Impact of amplitude and phase damping noise on quantum reinforcement learning: challenges and opportunities

Eu . Phys. J. Spec. Top.
h ps://doi.o g/10.1140/epjs/s11734-025-01760-3
THE EUROPEAN
PHYSICAL JOURNAL
SPECIAL TOPICS
Regula A icle
Impac o ampli ude and phase damping noise
on quan um ein o cemen lea ning: challenges
and oppo uni ies
Ma ´ıa Lau a Oli e a-A encio1,a, Lucas Lama a2,b,andJes´us Casado-Pascual1,3,c
1F´ısica Te´o ica, Uni e sidad de Se illa, Apa ado de Co eos 1065, 41080 Se ille, Spain
2Depa amen o de F´ısica A ´omica, Molecula y Nuclea , Uni e sidad de Se illa, 41080 Se ille, Spain
3Mul idisciplina y Uni o Ene gy Science, Uni e sidad de Se illa, 41080 Se ille, Spain
Recei ed 31 Ma ch 2025 / Accep ed 23 June 2025
©The Au ho (s) 2025
Abs ac Quan um machine lea ning (QML) is an eme ging field wi h significan po en ial, ye i emains
highly suscep ible o noise, which poses a majo challenge o i s p ac ical implemen a ion. While a ious
noise mi iga ion s a egies ha e been p oposed o enhance algo i hmic pe o mance, he impac o noise
is no ully unde s ood. In his wo k, we in es iga e he effec s o ampli ude and phase damping noise on
a quan um ein o cemen lea ning algo i hm. Th ough analy ical and nume ical analysis, we assess how
hese noise sou ces influence he lea ning p ocess and o e all pe o mance. Ou findings con ibu e o a
deepe unde s anding o he ole o noise in quan um lea ning algo i hms and sugges ha , a he han
being pu ely de imen al, una oidable noise may p esen oppo uni ies o enhance QML p ocesses.
1 In oduc ion
Quan um machine lea ning (QML) is a apidly g owing field wi hin quan um echnologies ha seeks o pe o m
machine lea ning asks mo e efficien ly han classical supe compu e s in e ms o ime, space, and ene gy esou ces
[1–5]. Le e aging quan um supe posi ion and en anglemen , he goal is o enable a mo e scalable implemen a ion
o a ious machine lea ning algo i hms using quan um compu e s [6].
A majo challenge in quan um compu ing, which also affec s QML, is he agili y o highly en angled many-
body quan um s a es. These s a es a e suscep ible o in e ac ions wi h unin ended quan um sys ems, leading o
decohe ence and he loss o compu a ional p ope ies necessa y o sol e a gi en p oblem. Howe e , an eme ging
pe spec i e in QML explo es decohe ence and dissipa ion no only as an obs acle bu also as a po en ial esou ce o
enhancing quan um lea ning [7–13]. This app oach is mo i a ed by he ac ha effec i e lea ning, bo h classical
and quan um, o en equi es some o m o nonlinea i y. Since isola ed quan um sys ems e ol e linea ly (i.e.,
uni a ily), some o m o coupling—whe he h ough quan um measu emen (p ojec i e, weak, e c.) o dissipa i e
and/o dephasing p ocesses go e ned by a mas e equa ion (Ma ko ian o non-Ma ko ian)—may play a c ucial
ole in enabling iche lea ning dynamics. In his con ex , ecen wo k has shown ha ca e ully uned ampli ude,
phase, and depola izing noise can imp o e he pe o mance o a ia ional quan um algo i hms, u he suppo ing
he idea ha noise can be ha nessed as a use ul ea u e in QML [14].
