A genetic algorithm to generate maximally orthogonal frames in complex space

Mach. Lea n.: Sci. Technol. 6(2025) 035022 h ps://doi.o g/10.1088/2632-2153/ad 53d OPEN ACCESS RECEIVED 28 May 2025 REVISED 9 July 2025 ACCEPTED FOR PUBLICATION 28 July 2025 PUBLISHED 5 Augus 2025 O iginal Con en om his wo k may be used unde he e ms o he C ea i e Commons A ibu ion 4.0 licence. Any u he dis ibu ion o his wo k mus main ain a ibu ion o he au ho (s) and he i le o he wo k, jou nal ci a ion and DOI. PAPER A gene ic algo i hm o gene a e maximally o hogonal ames in complex space Sebas ián Roca-Je a 1,2,∗and Juan Román-Roche1,3 1Ins i u o de Nanociencia y Ma e iales de A agón (INMA), CSIC-Uni e sidad de Za agoza, 50009 Za agoza, Spain 2Depa amen o de Física de la Ma e ia Condensada, Uni e sidad de Za agoza, 50009 Za agoza, Spain 3Depa amen o de Física Teó ica, Uni e sidad de Za agoza, 50009 Za agoza, Spain ∗Au ho o whom any co espondence should be add essed. E-mail: s oca@uniza .es Keywo ds: e olu iona y algo i hms, me ic geome y, quan um in o ma ion, ame heo y, packing p oblem, homson p oblem, sic po m Abs ac A ame is a gene aliza ion o a basis o a ec o space o a edundan o e spanning se whose ec o s a e linea ly dependen . F ames ind applica ions in signal p ocessing and quan um in o ma ion heo y. We p esen a gene ic algo i hm ha can gene a e maximally o hogonal ames (MOFs) o a bi a y size nin d-dimensional complex space. Fi s , we o malize he concep o MOF and demons a e ha i depends on he choice o an ene gy unc ion o weigh he di e en pai wise o e laps be ween ec o s. Then, we discuss he ela ion be ween di e en ene gy unc ions and well-known ame a ie ies such as igh and G assmannian ames and complex p ojec i e p-designs. Ob aining MOFs poses a global non-con ex minimiza ion p oblem. We discuss he ela ion wi h es ablished nume ical p oblems such as he Thomson p oblem and he p oblem o inding op imal packings in complex p ojec i e space. To ackle he minimiza ion, we design a hyb id gene ic algo i hm ha ea u es local op imiza ion o he pa en s. To assess he pe o mance o he algo i hm, we p opose wo isualiza ion echniques ha allow us o analyze he cohe ence and uni o mi y o high-dimensional ames. The gene ic algo i hm is able o p oduce highly-symme ic uni e sal ames, such as equiangula igh ames, symme ic, in o ma ionally comple e, posi i e ope a o - alued measu emen s and maximal se s o mu ually unbiased bases, o con igu a ions o up o d=6 and n=36, wi h un imes o he o de o se e al minu es on a egula desk op compu e o he la ges con igu a ions. 1. In oduc ion A he beginning o e e y in oduc o y cou se on linea algeb a, one is p esen ed wi h he concep o a basis o a ec o space. A basis is a collec ion o ec o s ha a e linea ly independen and o m a spanning se o he ec o space. In ac , he dimension o a ec o space is de ined as he numbe o elemen s o a basis. I he ec o space is u he endowed wi h an inne p oduc , he condi ion o linea independence is o en supe seded by he s onge condi ion o o hogonali y be ween he basis ec o s, hus de ining an o hogonal basis. A consequence o hese de ini ions is ha in an inne -p oduc ( ec o ) space o dimension done can only c ea e se s o a mos do hogonal ec o s (which would cons i u e o hogonal bases). In ligh o his ealiza ion, i is only na u al o wonde how o gene alize he concep o an o hogonal basis o se s o mo e han d ec o s. These o e -spanning se s a e ypically e e ed o as ames, which a e de ined and cha ac e ized in ame heo y [1,2]. The edundancy o ames p o ides obus ness agains e o s and is hus use ul in he encoding and e ie al o signals, wi h applica ions in elecommunica ions and (quan um) in o ma ion heo y [3]. Fo ins ance, equiangula igh ames (ETFs) wi h d2 ec o s p o ide symme ic, in o ma ionally comple e, posi i e ope a o - alued measu es (SIC-POVMs) ha a e op imal o quan um s a e omog aphy [4–9]. © 2025 The Au ho (s). Published by IOP Publishing L d Mach. Lea n.: Sci. Technol. 6(2025) 035022 S Roca-Je a and J Román-Roche Se e al amilies o ames wi h di e en p ope ies can be de ined, such as igh and G assmannian ames [10]. Many o hese p ope ies can be a ibu ed o pa icula ins ances o wha we call maximally o hogonal ames (MOFs): ames o n>d ec o s ha , al hough no comple ely o hogonal among hemsel es, a e maximally o hogonal in he sense ha hei pai wise inne p oduc s a e minimal. As o hogonali y is a pai wise p ope y, inc easing he o hogonali y be ween a gi en pai o ec o s will o en come a he expense o dec easing he o hogonali y be ween o he pai s o ec o s. Depending on how much alue one assigns o he o hogonali y o each pai , one will a o one ‘maximally o hogonal’ se o ec o s o ano he . Thus, he i s s ep in his pape is p o iding a o mal de ini ion o MOFs o a complex space o a bi a y dimension. Once a med wi h his de ini ion, we discuss hei p ope ies and es ablish hei ela ion wi h no able ame amilies such as igh and G assmannian ames, ETFs [10], SIC-POVMs [11], maximal se s o mu ually unbiased bases (MUBs) [12,13] and complex p ojec i e -designs [14,15]. We also show ha inding MOFs cons i u es, in mos cases, a nume ical op imiza ion p oblem. We discuss he ela ion wi h es ablished nume ical p