shorter circuits for its implementation, albeit with

lesser success rates.

I. INTRODUCTION

In recent years, there has been a tremendous progress in

achieving highly-precise theoretical predictions for particle

colliders, which has been possible because of the develop-

ment of new techniques in Quantum Field Theories (QFT)

as well as improved (classical) hardware. For instance,

from the theory point of view, the calculation of multi-

loop scattering amplitudes and Feynman integrals was

optimized by applying diﬀerent sophisticated techniques.

Some of these theoretical advancements include Mellin-

Barnes transformations [

–

], algebraic reduction of in-

tegrands [

–

], integration-by-parts identities [

contour deformation assisted by neural networks [

] and

sector decomposition [

–

], among several other highly

eﬃcient methods

. Special emphasis was recently put

in the development of four dimensional methods [

–

with the purpose of achieving a seamless combination of

algebraic/analytic strategies and numerical integration

directly in the four physical dimensions of the space-time.

∗Electronic address: giuseppe.clemente@desy.de

†Electronic address: arianna.crippa@desy.de

‡Electronic address: karl.jansen@desy.de

§Electronic address: selomit@iﬁc.uv.es

¶Electronic address: andres.renteria@iﬁc.uv.es

∗∗Electronic address: german.rodrigo@csic.es

††Electronic address: german.sborlini@usal.es

‡‡Electronic address: luizva@iﬁc.uv.es

For a complete review about currently available technologies for

QFT calculations, see Ref. [21] and references therein.

Overcoming the current precision frontier will certainly

require to push these methods to their limits.

Future collider experiments require challenging theoret-

ical predictions from full calculations at order

in the

perturbative expansion (N

LO), with

n≥

3. One fore-

seeable complication is directly related to the appearance

of several scales in multiloop multileg Feynman integrals.

These objects are analytically known for speciﬁc processes

and kinematic conﬁgurations up to two loops with a lim-

ited number of scales and very few examples of three-loop

amplitudes are starting to pop-up [

]. At this point,

it is very unlikely that all the required ingredients will be-

come analytically available and the development of novel

numerical approaches seems unavoidable.

Given present and future needs, a major progress in

solving (at least) two challenges is necessary. On one

side, new computational techniques must be developed,

exploiting the fundamental properties of QFT and the

mathematical concepts behind scattering amplitudes. On

the other side, these new methods must be implemented

in eﬃcient event generators, capable of overcoming the

bottlenecks of the currently available hardware.

Regarding the ﬁrst challenge, we focus on the Loop-

Tree Duality (LTD) formalism [

–

], which exhibits

very attractive mathematical properties and a manifestly

causal physical interpretation. Multiloop scattering am-

plitudes are transformed in LTD into the so-called dual

amplitudes by integrating out one component of each of

the loop momenta through the Cauchy’s residue theorem.

In this way, physical observables are expressed in terms

of Euclidean integrals that combine loop and tree-level

contributions (as well as renormalization counter-terms)

arXiv:2210.13240v3 [hep-ph] 14 Nov 2023

to achieve a fully local cancellation of singularities [

–

Besides the possibility of a local regularization, including

the simultaneous cancellation of infrared and ultraviolet

singularities, the LTD formalism fully exploits causal-

ity in QFT. A recent reformulation [

–

] showed that

multiloop scattering amplitudes and Feynman integrals

can be represented in terms of a subset of cut diagrams

that generalize the well-known Cutkosky’s rules [

]. In

particular, a manifestly causal representation can be di-

rectly obtained by decomposing the original Feynman

diagrams into binary connected partitions in the equiva-

lence class of topologies deﬁned by collapsing propagators

into edges (or multi-edges) [

]. As a result, a well

deﬁned geometrical algorithm is available and a strong

connection between causality and directed acyclic graphs

can be established [60].

The other challenge is related to an eﬃcient calcula-

tion of Feynman integrals, overcoming current hardware

limitations. In this direction, the development of novel

strategies for classically hard problems based on quantum

algorithms (QAs) is gaining momentum across diﬀerent

areas. For instance, there are several ideas that exploit

the potential speed-up of quantum computers, such as

database querying through Grover’s algorithm [

], the

famous Shor’s algorithm for factorization of large inte-

gers [

] or Hamiltonian minimization through quantum

annealing [

]. In the context of particle physics, QAs are

often applied to solve problems related to lattice gauge

theories [

–

]. Recent applications for high-energy col-

liders include jet identiﬁcation and clustering [

–

determination of parton densities (PDFs) [

], simulation

of parton showers [

], anomaly detection [

], and

integration of elementary particle processes [

]. This list

is rapidly growing, since QAs are suitable for several uses,

especially those involving minimization problems.