In p e ious wo ks, we ha e explo ed he ole o he mal dissipa ion in QML p o ocols [10]. Ou findings indica e
ha , a he han being pu ely de imen al, he mal dissipa ion can some imes enhance he lea ning p ocess. In his
pape , we build upon his esea ch by specifically analyzing phase damping noise (PDN) and ampli ude damping
noise (ADN) in a p o ocol o quan um ein o cemen lea ning. By in es iga ing hese ypes o noise, we aim o
deepen he unde s anding o when and how hese ypes o noise can be beneficial in quan um lea ning p o ocols.
The emainde o his wo k is s uc u ed as ollows. In Sec . 2, we p o ide a b ie desc ip ion o he p oblem
unde s udy, he ypes o noise conside ed, and he algo i hm o which hey a e applied. In Sec . 3,wep esen he
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Eu . Phys. J. Spec. Top.
nume ical esul s and analyze he condi ions unde which noise can enhance he algo i hm pe o mance. Finally,
in Sec . 4, we summa ize ou findings and conclusions.
2 P oblem s a emen
In he ein o cemen lea ning algo i hm p esen ed in Re . [15], he agen A is a known and con ollable quan um
sys em desc ibed by a s a e ec o |φ, o equi alen ly, by he associa ed densi y ope a o ρ=|φφ|. Fo he sake
o cla i y, we will es ic ou sel es o he simples case, whe e he sys em is a single qubi wi h compu a ional
basis {|0,|1}. The in e ac ion o he en i onmen E wi h he agen A o e a ime in e al τis cha ac e ized by
he uni a y ime e olu ion ope a o U(τ)=e−iHτ/, whe e His an unknown Hamil onian. This Hamil onian,
w i en in e ms o i s unknown exci ed s a e |eand g ound s a e |g, akes he o m H=ω
2(|ee|−|gg|),
whe e ωis a cha ac e is ic equency o he sys em. The goal o he algo i hm is o “lea n” how o cons uc , a
leas app oxima ely, ei he o he s a iona y s a es |eo |g, despi e he challenge posed by he ac ha hese
s a iona y s a es a e unknown. To his end, he algo i hm exploi s he ac ha , o τ=τn=2nπ/ω wi h n
being a na u al numbe , he s a iona y s a es |ee|and |gg|a e he only pu e s a es in a ian unde he uni a y
ime e olu ion, i.e., he only pu e s a es sa is ying he equa ion U(τ)|φφ|U†(τ)=|φφ|. Consequen ly, ac ions
compa ible wi h his p ope y a e ewa ded, while hose ha a e incompa ible a e penalized. The case τ=τnis an
excep ion since U(τn)=(−1)nI, wi h Ibeing he iden i y ope a o , making all pu e s a es sa is y he condi ion
U(τn)|φφ|U†(τn)=|φφ|. Fo a de ailed explana ion o why his algo i hm can be classified as a quan um
ein o cemen lea ning algo i hm, see Re s. [10,15].
In he p esence o noise, he ime e olu ion ha go e ns he in e ac ion be ween he en i onmen and he agen
is no longe uni a y. Specifically, o PDN and ADN— he cases analyzed in his wo k— he ime e olu ion o e an
in e al τ akes he o m
Eτ(ρ)=U(τ)E0(τ)ρE†
0(τ)+E1(τ)ρE†
1(τ)U†(τ), (1)
whe e {E0(τ), E1(τ)}a e K aus ope a o s [16] whose o m depends on he ype o noise conside ed [6]. Speci -