oblems such as he Thomson p oblem [16], he p oblem o inding op imal packings in complex p ojec i e space [17] and he p oblem o inding op imal e e ence s a es o quan um classi ie s [18,19]. Inspi ed by he use o a gene ic algo i hm o he Thomson p oblem [20], we p esen a gene ic algo i hm ha is capable o gene a ing MOFs o a bi a y size in a complex space o a bi a y dimension. Then, we isualize and measu e he quali y o he ames p oduced by he gene ic algo i hm. In he cases whe e he p oblem o inding MOFs is equi alen o he Thomson p oblem o he p oblem o inding op imal packings in complex p ojec i e space, we compa e he gene ic algo i hm wi h s a e-o - he-a (SOTA) nume ical me hods. The emainde o he manusc ip is o ganized as ollows. Sec ion 2is dedica ed o de ining MOFs, discussing hei p ope ies, and ela ing hem o no able ame amilies. In sec ion 3we discuss ela ed nume ical p oblems. The gene ic algo i hm is p esen ed in sec ion 4. In sec ion 5, we isualize and discuss he esul s ob ained wi h he algo i hm and i s pe o mance. We end he pape wi h he conclusions and ou look o ou wo k in sec ion 6. We also p o ide echnical de ails and complemen a y esul s in he appendices. 2. MOFs 2.1. De ini ion The cen al objec o ou discussion is a se o ec o s Φd,n={|ϕi⟩}n i=1⊂Cdwi h Cda ini e complex coo dina e space o dimension d. The canonical sesquilinea inne p oduc o wo ec o s |ϕ1⟩,|ϕ2⟩is deno ed ⟨ϕ1|ϕ2⟩. De ini ion 2.1. [1]. A se o ec o s Φd,n={|ϕi⟩}n i=1⊂Cdwi h n⩾dis a ame o Cdi he e exis ame bounds 0 <A⩽B<∞such ha o e e y | ⟩∈Cd A∥ ∥⩽ n ∑ i=1|⟨ |ϕi⟩|2⩽B∥ ∥.(1) I one can se A=Bin equa ion (1), he ame is said o be igh [21]. Fo n=d, a ame is jus a basis. I |⟨ϕi|ϕi⟩|=1∀ϕi∈Φd,n,Φd,nis a uni no m ame. Wi hou loss o gene ali y, we will assume ha all ec o s a e o uni no m in he ollowing. In a ec o space o ini e dimension, such as he ones conside ed in his pape , a ame is essen ially a ini e spanning se . We can o mula e an ene gy associa ed o a se EW(Φd,n) = ∑ i=j W(|⟨ϕi|ϕj⟩|),(2) wi h W, he weigh ing unc ion, an a bi a y eal- alued and inc easing unc ion well-de ined in he in e al [0,1]. Such an ene gy g ows whene e he o e lap ( he modulus o he inne p oduc ) o wo ec o s inc eases, ce e is pa ibus, and hus i will be minimized by a se wi h la ge o e -all o hogonali y. We deno e a minimizing se as ¯ Φd,n. P oposi ion 2.1. A se o uni no m ec o s ¯ Φd,n={|ϕi⟩}n i=1⊂Cd ha minimizes EW(2) is a uni no m ame. P oo . We ha e o show ha ¯ Φd,nis a spanning se o Cd. Fi s , i n=dany se ha minimizes EWis an o hogonal basis and hus spans Cd. In he ollowing we assume n>dand p o e he p oposi ion by con a- dic ion. Le us assume ha ¯ Φd,nis no spanning and deno e i s o hogonal complemen as ¯ Φ⊥ d,n. No e ha W(0)⩽W(|⟨ϕi|ϕj⟩|)⩽W(1), wi h equali y in he lowe bound i and only i |ϕi⟩and |ϕj⟩a e o hogonal o each o he . Le us de ine ϕ⊥ k={|ϕj=k⟩∈ ¯ Φd,n|W(|⟨ϕk|ϕj⟩|)>W(0)}.(3) 2 Mach. Lea n.: Sci. Technol. 6(2025) 035022 S Roca-Je a and J Román-Roche Le us choose a ec o |ϕk⟩∈ ¯ Φd,nsuch ha ϕ⊥ kis no he emp y se . A non-spanning se wi h n>d ec o s mus con ain al leas h ee such ec o s. Then, he ame ¯ Φ′ d,n o med by eplacing |ϕk⟩by |ϕ′ k⟩∈ ¯ Φ⊥ d,nin ¯ Φd,n has ene gy EW(¯ Φ′ d,n) = EW(¯ Φd,n)−ϵwi h ϵ=2∑ j∈ϕ⊥ k(W(|⟨ϕk|ϕj⟩|)−W(0))>0,(4) which implies ha ¯ Φd,ndoes no minimize EW. The e o e we e e o any se ha minimizes EWas a MOF. Depending on he unc ional o m o he weigh ing unc ion, W, he impac o la ge and small o e laps on he ene gy can a y and he MOF will depend on W. We deno e i ¯ Φd,n[W]. Thus, he e is no a unique de ini ion o a MOF. 2.2. P ope ies o he di e en MOFs 2.2.1. Weigh ing unc ions: p- ame po en ial and Riesz s-ene gy We conside wo main amilies o weigh ing unc ions: o W(x) = x2p he ene gy becomes a p- ame po en ial [4,22,23] FPp(Φd,n) = ∑ i=j|⟨ϕi|ϕj⟩|2p(5) and o W(x) = D−s(x), wi h D(x) he ace o cho dal dis ance D(x) = 2√1−x2,(6) he ene gy becomes a p ojec i e Riesz s-ene gy [23,24] REs(Φd,n) = ∑ i=j(2√1−|⟨ϕi|ϕj⟩|2)−s .(7) To be p ecise, D−s(x)is no pe se a alid weigh ing unc ion as i is no de ined in he limi x→1. This can be ixed, by imposing ha D−s(1)=+∞, which would co espond o aking he limi om he le , o by de ining he unc ion by pa s, such ha D−s(1−ϵ⩽x⩽1) = D−s(1−ϵ)wi h ϵas a ee pa ame e . In p ac ice, we do he la e . In he ollowing, we discuss he ela ion be ween he p- ame po en ial and he Riesz s-ene gy and he p ope ies o he ames ha minimize hem. 2.2.2. Riesz s-ene gy and asymp o ic uni o mi y The main in e es o he Riesz s-ene gy is ha i allows us o eason in e ms o he poppy-seed bagel heo em, which s a es ha he minimal-ene gy a angemen o npa icles cons ained o a bounded D-dimensional su ace and subjec o a po en ial o he o m −s, whe e is he dis ance be ween pa icles, ends o be uni o mly dis ibu ed o la ge nwhen s⩾D[25]. To unde s and his esul , one mus hink physically in e ms o sho - and long- ange in e ac ions. Fo la ge s, la ge o e laps (sho - ange in e ac ions) a e weigh ed mo e hea ily in he ene gy ela i e o small o e laps (long- ange in e ac ions). This implies ha in he minimiza ion p ocess ec o s p io i ize dec easing he ew la ge o e laps wi h hei nea es neighbo s be o e ying o minimize a la ge numbe o small o e laps wi h dis