With this panorama in mind, the purpose of this article

is to explore the application of QAs for unveiling the

causal structure of multiloop scattering amplitudes and

Feynman integrals, or equivalently acyclic conﬁgurations

of directed graphs. Our strategy consists in exploiting

the properties of the adjacency matrix in graph theory

to build a Hamiltonian which weights the cost of diﬀer-

ent momentum ﬂow conﬁgurations. Explicitly, causal

conﬁgurations are those with minimum energy, so we se-

lect them by identifying the minima of the Hamiltonian.

This identiﬁcation is implemented through a Variational

Quantum Eigensolver (VQE) [

–

], namely a hybrid

quantum-classical algorithm that seeks the ground state

of a given operator.

The classical problem of identifying or counting directed

acyclic graphs is #P-hard, see Ref. [

]. Explicitly, given

a graph

= (

V, E

) with

vertices and

edges, there

are 2

|E|

possible directed graphs that must be checked

for cycles. Eﬃcient classical algorithms have been pro-

posed in the literature: e.g., Ref. [

] gives an example

of a Fixed Parameter Tractable algorithm, whose scaling

depends on a single structural parameter of the graph,

see also references therein. This algorithm consists of a

Binary Decision Diagram algorithm [

], which performs

asymptotically in

w/4

) running time per solution,

with the path-width

always smaller than the input size

of the graph, becoming closer to it for dense graphs (i.e.,

maximally connected graphs). Instead, the motivation of

our paper is to encode the complexity of the graph by

means of a Hamiltonian, i.e. a cost function to be mini-

mized in order to ﬁnd the directed acyclic conﬁgurations.

Since VQE is expected to oﬀer a speed-up in minimiza-

tion problems compared to purely classical algorithms,

we explore up to what extent this speed-up remains in

the detection of directed acyclic graphs.

The outline of this paper is the following. In Sec. II,

we present a brief introduction to LTD, and we oﬀer a de-

scription of its manifestly causal representation. Then, in

Sec. III, we discuss the geometrical aspects of the causal

conﬁgurations, presenting explicit examples in App. A.

After that, we introduce the Hamiltonian approach in

Sec. IV, putting special emphasis in the connection with

the adjacency matrix of directed acyclic graphs. We of-

fer a deﬁnition of the loop Hamiltonian and its classical

reconstruction algorithm in Secs. IV A and IV B, respec-

tively. We discuss subtleties of the encoding of vertex

registers in App. B. Then, we discuss the implementation

of the VQE in Sec. V, presenting an explicit example

for a representative two-loop topology in Sec. V A. The

full list of Hamiltonians for the topologies studied in this

article is given in App. C. Right after in Sec. V B, we

discuss an improved strategy based on multiple runs of

the VQE to collect, step by step, all the possible causal

solutions. In Sec. VI, we carefully compare the VQE

approach with the Grover’s based algorithm described in

Ref. [

], paying attention to the resources required for

a successful implementation in (real) quantum devices.

Finally, conclusions and further research directions are

presented in Sec. VII.

II. LOOP-TREE DUALITY AND CAUSALITY

To reach highly precise theoretical predictions, it is

necessary to deal with multiloop scattering amplitudes

and the corresponding Feynman integrals. In the Feyn-

man representation, the most general

-loop scattering

amplitude with Pexternal particles is given by

A(L)

F=Zℓ1...ℓL

N{ℓs}L,{pj}Pn

i=1

GF(qi),(1)

where

with

i∈ {

, . . . , n}

are the momenta ﬂowing

through each Feynman propagator,

(

) = (

i−m2

−1

, and

represents a numerator, whose speciﬁc form

depends on the topologies of the diﬀerent diagrams, the

interaction vertices and the nature of the particles that

propagate inside the loops. Regarding the momenta of

the internal particles, they are linear combinations of

the primitive loop momenta associated to the integra-

tion variables in the loop,

ℓs

with

s∈ {

, . . . , L}

), and

the external momenta,

with

j∈ {

, . . . , P }

. Given a

Feynman diagram, we can group the diﬀerent internal

momenta into sets: two lines are said to belong to the

same set if their propagators involve the same linear com-

bination of primitive loop momenta [

]. This is useful for

achieving an eﬃcient classiﬁcation of diagrams according

to their causal structure [47–49].

The denominator of Eq. (1) characterizes the singular

structure of the scattering amplitude, and singularities

provide important information to simplify loop calcula-

tions. In this direction, the Loop-Tree Duality (LTD)

[

] makes use of Cauchy’s residue theorem (CRT) to

reduce the dimensionality of the integration domain, en-

abling the possibility of transforming it into an Euclidean

space. By means of an iterated application of CRT, we

can get rid of one integration variable per loop: this is

equivalent to say that the LTD representation of a

-loop

amplitude is obtained by cutting (or setting on-shell)

internal lines. The eﬀect of cutting a line is the modiﬁca-

tion of the inﬁnitesimal complex prescription, leading to

the so-called dual propagators [

]. It is important to

highlight that this modiﬁed prescription plays a crucial

role to preserve causality, as we will discuss later.