ically, in he case o PDN, he K aus ope a o s and he co esponding ime e olu ion ake he o m E0(τ)=
|gg|+e−τ/TD|ee|,E1(τ)=√1−e−2τ/TD|ee|,and
Eτ(ρ)=ρee|ee|+ρgg|gg|+e−τ/TDe−iωτ ρeg|eg|+eiωτ ρge|ge|, (2)
whe e ραβ =α|ρ|β, wi h α,β∈{e,g},andTDis a pa ame e known as he decohe ence ime [17,18]. Fo ADN,
he K aus ope a o E0(τ) emains he same as o PDN, while E1(τ)=√1−e−2τ/TD|ge|, leading o he ime
e olu ion
Eτ(ρ)=|gg|+ρeee−2τ/TD(|ee|−|gg|)
+e−τ/TDe−iωτ ρeg|eg|+eiωτ ρge|ge|.(3)
In his case, in addi ion o he supp ession o he off-diagonal elemen s o he densi y ope a o in he eigens a e
basis o H(decohe ence), he e is also a decay o he exci ed s a e |e o he g ound s a e |gwi h a mean decay
ime o TD/2. To some ex en , his ype o noise can be ega ded as a specific ins ance o he he mal dissipa ion
analyzed in Re . [10], bu a absolu e empe a u e equal o ze o. No e ha , as in he uni a y ime e olu ion wi hou
noise, he s a iona y s a es |ee|and |gg|a e he only pu e s a es in a ian unde he non-uni a y e olu ion
induced by PDN [Eq. (2)]. In con as , unde ADN [Eq. (3)], only he g ound s a e |gg| emains in a ian . This
asymme y will be c ucial o he algo i hm pe o mance in he p esence o ADN, as discussed la e . Nex , we
p o ide a de ailed desc ip ion o how he algo i hm wo ks in he p esence o hese ypes o noise.
The algo i hm consis s o a la ge numbe o i e a ions, indexed by a na u al numbe k. The uni a y ans o -
ma ion gene a ed in he k h i e a ion is deno ed by Dk. Mo eo e , in each i e a ion, a nume ical alue wi hin
he in e al [0, 1] is assigned o a pa ame e wk, called he explo a ion pa ame e . The goal is o Dk|φφ|D†
k,
wi h |φφ|being a gi en s a e, o g adually app oach ei he o he a ge s a es |ee|o |gg|as he numbe
o i e a ions inc eases. S a ing wi h he ini ial alues D0=Iand w0= 1, he alues o Dk+1 and wk+1 a e
i e a i ely upda ed om he p e ious alues Dkand wkacco ding o he ollowing s eps:
1. The uni a y ans o ma ion Dkis applied o one o he compu a ional basis s a e, say, he s a e |00|, o
cons uc he s a e ρk=Dk|00|D†
k.
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Eu . Phys. J. Spec. Top.
2. The esul ing sys em e ol es o a ime τ, yielding he ans o med s a e ρ
k=Eτ(ρk), whe e Eτis gi en by
Eq. (2)o (3), depending on he ype o noise.
3. The ini ial uni a y ans o ma ion is hen e e sed, yielding ρ
k=D†
kρ
kDk. No e ha , in he p esence o PDN,
i ρkhad eached one o he a ge s a es |ee|o |gg|, hen, a e he ime e olu ion, ρ
kwould emain in
ha s a e and, consequen ly, ρ
kwould be equal o |00|. In he p esence o ADN, his would only hold i ρk
had eached he g ound s a e |gg|.
4. A measu emen in he compu a ional basis is pe o med on he sys em ob ained in he p e ious s ep, yielding a
esul mk∈{0, 1}. As deduced om he p e ious discussion, he measu emen ou come mk= 0 is compa ible
wi h ρkha ing eached one o he a ge s a es, while he ou come mk= 1 is incompa ible wi h his.