an neighbo s. The esul ing a angemen is uni o m because each ec o is mos ly conce ned wi h being apa om i s nea es neighbo s and almos insensi i e o he global a angemen o ec o s. Fo small s, he opposi e is ue, ec o s p io i ize maximizing he numbe o ec o s ha hey ha e a small o e lap wi h, which may come a he cos o inc easing he o e lap wi h a ew nea es neighbo s, leading o c owding and lack o uni o mi y. The Riesz s-ene gy is in a ian unde changes in he global phase o he ec o s, so we can iden i y equi alence classes o all uni ec o s di e ing by a global phase, which essen ially de ines he complex p ojec i e space CPd−1. A pa ame e iza ion o CPd−1in eal coo dina es o ms a mani old o dimension 2(d−1). Following he poppy-seed bagel heo em, we will ha e o se s⩾2(d−1) o gene a e MOFs, ¯ Φd,n[s], ha a e asymp o ically uni o m, in he sense o uni o mly dis ibu ed on he mani old ha pa ame izes CPd−1, o n→∞[23]. The same easoning can be applied o he ame po en ial (1). Al hough we canno in oke he poppy-seed bagel heo em o p oduce p ecise egimes depending on he alue o plike we ha e jus done o he Riesz s-ene gy, he a gumen on he basis o compe ing sho - and long- ange in e ac ions emains alid. Small alues o pgi e mo e ela i e impo ance o long- ange in e ac ions and hus a o less uni o m ames, and ice e sa. 3 Mach. Lea n.: Sci. Technol. 6(2025) 035022 S Roca-Je a and J Román-Roche 2.2.3. Complex p ojec i e p-designs and uni o mi y Uni o mi y is only p ope ly de ined in he dis ibu ion sense, o n→∞. In his limi , i is use ul o iew uni o mi y as he p ope y o a ame ha in eg a ing a unc ion o e he Haa measu e o he co esponding mani old is he same as a e aging he unc ion o e he ame elemen s. Fo a ini e ame, an a e age o e he ame elemen s will only p o ide an app oxima ion o he in eg al. In his con ex , a ame is said o be a complex p ojec i e p-design i a e aging a polynomial o deg ee po less o e he ame elemen s is equi alen o in eg a ing he polynomial o e he Haa measu e o he co esponding mani old [5]. In layman’s e ms, ini e ames canno be uni o m, bu he ame ec o s can be su icien ly e enly dis ibu ed on he mani old ha a e aging simple enough unc ions o e hem is equi alen o a e aging o e he ull mani old. The p- ame po en ial obeys he Welch bound FPp(Φd,n)⩾n2 (d+p−1 p)−n,(8) wi h equali y when he ame is a complex p ojec i e p-design [26,27]. Thus, he uni o mi y o a ini e ame can be quan i ied by compu ing he la ges p o which i sa u a es he Welch bound. This also implies ha minimize s o he p- ame po en ial end o be p ojec i e p-designs. 2.2.4. One-designs a e igh The ame po en ial was in oduced by Bennede o and Fickus, o iginally wi h p=1, o he s udy o ini e no malized igh ames [28]. They p o ed ha FP1has se e al con enien p ope ies: i s global minima a e igh ames and sa u a e he Welch bound FP1(¯ Φd,n[p=1]) = n2/d−n, and all i s local minima a e degene a e, i.e. hey a e global minima and hus igh ames and one-designs. This ells us ha we can see igh ames as a pa icula ins ance o MOFs as de ined in p oposi ion 1. In ac , Benede o and Fickus al eady e e o a se o ec o s ha minimize he ame po en ial FP1as maximally o hogonal. Thus, hey equa e MOFs wi h igh ames. As a gued abo e, i is na u al o elax his equa ion and conside igh ames as jus one case o many MOFs. Jus like FP1leads o MOFs ha a e igh , o he weigh ing unc ions a e equally alid and endow he co esponding MOFs wi h o he use ul p ope ies, such as uni o mi y o low cohe ence. We men ion in passing ha ano he consequence o Benede o and Fickus’s wo k is ha inding igh ames is an easy ask. A local op imiza ion om an a bi a y ini ial condi ion su ices o cons uc igh ames nume ically, al hough cons uc i e analy ical me hods ha e also been de ised [29,30]. O he weigh ing unc ions do no lead o such an easily minimizable ene gy unc ion, and hus ob aining he co esponding MOFs poses a nume ical challenge. We a e now in he posi ion o men ion ha igh ames a e no , jus by i ue o being igh , uni o m. This is o be expec ed since hey a e he minimize s o FP1and hus, gene ally, jus p ojec i e one-designs, he lowes ank o uni o mi y o ini e ames. No e also ha he e mus exis a igh ame o e e y degene a e global minima o FP1, so a igh ame is no a unique objec o a gi en dand n, whe eas we iden i y a uni o m ame as a highly-symme ic unique a angemen , modulo o hogonal ans o ma ions. In ac , o a gi en con igu a ion o dand n igh ames o m ei he a mani old o a disjoin union o mani olds [31]. 2.2.5. Cohe ence: he p →∞and s →∞limi s Fo s→∞, we can w i e lim s→∞ (REs)1/s=m1/smax i=j 1 D(|ϕi⟩,|ϕj⟩),(9) whe e mis he mul iplici y o he la ges o e lap. The e o e, only he maximum o e lap(s) (minimum dis ance(s)) con ibu e o he ene gy and se s ha only di e in submaximal o e laps will be degene a e. In his se ing, he MOF will be gi en by ¯ Φd,n[s→∞] = a gmin{Φd,n}{max i=j|⟨ϕi|ϕj⟩|}.(10) No e ha he same MOF is ob ained by minimizing he p- ame po en ial in he limi p→∞, he wo weigh ing unc ions a e equi alen in his limi . Equa ion (10) indica es ha ¯ Φd,n[s→∞]is he se ha minimizes he maximum o e lap o , equi alen ly, ha maximizes he minimum ( ace) dis ance (6) be ween ec o s. P oblems o his so a e ypically e e ed o as packing p oblems [32]. I is also impo an o iden i y he cohe ence µas he quan i y ha is minimized in equa ion (10) µ(Φd,n) = max i=j|⟨ϕi|ϕj⟩|.