The traditional formulation of LTD leads to the so-

called dual representation, in which there is one contribu-

tion for each possible connected tree obtained by setting

on-shell

internal propagators. If we study the structure

of the denominators of each dual term, we immediately

realize the presence of divergences that do not correspond

to any physical conﬁguration. These unphysical singu-

larities are spurious, and they vanish when we add all

the dual contributions together. This is the so called

manifestly causal representation within LTD [46,88].

Before moving on, let us recall one crucial fact about the

Feynman propagators. If we deﬁne the positive on-shell

energy

q(+)

i,0=q⃗qi2+m2

i−ı0,(2)

then the propagator can be written as

GF(qi) = 1

qi,0−q(+)

i,0

×1

qi,0+q(+)

i,0

.(3)

This decomposition suggests that each Feynman propa-

gator encodes a quantum superposition of two on-shell

modes, with positive and negative energy respectively.

The so-called causal representations are obtained by con-

sistently aligning these modes. Furthermore, this super-

position also motivates the identiﬁcation of each internal

propagator with a qubit, as we will explain later.

As carefully discussed in Refs. [

], the calculation

of the nested residues within the LTD formalism leads to

a manifestly causal representation of multiloop multileg

scattering amplitudes. Thus, it can be shown that Eq. (1)

is equivalent to

A(L)

D=Z⃗

ℓ1...⃗

ℓL

xnX

σ∈Σ

Nσ

n−L

i=1

λhσ(i)

σ(i)

+(λ+

p↔λ−

p),(4)

with xn=Qn2q(+)

i,0,hσ(i)=±1, and

Z⃗

ℓs

=−µ4−dZdd−1ℓs

(2π)d−1,(5)

the integration measure in the loop three-momentum

space. It turns out that Eq. (4) only involves denom-

inators with on-shell energies, added together in same-

sign combinations within the so-called causal propagators,

1/λhσ(i)

σ(i), with

λhσ(i)

σ(i)≡λ±

p=X

i∈p

q(+)

i,0±kp,0,(6)

where

(

) includes information about the partition

the set of on-shell energies and the corresponding orienta-

tion of the energy components of the external momenta,

kp,0

. Each causal propagator is in a one-to-one corre-

spondence with any possible threshold singularity of the

amplitude

A(L)

, which contains overlapped thresholds.

These are known as entangled causal thresholds:

(

)

indicates the set of causal propagators that can be simul-

taneously entangled. The set Σ in Eq. (4) contains all

the combinations of allowed causal entangled thresholds,

which involves overlapped cuts with aligned momenta

ﬂow.

An important advantage of the representation shown in

Eq. (4) is the absence of spurious nonphysical singularities.

In fact, since causal propagators involve same-sign com-

binations of positive on-shell energies, the only possible

singularities are related to IR/UV and physical thresholds.

More details about the manifestly causal LTD representa-

tion and its beneﬁts can be found in Refs. [

III. GEOMETRIC INTERPRETATION OF

CAUSAL FLOWS

In order to identify the causal conﬁgurations of a multi-

loop Feynman diagram in a quantum device, it is necessary

to translate the problem into a suitable language. The

geometrical formulation of the manifestly causal LTD

representations [

] provides a clear and intuitive inter-

pretation. In the following, we brieﬂy describe the most

relevant features of this formalism, recalling some basic

concepts and deﬁnitions as presented in Refs. [

Any scattering amplitude can be represented starting

from Feynman diagrams built from interaction vertices

and internal lines (or propagators) connecting those ver-

tices. Regarding causality, those lines that connect the

same vertices can be merged into a single edge, leading

to reduced Feynman graphs because the only allowed con-

ﬁgurations which are causal are those in which all the

momentum ﬂows of these propagators are aligned in the

same direction. Then, reduced graphs are built from ver-

tices and edges, as described in Refs. [

–

], and their

causal structure turns out to be equivalent to that of the

dual representation of the original Feynman diagrams [

The number of loop integration variables in the LTD rep-

resentation given by Eq. (4) is directly related to the

topological independent loops present at the level of the

Feynman diagram. However, the reduced Feynman graph

has a fewer number of graphical loops, also called eloops,

since bunches of propagators connecting the same vertices

were collapsed to a single edge. Furthermore, it turns

out that vertices, edges and eloops are the only required

ingredients to obtain the causal representation of any

multiloop scattering amplitude [50,51].