5. Depending on he ou come o mk, he ollowing p ocedu e is applied: •I mk= 0, since he ou come is
compa ible wi h ha ing eached one o he a ge s a es, a ewa d is g an ed by educing he explo a ion
pa ame e acco ding o wk+1 = wk, whe e ∈(0, 1) is a pa ame e known as he ewa d a e. Addi ionally,
he uni a y ans o ma ion Dk emains unchanged, i.e., Dk+1 =Dk, and he p ocess e u ns o s ep 1. •I
mk= 1, since he ou come is incompa ible wi h ha ing eached one o he a ge s a es, a punishmen is
applied by inc easing he explo a ion pa ame e o wk+1 = min(pwk, 1), whe e p>1 is a pa ame e known as
he punishmen a e and he min unc ion ensu es ha wk+1 does no exceed 1. Addi ionally, h ee pseudo-
andom numbe s αk,βk,andγka e gene a ed, uni o mly dis ibu ed wi hin he explo a ion in e al [−wkπ,
wkπ], and used o cons uc he pseudo- andom o a ion
Rk=Dke−iβkY/2e−iγkZ/2e−iαkX/2D†
k, (4)
whe e X=|01|+|10|,Y=−i(|01|−|10|), and Z=|00|−|11|a e he Pauli ope a o s. Finally, he
uni a y ans o ma ion Dkis upda ed o Dk+1 =RkDk=Dke−iβkY/2e−iγkZ/2e−iαkX/2, he qubi is es o ed
o i s o iginal s a e |00|by applying he uni a y ans o ma ion X, and he p ocess e u ns o s ep 1.
The p e iously desc ibed algo i hm is conside ed o con e ge i he explo a ion pa ame e wkapp oaches ze o as
kinc eases. In his case, he pseudo- andom o a ions Rk end o he iden i y ope a o , and consequen ly, he
uni a y ans o ma ions Dkcon e ge o a cons an alue. Mo eo e , he as e wkapp oaches ze o, he as e he
algo i hm con e ges.
3Resul s
To examine he impac o he p e iously discussed noise sou ces on he algo i hm om he p eceding sec ion, we
ha e implemen ed i using a Hamil onian o he o m
H=ω
4(√3X−Z), (5)
which co esponds o se ing |e=(|0+√3|1)/2and|g=(−√3|0+|1)/2 in he Hamil onian exp ession
in oduced ea lie . To wo k wi h dimensionless quan i ies, we define he pa ame e s ˜τ=ωτ and ˜
TD=ωTD.The
measu emen p ocess desc ibed in i em 4 o he p e ious sec ion is simula ed as ollows: a each i e a ion, we
calcula e he p obabili y o ob aining 0 as he measu emen ou come using he exp ession Pk(0) = T (|00|ρ
k)=
T [|00|D†
kEτ(ρk)Dk]=T [ρkEτ(ρk)], wi h Eτgi en by Eq. (2)o (3), depending on he ype o noise conside ed.
Then, a pseudo- andom numbe χkis gene a ed, uni o mly dis ibu ed in he in e al [0, 1]. I χk≤Pk(0), he
measu emen ou come is mk= 0; o he wise, i is mk=1.
Thanks o he ac ha he exci ed and g ound s a es a e known in he conside ed example, we can use his
knowledge o assess he accu acy o he algo i hm desc ibed in he p e ious sec ion. To his end, a each i e a ion,
we compu e he squa e oo fideli y be ween he s a e ρkand he s a iona y s a es |ee|and |gg|, gi en by
(e)
k=|e|Dk|0| and (g)
k=|g|Dk|0|, espec i ely. Since, a p io i, i is no known which o he wo s a iona y
s a es is close o ρk, i is also con enien o conside he highes fideli y be ween (e)
kand (g)
k, i.e., k= max( (e)
k,
(g)
k). The close he alue o kapp oaches 1 as kinc eases, he mo e accu a e he algo i hm es ima ion o one o
he s a iona y s a es will be.
I is wo h men ioning ha he quan i ies wk, (e)
k, (g)
k,and ka e andom a iables, whose alues will a y om
one ealiza ion o he algo i hm o ano he . The andomness o hese a iables a ises om wo ac o s: fi s , he
inhe en s ochas ic na u e o he measu emen ou come mk, and second, he pseudo- andom selec ion o he angles
αk,βk,andγk. Fo his eason, i is con enien o pe o m a la ge numbe No ealiza ions (in ou calcula ions,
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Eu . Phys. J. Spec. Top.