(11) 4 Mach. Lea n.: Sci. Technol. 6(2025) 035022 S Roca-Je a and J Román-Roche The minimize s o cohe ence a e called G assmannian ames [10]. Low cohe ence and ‘ igh ness’ a e dual p ope ies in e ms o he encoding capabili ies o a ame. A igh ame allows ‘pe ec econs uc ion’ o he encoded in o ma ion, whe eas a G assmannian ame maximizes ‘ obus ness agains e o ’ [33]. In gene al, i is impossible o op imize bo h simul aneously. This is e iden om he ac ha each is associa ed wi h minimizing a di e en ene gy unc ion, FP1 o igh ness and RE∞o FP∞ o low cohe ence. Simila ly o he Welch bound o he p- ame po en ial, he e exis bounds o he cohe ence. Namely, •Bukh–Cox bound [34] µ(Φd,n)⩾(n−d)2 n[1+ (n−d−1)√n−d+1]−(n−d)2i n>d.(12) •Welch–Rankin bound [26,35] µ(Φd,n)⩾√n−d d(n−1)i n>d.(13) •O hoplex bound [36] µ(Φd,n)⩾1 √di n>d2.(14) •Le ens ein bound [37] µ(Φd,n)⩾√2n−d(d+1) (n−d)(d+1)i n>d2.(15) F ames ha sa u a e he maximum applicable bound a e op imal packings, also known as op imal G assmannian ames. 2.2.6. Cohe ence and degene acy To unde s and he geome ic p ope ies o G assmannian ames, le us discuss he pa icula case o d=2, n=5. Because ec o s in d=2 equi alen unde global phase changes o m CP1, which is di eomo phic o he 2-sphe e S2, hey can be isualized on he Bloch sphe e. Fu he mo e, o d=2 he ace dis ance be ween ec o s (6) is equal o he Euclidean dis ance be ween hei co esponding poin s on he Bloch sphe e. Fo la ge d he nice isualiza ion is los , al hough some in ui ion abou he opology o CPd−1can s ill be ob ained [38]. Using he Bloch sphe e, we can see ha he e is a con inuum o degene a e ames ha minimize he cohe ence, {¯ Φ2,5[p→∞]}, which a e illus a ed in igu e 1. They a e degene a e because hei ene gy is de e mined by he la ges o e lap, which is he o e lap be ween he ec o s a he Equa o o he Bloch sphe e and he ec o s a he No h and Sou h poles. The h ee ec o s a ound he Equa o can be edis ibu ed be ween o ming igh angles be ween hemsel es, and hus hei o e laps among hemsel es being equal o hei o e lap wi h he No h- and Sou h-pole ec o s, and o ming an equila e al iangle. Howe e , his edis ibu ion only dec eases he submaximal o e laps, and does no lowe he cohe ence. Al e na i ely, one may e alua e he ene gy o hese se s unde , e.g. he 6- ame po en ial. Because o p=6 all o e laps con ibu e o he ene gy, he degene acy is b oken and he se in which he ec o s a he Equa o o m an equila e al iangle is lowes in 6- ame po en ial. In ac , his se is p ecisely he se ha minimizes he 6- ame po en ial and is he e o e he MOF o his weigh ing unc ion, ¯ Φ2,5[p=6]. No e ha , as we jus explained, ¯ Φ2,5[p=6]∈{¯ Φ2,5[p→∞]}. Because he Bloch sphe e g an s us a g aphical in e p e a ion o he di e en se s, i is easy o see in igu e 1 ha he ec o s in ¯ Φ2,5[p=6]a e mo e uni o mly dis ibu ed on he sphe e han he ec o s o he o he se s in {¯ Φ2,5[p→∞]}. The phenomenon ha we jus desc ibed o d=2 and n=5 is also obse ed o o he alues o dand nbecause i is a consequence o he ac ha he cohe ence dis ega ds submaximal o e laps (o he examples appea in sec ion 5). No e ha ou a gumen in he p e ious pa ag aph is cen e ed on he appea ance o sys ema ic degene acies in he cohe ence. We a e dis ega ding degene acies a ising om he o hogonal symme y in insic o MOFs. 2.2.7. Uni e sal MOFs We ha e iden i ied igh ness and low cohe ence as associa ed wi h di e en weigh ing unc ions and hus gene ally incompa ible in a MOF. We ha e also discussed he concep o complex p ojec i e p-designs and he uni o mi y o ames. Uni o mi y is o be expec ed o MOFs ha a e he minimize s o an ene gy ha 5 Mach. Lea n.: Sci. Technol. 6(2025) 035022 S Roca-Je a and J Román-Roche Figu e 1. Degene acy o he cohe ence. Bloch-sphe e ep esen a ion ( op) and o e laps among ec o s (bo om) o he di e en G assmannian ames o d=2 and n=5. Column (a) shows he dis ibu ion o ec o s ob ained by sub ac ing one ec o om he equa o om ¯ Φ2,6[s→ ∞]. The ames depic ed in columns (b) and (c) a e ob ained by con inuously sp eading he emaining h ee ec o s a he equa o un il hey o m an equila e al iangle. No e ha , o each jin all h ee panels, he e a e i e do s in he plo , each co esponding o he o e lap o ec o jwi h e e y ec o in he ame (including i sel ). Visually, he e appea s o be only h ee do s o each jbecause he e a e only h ee dis inc o e laps and so h ee o he do s coincide pe ec ly. a o s sho - ange in e ac ions, wi hou ocusing only on nea es -neighbo in e ac ions like cohe ence does. Thus nei he igh no G assmannian ames a e, gene ally, maximally uni o m gi en hei size. The e can exis , howe e , o ce ain alues o dand n, uni e sal MOFs, i.e. ames ha a e simul aneously igh , minimally cohe en and maximally uni o m gi en hei size. ETFs a e he mos p ominen example [10]. ETFs a e igh ames whe e e e y ec o has he same o e lap wi h e e y o he ec o in he ame. Thei exis ence is es ic ed o n⩽d2[39]. ETFs wi h n=d2a e known in quan um mechanics as SIC-POVMs. The cha ac e is ics ha gi e hem hei name make SIC-POVMs use ul o pe ec econs uc ion o inpu s a es om measu emen da a [4–9]. Thei exis ence o any dimension, dwas conjec u ed by Zaune [40]. SIC-POVMs a e also complex p ojec i e wo-designs [4]. The ac ha no ETFs exis o n>d2 ells us ha d2ma ks he h eshold beyond which he ame ec o s s a o c owd he ec o space. Since he ame canno be equiangled while minimizing he ene gy, some ec o s will necessa ily ha e bo h sho - and long- ange neighbo s. This gi es ise o he dynamics a o ded by he compe i ion be ween sho - and long- ange in e ac ions ha we discussed be o e. I is o n>d2 ha igh ness, low-cohe ence and uni o mi y s a o be incompa ible. Ne e heless, he e do exis highly symme ic ames o n>d2. A no able example a e se s o MUB [12,13]. A pai o o hono mal bases {|ϕi⟩}d i=1,{|ψj⟩}d j=1o Cda e mu ually unbiased i |⟨ϕi|ψj⟩|2=1 d.