Speciﬁcally, given a reduced Feynman graph, the asso-

ciated causal propagators 1

/λ±

correspond to the set of

connected binary partitions of vertices. Also, they can be

graphically interpreted as lines cutting the diagrams into

two disconnected pieces, in a full analogy with Cutkosky’s

formulation [

]. Then, the representation in Eq. (4) can

be constructed from all the possible causal compatible

combinations of

causal propagators, the so-called causal

entangled thresholds. The number

is known as the order

of the diagram and deﬁned as

V−

1 [

]. It

naturally induces a topological classiﬁcation of families

of Feynman diagrams, or equivalently, reduced Feynman

graphs. The family with

= 1 is known as Maximal

Loop Topology (MLT) [

] and only involves two vertices;

a Next-to-Maximal Loop Topology (NMLT) has three

vertices, and so on.

It was shown that causal entangled thresholds can be

identiﬁed by imposing geometrical selection rules. These

rules are deeply connected to the algebraic formulation

presented in Refs. [50,90], and they establish that:

When considering all the

causal thresholds that

can be simultaneously entangled, all the edges can

be set on shell simultaneously. This is equivalent to

impose that the causal entangled thresholds depends

on the on-shell energies q(+)

i,0of all the edges.

Two diﬀerent causal propagator

λi

and

λj

can be

simultaneously entangled if they do not cross each

other. In other words, this means that the asso-

ciated partition of vertices do not intersect or are

totally included one in the other.

The momentum of the edges that crosses a given

binary partition of vertices must be consistently

aligned, i.e., momentum must ﬂow from one parti-

tion to a diﬀerent one.

A careful explanation of these causal rules and their geo-

metric interpretation was presented in Ref. [

], including

pedagogical examples and practical cases of use. By im-

posing these three rules on the set of all the possible

combinations of

thresholds, we construct Σ, namely the

set of all causal entangled thresholds leading to Eq. (4).

Furthermore, as previously shown in Ref. [

], the third

condition is equivalent to ordering the internal edges in

such a way that the reduced Feynman diagram is a di-

rected acyclic graph. This means that there cannot be

closed cycles, allowing the information to propagate con-

sistently from one partition of the diagram to the other;

this condition is necessary for having a causal-compatible

partition. For this reason, given a reduced Feynman

graph, it is crucial to eﬃciently detect all the associated

directed acyclic graphs, since they are a vital ingredient to

reconstruct the LTD causal representation. To clarify this

discussion, we present an explicit application example in

App. A.

IV. HAMILTONIAN FORMALISM FOR

CAUSAL-FLOW IDENTIFICATION

The ﬁrst proof-of-concept of a quantum algorithm for

Feynman loop integrals was presented in Ref. [

], man-

aging to successfully unfold the causal conﬁgurations of

multiloop Feynman diagrams. The implementation fol-

lows a modiﬁed Grover’s quantum algorithm to identify

the presence of directed acyclic conﬁgurations, represent-

ing the causal solutions, over diﬀerent subloops of the

multiloop topologies.

The aim of this Section is to deﬁne a Hamiltonian

whose ground state corresponds to a superposition of

all the acyclic conﬁgurations of a given multiloop Feyn-

man diagram2. We ﬁrst discuss the construction of such

Hamiltonian based on direct inspection of the multiloop

topology, and then we discuss a more general approach

based on the adjacency matrix of a graph, an object that

concentrates all the information regarding the orientation

of edges and the vertices that they connect within a re-

duced Feynman graph (as deﬁned in Sec. III). We also

give a more formal presentation, relying on concepts and

notations from graph theory.

A. Cycle detection via Hamiltonian optimization

Let us consider a generic undirected graph

= (

V, E

with

a set of distinct vertices and

E⊂V×V

a set of

edges (also known as links). Given the graph

, one can

build a set

of 2

|E|

possible directed graphs where we

associate a speciﬁc direction to each link. We will denote

the subset of directed acyclic graphs as e

DG⊂ DG.

In order to formulate this problem as a Hamiltonian

optimization, we ﬁrst ﬁx the encoding for all the possible

(classical) solutions as elements of the computational basis

To avoid any possible confusion, we would like to emphasize that

this is not the Hamiltonian of the theory from which the Feynman

rules are extracted, but rather a function whose minimization

sets by construction the causal orientations of the diagram.

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VariationalquantumeigensolverforcausalloopFeynmandiagramsanddirectedacyclicgraphsGiuseppeClemente,1,∗AriannaCrippa,1,†KarlJansen,1,‡SelomitRam´ırez-Uribe,2,3,4,§Andr´esE.Renter´ıa-Olivo,2,¶Germ´anRodrigo,2,∗∗GermanF.R.Sborlini,5,6,††andLuizValeSilva2,‡‡1DeutschesElektronen-SynchrotronDESY,Platanenal...

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Variational quantum eigensolver for causal loop Feynman diagrams and directed acyclic graphs Giuseppe Clemente1Arianna Crippa1Karl Jansen1Selomit Ram ırez-Uribe2 3 4 Andr es

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