Fig. 1 Mean fideli y Fkas
a unc ion o he numbe o
i e a ions kin he p esence
o PDN (le panels) and
ADN ( igh panels). Resul s
a e shown o diffe en
dimensionless decohe ence
imes, namely, ˜
TD=1( ed
do ed lines), ˜
TD= 10 (blue
dashed lines), ˜
TD= 100
(g een dashed-do ed lines),
and ˜
TD=∞(black solid
lines), and o wo alues o
he dimensionless e olu ion
ime, specifically, ˜τ= 1 ( op
panels) and ˜τ=2π(bo om
panels). The case ˜
TD=∞
co esponds o he scena io
wi h no noise, whe e he
e olu ion is uni a y
we use N= 1000) and conside he a i hme ic mean o he alues o wk, (e)
k, (g)
k,and kob ained in each
ealiza ion, which will be deno ed as Wk,F(e)
k,F(g)
k,andFk, espec i ely.
In Fig. 1, we depic he mean fideli y Fkas a unc ion o he numbe o i e a ions kin he p esence o PDN
(le panels) and ADN ( igh panels). The esul s a e shown o se e al dimensionless decohe ence imes, namely,
˜
TD= 1 ( ed do ed lines), ˜
TD= 10 (blue dashed lines), ˜
TD= 100 (g een dashed-do ed lines), and ˜
TD=∞(black
solid lines), and o wo alues o he dimensionless e olu ion ime, specifically, ˜τ= 1 ( op panels) and ˜τ=2π
(bo om panels). The decohe ence ime ˜
TD=∞co esponds o he case wi hou noise, whe e he e olu ion is
uni a y. Fo all pa ame e alues conside ed, we ha e e ified he con e gence o he algo i hm by checking ha
he mean explo a ion pa ame e Wkdec eases o ze o as he numbe o i e a ions kinc eases sufficien ly, al hough
hese esul s a e no shown in he figu e. As obse ed in he op le panel, o ˜τ= 1, he effec o PDN on he
algo i hm is minimal, allowing i o pe o m well e en when he decohe ence ime is compa able o he sys em’s
cha ac e is ic imescales. In ac , o ˜
TD= 1, he algo i hm pe o ms sligh ly be e han in he absence o noise.
In con as , o ˜τ=2π, he p esence o PDN conside ably enhances he algo i hm pe o mance o dimensionless
decohe ence imes ha a e no oo la ge, specifically o ˜
TD=1and ˜
TD= 10, wi h be e pe o mance o smalle
decohe ence imes (see bo om le panel). This occu s because, as men ioned in Sec . 2, he uni a y e olu ion
ope a o U(τ) becomes equal o minus he iden i y ope a o o τ=2π/ω, making all s a es in a ian unde he
uni a y e olu ion o e a ime 2π/ω. As a esul , o ˜τ=2πand in he absence o noise, he algo i hm is unable
o dis inguish be ween s a iona y and non-s a iona y s a es, causing i o ail. This is no he case in he p esence
o PDN, in which he s a iona y s a es |ee|and |gg|a e he only pu e s a es ha emain in a ian unde he
non-uni a y e olu ion desc ibed by Eq. (2), e en o ˜τ=2π. This allows he algo i hm o dis inguish be ween
s a iona y and non-s a iona y s a es, esul ing in a significan imp o emen in i s pe o mance wi h espec o he
noise- ee case. In he p esence o ADN ( igh panels), he beha io is consis en wi h ha obse ed o PDN, and
he explana ion o he imp o ed pe o mance in he noisy case compa ed o he noise- ee case in he bo om
igh panel emains applicable. Howe e , in his case, only he s a iona y s a e |gg| emains in a ian unde he
non-uni a y ime e olu ion gi en by Eq. (3), unlike in he PDN case, whe e bo h s a iona y s a es we e in a ian .