(16) The maximum numbe o MUB M(d)obeys M(d)⩽d+1, wi h equali y only i dis an in ege powe o a p ime numbe . Thus, i dis an in ege powe o a p ime i is possible o cons uc highly symme ic ames o d(d+1) ec o s, by combining he d+1 MUB, whe e e e y ec o has only nea es -neighbo s wi h o e lap 1/√d( he ec o s o he o he bases) and nex -nea es -neighbo s wi h null o e lap ( he o he ec o s o he same basis). Despi e no being ETFs, ames buil om d+1 se s o MUB a e G assmannian and igh [17] and complex p ojec i e wo-designs [27]. 3. Rela ed nume ical op imiza ion p oblems F om he de ini ion o MOFs one can see ha inding hem cons i u es a mul i a ia e minimiza ion p oblem. Finding he global minima o a non-con ex mul i a ia e unc ion is a complex ask, wi hou gua an ee o success in mos cases. A no able excep ion is he inding o igh ames, FP1, whe e he local minima a e all degene a e and hus global minima a e commonplace [28]. In he emaining cases, inding MOFs can be ackled wi h nume ical op imiza ion. In his sec ion, we discuss he ela ion be ween he p oblem o inding MOFs and o he es ablished nume ical p oblems. 6 Mach. Lea n.: Sci. Technol. 6(2025) 035022 S Roca-Je a and J Román-Roche 3.1. The Thomson p oblem The Thomson p oblem aims o de e mine he minimum ene gy a angemen o Nelec ons cons ained o he su ace o he uni sphe e and subjec o elec os a ic epulsion [16,41]. Thus, i ij =∥ i− j∥is he dis ance be ween each pai o elec ons, he Thomson p oblem o Nelec ons is equi alen o inding he a angemen { i|∥ i∥=1}N i=1 ha minimizes E=∑ i=j 1 ij .(17) Iden i ying he elec on coo dina es on he uni sphe e wi h he coo dina es o complex ec o s o d=2 on he Bloch sphe e, i is clea ha he Thomson p oblem is equi alen o he p oblem o inding MOFs in d=2 by minimizing he Riesz one-ene gy. The Thomson p oblem is a challenging nume ical minimiza ion p oblem, as he numbe o local minima is obse ed o g ow exponen ially wi h n. In ac , i has been used as a benchma k o global op imiza ion algo i hms. The e is ag eemen be ween all nume ical and heo e ical me hods on wha is he global minimum o a angemen s o n⪅100 and pu a i ely global minima ha e also been ob ained o n⪅1000 [42,43]. O he di e en me hods employed, a gene ic algo i hm combined wi h local op imiza ion a each gene a ion has p o en pa icula ly e ec i e, eaching a angemen s o n⩽200 [20]. The gene ic algo i hm ha we p esen in he nex sec ion can be seen as a gene aliza ion o he gene ic algo i hm employed on he Thomson p oblem [44]. 3.2. Op imal packings in complex p ojec i e space The p oblem o inding op imal packings in complex p ojec i e space consis s o inding a se o lines h ough he o igin o Cd ha a e geome ically as sp ead apa as possible, i.e. wi h maximum angle dis ances be ween hem. In p ac ice, he lines a e ep esen ed by uni ec o s, such ha a se is ep esen ed by a uni no m ame and he geome ic sp ead is quan i ied by he cohe ence (11) o he ame. Since he cohe ence is in a ian unde changes in he global phase o he ec o s, we can iden i y equi alence classes o all uni ec o s di e ing by a global phase, so he se o lines in Cdis ul ima ely equi alen o a se o elemen s o complex p ojec i e space CPd−1. Finding op imal packings in complex p ojec i e space is hus equi alen o inding MOFs o FP∞o RE∞, i.e. G assmannian ames. The e exis a numbe o heo e ical and nume ical esul s on G assmannian ames, which a e nea ly summa ized in he e iew by Jaspe e al [17]. In he e iew, he au ho s ad e ise a websi e by he name o Game o Sloanes (in hono o Neil Sloane, who hos s a websi e wi h he bes known packings in Euclidean space) ha hos s an open compe i ion o ind pu a i ely op imal esul s o packings in complex p ojec i e space [45]. The nume ical me hod used o ob ain he bes known packings lis ed on he Game o Sloanes consis s on sequen ially applying wo algo i hms: local op imiza ion on he G assmannian mani old [46,47] ollowed by al e na ing p ojec ion [48]. As i s name implies, he algo i hm based on local op imiza ion on he G assmannian mani old is speci ic o his p oblem. On he con a y, al e na ing p ojec ion is a he gene al and can be used in a a ie y o se ings. In any case i does no wo k ha well on i s own, mainly because i bene i s om an adequa e ini ial condi ion, like he one ou pu ed by he local op imiza ion. Fo his eason, i is in e es ing o see how a comple ely gene al app oach like a gene ic algo i hm a es agains hese specialized echniques. To his ega d, i mus be no ed ha al hough he sea ch o a G assmannian ame is oo ed in he minimiza ion o he cohe ence (11), in p ac ice, o ob ain a smoo h unc ion, he cohe ence is de ined as he limi µ(Φ) ≡lim p→∞ (FPp(Φ))1 2p(18) Then, a ela i ely small alue o pis used ini ially, and se e al local op imiza ions a e pe o med sequen ially. A e each local op imiza ion, he alue o pis inc eased and a new local op imiza ion begins. This p ocess is epea ed un il con e gence is eached [47]. Due o he ini e alues o pused in p ac ice, he ac ual nume ical minimiza ion is equi alen o he p oblem o inding MOFs o he p- ame po en ial wi h ini e p. Howe e , in his case, pac s as a nume ical hype pa ame e ha is uned o op imize esul s wi h espec o a igu e o me i : he cohe ence (11). 3.3. Op imal e e ence s a es in quan um classi ie s The concep o maximally o hogonal s a es is in oduced in he con ex o a single-qubi classi ie o e e o a se o h ee o mo e s a es in he Bloch sphe e ha would op imally se e as e e ence s a es o each class in he classi ica ion p ocess [18]. In an n-pa i e classi ica ion one would ha e o selec ns a es o he qubi as 7 Mach. Lea n.: Sci. Technol. 6(2025) 035022 S Roca-Je a and J Román-Roche e e ence s a es, assigning each o hem o one o he classes. Then, he aining p ocess would s i e o lea n o assign o each da a en y a s a e on he Bloch sphe e ha was closes , i.e. wi h he la ges o e lap, o he e e ence s a e o he co esponding class. This p ocess is acili a ed i he e e ence s a es a e e enly dis ibu ed in he s a e space o he qubi . This is so ha he basin o s a es ha a e closes o a gi en e e ence s a e is o equal size o all e e ence s a es, p e en ing de aul biases in he classi ie ha would ha e o be o e come du ing he lea ning p ocess. In sec ion 2we discussed ha uni o mi y is a p ope y only o some MOFs ha a e he minimize s o an adequa e ene gy unc ion. Howe e , as we discussed in ha same sec ion, o ames wi h ew elemen s, he e exis uni e sal MOFs ha achie e igh ness, minimal cohe ence and uni o mi y. Consequen ly, in a qubi , and o small se s, coming up wi h se s o maximally o hogonal s a es is ela i ely s aigh o wa d, wi h some imagina ion i can e en be done g aphically. This is he case in e e ence [18] whe e only se s o up o six maximally o hogonal s a es a e conside ed and he challenge o ob aining la ge se s is no discussed. The idea is u he de eloped in e e ence [19], whe e hey s udy a single-qudi classi ie as he na u al gene aliza ion o he single-qubi classi ie . In o de o gene a e la ge se s o op imal e e ence s a es o a qudi i was necessa y o o malize he concep o maximal o hogonali y, whe e he nuanced dis inc ion be ween maximum o hogonali y and maximum uni o mi y becomes appa en , and o de elop he gene ic algo i hm ha we p esen in his pape . The equi alence be ween maximally o hogonal s a es o a qudi and MOFs is clea om he ac ha he coo dina es o all possible no malized pu e s a es o a d-dimensional qudi o m he complex p ojec i e space CPd−1. 4. The gene ic algo i hm A gene ic algo i hm is a gene al op imiza ion me hod inspi ed by Da winian na u al selec ion. A popula ion o candida e solu ions o he op imiza ion p oblem is e ol ed ac oss ime in successi e gene a ions. A each gene a ion, he i es indi iduals among he popula ion a e selec ed and h ough ecombina ion, mu a ion and su i al, hey gene a e he nex popula ion. The p ocess is i e a ed un il con e gence is eached. The p ocess o ecombina ion a emp s o s ochas ically combine he good cha ac e is ics o one indi idual wi h hose o ano he , gi ing ise o an o e -all be e indi idual. The p ocess o mu a ion in oduces noise, allowing a andom explo a ion o he s a e space. The p ocess o su i al in oduces de e minism in he algo i hm. Because he i es indi iduals a e allowed o su i e, i ensu es a s eady low owa d be e solu ions, wi hou s ochas ic se backs. The p oblem o inding MOFs lends i sel o be implemen ed as a gene ic algo i hm because he ecombina ion p ocess can be implemen ed s aigh o wa dly by combining subse s o ec o s o he wo pa en ames. In ou pa icula se ing, o a gi en dand n, a popula ion, Pd,n={Φk d,n}N k=1, will be a collec ion o pu a i e MOFs, he indi iduals, wi h N he size o he popula ion. The ec o s in each ame a e no malized, ⟨ϕk i|ϕk i⟩=1∀i,k, and encoded as a coo dina e ec o in he o hono mal basis o Cd. To elimina e degene acies due o symme y, we ix he i s ec o o each ame o be |ϕk 1⟩= (1,0,...,0)∀k. In all o he ec o s, he global phase is elimina ed |ϕk i=1⟩= (zk,1,...,zk,d), wi h zk,1∈Rand zk,l=1∈C. Gene ic algo i hms a e ypically o mula ed as a maximiza ion p oblem, whe e he i ness Fis he unc ion o maximize. He e, we de ine i ness as he nega i e ene gy Riesz s-ene gy: F(Φ) = −REs(Φ). A each gene a ion, we so he indi iduals by i ness and selec a numbe o hem o se e as pa en s o he nex gene a ion. I is common in gene ic algo i hms o selec he pa en s s ochas ically, wi h a p obabili y p opo ional o hei i ness, o delay he con e gence p ocess and e icien ly explo e he pa ame e space. He e we employ a udimen a y de e minis ic al e na i e. We ix a i ness gap, ∆F, and selec ou pa en s in dec easing o de o i ness, s a ing om he i es indi idual and ensu ing ha he nex