To analyze how his diffe ence is eflec ed in he algo i hm pe o mance, Fig. 2shows he mean fideli ies asso-
cia ed wi h he g ound s a e, F(g)
k( op panels), and he exci ed s a e, F(e)
k(bo om panels), as a unc ion o he
numbe o i e a ions k o he PDN case (le panels) and he ADN case ( igh panels). As seen in his figu e, o
PDN, he esul s o F(g)
kand F(e)
kshow li le dependence on he dimensionless decohe ence ime ˜
TDand emain
close o hose ob ained in he noise- ee case, i.e., o ˜
TD=∞. In con as , in he p esence o ADN, he fideli y
F(g)
kinc eases significan ly as ˜
TDdec eases, eaching alues close o 1 o ˜
TD=1and ˜
TD= 10, while F(e)
kexhibi s
a subs an ial decline. This asymme y in he beha io o F(g)
kand F(e)
k, which is obse ed o ADN bu absen in
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Eu . Phys. J. Spec. Top.
Fig. 2 Mean fideli ies
associa ed wi h he g ound
s a e, F(g)
k( op panels),
and he exci ed s a e, F(e)
k
(bo om panels), as a
unc ion o he numbe o
i e a ions k o phase
damping noise (le panels)
and ampli ude damping
noise ( igh panels). The
alues o he dimensionless
decohe ence imes ˜
TDa e
hesameasinFig.1,and
he dimensionless e olu ion
ime is ˜τ=1
he PDN case, s ems om he dis inc p ope ies o he s a iona y s a es unde each ype o noise. As p e iously
men ioned, unde PDN, bo h s a iona y s a es, |ee|and |gg|, emain in a ian unde he non-uni a y ime
e olu ion gi en by Eq. (2). As a esul , he algo i hm con e ges o ei he o hese s a es wi h simila p obabili y,
leading o he compa able alues o F(g)
kand F(e)
kobse ed in he le panels o Fig. 2. In con as , unde ADN
[Eq. (3)], only he g ound s a e |gg| emains in a ian . Consequen ly, he algo i hm p e e en ially con e ges o
his s a e, leading o a significan enhancemen o F(g)
kwhile F(e)
kdec eases acco dingly. As he dimensionless
decohe ence ime ˜
TDinc eases, he non-uni a y ime e olu ion (3) g adually app oaches he uni a y case ˜
TD=∞,
whe e bo h |ee|and |gg|a e once again in a ian . This p og essi ely educes he asymme y obse ed o lowe
alues o ˜
TD.
Acco ding o he p e ious discussion, one migh conclude ha he p esence o ADN would be ad an ageous
p ima ily when aiming o cons uc he g ound s a e |gg|, while i would be de imen al i he goal we e o
cons uc he exci ed s a e |ee|. Howe e , he uni a y ans o ma ions Dkob ained h ough he p e iously
p esen ed algo i hm also allow o an app oxima e p epa a ion o he exci ed s a e by applying hem o he
compu a ional basis s a e |11|ins ead o |00|. Indeed, due o he uni a y na u e o he ope a o s Dk, he s a e
ec o s Dk|1and Dk|0a e o hogonal. Consequen ly, i Dk|00|D†
kis close o he g ound s a e |gg|, hen
Dk|11|D†
kwill be close o he exci ed s a e |ee|. To confi m his, Fig. 3displays he mean fideli ies associa ed
wi h he g ound s a e ( op panels) and he exci ed s a e (bo om panels) as unc ions o he numbe o i e a ions
k. The le panels co espond o he s a es Dk|00|D†
k, while he igh panels co espond o he s a es Dk|11|D†
k.
The alues o ˜
TDand ˜τa e he same as in Fig. 3. As seen in he figu e, while he s a es Dk|00|D†
kg adually
app oach he g ound s a e |gg|as kinc eases o ˜
TD=1and ˜
TD= 10 ( op le panel), he s a es Dk|11|D†
k
simila ly con e ge o he exci ed s a e |ee|(bo om igh panel). In summa y, he uni a y ans o ma ions Dk
allow o he calcula ion o bo h he g ound and exci ed s a es by simply applying hem o diffe en compu a ional
basis s a es.