pa en has a i ness a leas ∆Fsmalle han he p e ious pa en . The i ness gap is de e mined as a ac ion o he i ness o he i es indi idual in ha gene a ion ∆F=αdi Fmax. The di e si y a io, αdi , is a hype pa ame e o he algo i hm ha we se o αdi =0.1. Recombina ion is implemen ed as a andom c osso e o ec o s om wo indi iduals. A i ial ecombina ion could consis on gene a ing a andom in ege ibe ween 1 and n o c ea e a child wi h he i i s ec o s o pa en A and he las n−i ec o s o m pa en B. Howe e , he ec o s wi hin a ame a e, a p io i, no so ed in any pa icula ashion, i.e. hei o de does no e lec any geome ical s uc u e in he mani old ha pa ame izes he ec o s. The e o e, he subse o ec o s inhe i ed om each pa en would be andom, consis ing o ec o s om pa en A ha could be a bi a ily close o ec o s om pa en B, and hus h ow away mos o he o hogonali y gained o e he cou se o he op imiza ion p ocess. Because o his, a i ial ecombina ion like he one jus desc ibed would, mos likely, c ea e a child wi h a i ness much lowe han ha o i s pa en s. To p e en his, we so he ec o s o each pa en by dis ance o he i s ec o o each se (which is ixed and he same o all se s). A e he so , he o de o he ec o s in a se does e lec some s uc u al p ope ies. Namely, he i s ec o o he se ma ks a e e ence poin in he mani old ha is common o all se s. A e so ing, we apply he andom ecombina ion by d awing he i s i ec o s om 8 Mach. Lea n.: Sci. Technol. 6(2025) 035022 S Roca-Je a and J Román-Roche pa en A, which a e expec ed o be ela i ely close o he e e ence poin , and he las n−i ec o s om pa en B, which a e expec ed o be ela i ely a om he e e ence poin . To be e unde s and his p ocess, le us conside he case d=2. Now he mani old is he Bloch sphe e and he e e ence poin is gi en by he ec o (1, 0), which ma ks he no h pole. Selec ing he i s i ec o s om pa en A co esponds o selec ing ec o s co e ing a sphe ical cap a ound he no h pole wi h an app oxima e heigh de e mined by he a io i/n. Then, selec ing he n−i ec o s om pa en B co esponds o selec ing ec o s co e ing a sphe ical cap a ound he sou h pole wi h an app oxima e heigh de e mined by he a io (n−i)/n. Al hough no hing gua an ees ha he wo caps do no o e lap o , con a ily, lea e blank a sphe ical s ip in be ween hem, a leas he geome ical s uc u e wi hin each cap is p ese ed and he i ness o he child will only depend on he seam be ween he wo caps. A simila ecombina ion s a egy is used in e e ence [20]. Following he same easoning, we implemen mu a ions by so ing a se by dis ance o he i s ec o , gene a ing a andom in ege ibe ween 1 and nand a andom phase θ∈[0,2π)and o a ing he las n−i ec o s o he so ed se by he ela i e phase, i.e. (z1,z2,...,zd)→(z1,z2eiθ,...,zdeiθ). Fo d=2, his amoun s o o a ing he las n−i ec o s o he se an angle θa ound he no h pole. In addi ion o he usual ope a ions o a gene ic algo i hm: selec ion o he i es , ecombina ion, and mu a ion, we also implemen local op imiza ion. A each i e a ion, a e he ou pa en s a e selec ed and be o e ecombina ion and mu a ion, hey a e each subjec ed o a local op imiza ion p ocess. To implemen he op imiza ion, we encode a se as a (n−1)×d×2 dimensional ec o whe e he i s elemen is he eal pa o he second elemen o he i s ec o , he second elemen is he imagina y pa o he second elemen o he i s ec o , he hi d elemen is he eal pa o he hi d elemen o he i s ec o and so on, i.e. Φd,n→(ℜ(z2,1),ℑ(z2,1),...,ℑ(z2,d),ℜ(z3,1),...). Due o he necessi y o en o ce no maliza ion on he ec o s composing each se h oughou he op imiza ion p ocess, his is a case o mul i a ia e cons ained op imiza ion which we ackle using sequen ial leas -squa es p og amming. The local op imiza ion uns un il ei he con e gence o a maximum numbe o i e a ions, which cons i u es ano he hype pa ame e , a e eached. F om a gene ic pe spec i e, applying a local op imiza ion o he pa en s ha is la e inhe i ed by he nex gene a ion in oduces a Lama ckian aspec o an o he wise Da winian e olu ion p ocess. Gene ic algo i hms based on his hyb id app oach a e ypically e med meme ic algo i hms [49]. In summa y, he algo i hm uns as ollows: 1. Selec ene gy unc ion, i.e. selec he i ness unc ion as he nega i e o he ene gy unc ion. 2. Gene a e ini ial popula ion. We gene a e Nse s o nno malized andom ec o s d awn om he Haa dis ibu ion. 3. Selec he ou ‘ i es ’ indi iduals as he pa en s ( espec ing he selec ed i ness gap, ∆F). 4. Apply local op imiza ion o he pa en s. 5. Gene a e wel e child en by ecombining e e y pa en wi h e e y o he pa en (no e ha each pai o pa en s ma es wice, exchanging he oles o pa en s A and B in each case). 6. Gene a e ou o he child en by mu a ing each pa en once. 7. Fo m a new popula ion wi h he ou pa en s and he six een child en and e alua e he i ness o he i es indi idual. 