4 Conclusions
In his wo k, we ha e analyzed he impac o wo common ypes o noise—PDN and ADN—on he ein o cemen
lea ning quan um algo i hm p oposed in Re . [15]. Th ough he s udy o specific examples, we ha e shown ha
he p esence o noise does no necessa ily hinde he algo i hm pe o mance; in some cases, i can e en ha e a
beneficial effec .
123

Eu . Phys. J. Spec. Top.
Fig. 3 Mean fideli ies
associa ed wi h he g ound
s a e, F(g)
k( op panels),
and he exci ed s a e, F(e)
k
(bo om panels), as a
unc ion o he numbe o
i e a ions k. The le panels
co espond o he s a es
Dk|00|D†
k, while he igh
panels co espond o he
s a es Dk|11|D†
k.The
alues o ˜
TDand ˜τa e he
same as in Fig. 2
In pa icula , we ha e demons a ed ha o ce ain alues o he e olu ion ime τ, he p esence o noise has
li le impac when he algo i hm accu acy is assessed using he mean fideli y Fk. Howe e , o o he alues o τ,
noise can significan ly enhance he algo i hm pe o mance. Fu he mo e, we ha e shown ha he wo ypes o
noise affec he s a iona y-s a e fideli ies, F(g)
kand F(e)
k, in ma kedly diffe en ways. While PDN influences bo h
fideli ies symme ically, ADN in oduces an asymme y, a o ing con e gence o he g ound s a e o e he exci ed
s a e. We ha e explained his diffe ence by analyzing he pu e s a es ha emain in a ian unde he non-uni a y
e olu ion associa ed wi h each ype o noise.
Al hough his asymme y migh sugges ha ADN enhances he p epa a ion o he sys em in he g ound s a e
compa ed o he noise- ee case, we ha e also demons a ed ha he uni a y ans o ma ion gene a ed by he
algo i hm also enables he p epa a ion o he exci ed s a e simply by applying i o a diffe en compu a ional basis
s a e.
In his wo k, o he sake o cla i y and simplici y in he desc ip ion, we ha e es ic ed ou sel es o he case o
a single qubi . Howe e , in u u e esea ch, we aim o analyze how he esul s desc ibed he e gene alize when he
numbe o qubi s is inc eased.
Acknowledgemen s The au ho s acknowledge p ojec PID2022-136228NB-C22 unded by MCIN/AEI/
10.13039/501100011033 and by “ERDF A way o making Eu ope”, EU. Fu he mo e, his wo k has been pa ially
financially suppo ed by he Minis y o Economic Affai s and Digi al T ans o ma ion o he Spanish Go e nmen
h ough he QUANTUM ENIA p ojec call—Quan um Spain p ojec , and by he Eu opean Union h ough he Reco e y,
T ans o ma ion and Resilience Plan—Nex Gene a ionEU wi hin he amewo k o he “Digi al Spain 2026 Agenda”. I
has also been co-financed by EU, Minis e io de Hacienda y Funci´on P´ublica, FEDER and Jun a de Andaluc´ıa (p ojec
SOL2024-31833).
Funding Funding o open access publishing: Uni e sidad de Se illa/CBUA.
Da a a ailabili y All simula ion sc ip s used o p oduce he esul s p esen ed in his pape a e a ailable a h ps://gi
hub.com/MLOA25/YQS.
Open Access This a icle is licensed unde a C ea i e Commons A ibu ion 4.0 In e na ional License, which pe mi s
use, sha ing, adap a ion, dis ibu ion and ep oduc ion in any medium o o ma , as long as you gi e app op ia e c edi
o he o iginal au ho (s) and he sou ce, p o ide a link o he C ea i e Commons licence, and indica e i changes we e
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123
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