8. I he maximum numbe o i e a ions o he con e gence c i e ion a e eached, hal , o he wise go o s ep 3 wi h he new popula ion. To induce he hal ing o he algo i hm, one can ix a maximum numbe o i e a ions o a con e gence c i e ion based on he a e o change o he i ness ac oss gene a ions. We a bi a ily ix he numbe o pa en s o ou , he numbe o ecombina ions o wel e, he numbe o mu a ions o ou and consequen ly he popula ion size o wen y (inhe i ed om e e ence [20]), hese numbe s could be le as hype pa ame e s. Mo e nume ical de ails can be ound in appendix A 5. Resul s and discussion 5.1. The Thomson p oblem As p esen ed in sec ion 3.1, o d=2 and using he Riesz one-ene gy as he weigh ing unc ion, he p oblem o inding MOFs is equi alen o he Thomson p oblem. The e o e, we use he Thomson p oblem as a benchma k o he GA. Figu e 2(a) shows he ela i e ene gy di e ence, ϵ el =|REGA 1−RE e 1|/|RE e 1|, be ween he ames p oduced by he GA o he Thomson p oblem and he bes known a angemen s epo ed in e e ence [50] o con igu a ions o up o n=100 poin s. The GA p oduces op imal o nea -op imal ames wi h a ela i e e o unde ∼0.1% o n⩽100. 9 Mach. Lea n.: Sci. Technol. 6(2025) 035022 S Roca-Je a and J Román-Roche Table A1. Run imes o he gene ic algo i hm o di e en con igu a ions. The numbe o gene a ions was capped a 15. d n Run ime Gene a ions Time/gene a ion 2 6 0.6s 5 0.12s 2 24 99.9s 15 6.6s 4 16 28.7s 7 4.1s 4 40 9min6s 9 1min7s Figu e 6. Dependence o he mesh no m wi h he numbe o Haa ec o s NH o d=4 and di e en alues o n. A.2. Compu ing he mesh no m and he sha e o he ec o space The mesh no m and he sha e o he ec o space can only be compu ed app oxima ely by pe o ming a ini e sampling o NH ec o s om he Haa dis ibu ion o each dimension d. In igu e 6we show how he mesh no m depends on NH o di e en con igu a ions. We ind ha a alue o NH=107is su icien o esol e changes in he mesh no m ⩾10−2, which is su icien o compa e, e.g. he ames ob ained by op imizing di e en me ics in igu e 4. Appendix B. Visualizing uni o mi y: sha e o he ec o space In sec ion 5.2.1 we de ined he no malized sha e o he ec o space βjas a way o isualize he uni o mi y o a ame. By de ini ion, he sha e and he mesh no m a e ela ed. The mesh no m measu es he adius o he la ges gap le in he ec o space by ec o s o he ame. The sha e measu es he po ion o he ec o space ha is closes o each ec o o he ame. The ela ion s ems om he ac ha he ec o s o he ame su ounding he la ges gap will ha e a la ge sha e. The e o e, he mesh no m is posi i ely co ela ed wi h he la ges alues o he sha e. This ela ion is simila o he ela ion be ween cohe ence and o e laps, whe e cohe ence is he maximum o e lap. No e also ha he sha e o he ec o space is he me ic o op imize when looking o op imal e e ence s a es o a quan um classi ie (see sec ion 3.3). A non-uni o m dis ibu ion o he sha es is associa ed wi h de aul biases o he quan um classi ie . In igu es 7and 8we isualize he uni o mi y o he ames ob ained wi h he GA o d=4. Figu e 7is complemen a y o igu e 3. In igu e 7we show he ames ob ained o d=4 and n∈[4,43]. Each panel displays he sha e o he ec o space βjo each ec o |ϕk⟩∈ ¯ Φ4,n[s=8]. In addi ion, he s anda d de ia ion o he sha es is indica ed wi hin each panel. Uni o m ames ha e βj≈1 o all ec o s and a small s anda d de ia ion. I is in e es ing o no e ha all ames up o he SIC-POVM o n=16 a e ela i ely uni o m. App eciable non-uni o mi ies only appea beyond his poin . Ne e heless, highly symme ic ames such as he se o MUB o he h ee-design a n=40 a e again ela i ely uni o m. Figu e 8is complemen a y o igu e 4. In igu e 8we plo he sha e o he ec o space o h ee di e en con igu a ions, o d=4 and n=23,40,43. Fo each con igu a ion we plo he ame wi h lowes cohe ence, highes mesh no m and lowes looseness (closes o being igh ). Fo n=40, gi en he high symme y o he h ee-design, he e a e ba ely any di e ences be ween he h ee cases. In con as , o n=23 and n=43 he e a e clea di e ences in he sha es be ween he ames ha op imize each me ic. F ames ha minimize cohe ence end o display sha es g ouped in wo o mo e dis inc alues. On he o he hand, ames ha 16 Mach. Lea n.: Sci. Technol. 6(2025) 035022 S Roca-Je a and J Román-Roche Figu e 7. Maximally o hogonal ames o d=4 and s=8. Each panel shows he no malized sha e o he ec o space βj(21) o each ec o he bes MOF ou o en uns o he GA o a ce ain size n. The ho izon al black line is a guide o he eye ma king he alue 1. The panel i le indica es whe he i is a high-symme y ame such as an ETF, maximal se o MUB o SIC-POVM. The s anda d de ia ion o he sha e σβis indica ed wi hin each panel. Figu e 8. Gene ic algo i hm s ochas ic esul s. No malized sha e o he ec o space βj(21) o he maximally o hogonal ames o d=4 and s=8 and n=23,40,43. Each column shows he esul o a di e en un o he GA ha op imizes a di e en me ic. The ho izon al black line is a guide o he eye ma king he alue 1. The s anda d de ia ion o he sha e σβis indica ed wi hin each panel. 17 Mach. Lea n.: Sci. Technol. 6(2025) 035022 S Roca-Je a and J Román-Roche maximize he mesh no m o minimize looseness ha e mo e e enly dis ibu ed sha es. This is e lec ed in he s anda d de ia ion o he sha e, which is highe o he ames ha minimize cohe ence. We no e ha o se e al con igu a ions in igu es 7and 8 he dis ibu ion o sha es is mos ly concen a ed a ound 1, wi h one o a ew ou lie s wi h a alue signi ican ly di e en han 1. In hese cases, he s anda d de ia ion may no e lec he ac ual uni o mi y o he ame and i could be in e es ing o inco po a e es ima o s o ku osis o mul imodali y o he dis ibu ion o sha es. 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