Theo y o Ca ego ical Equi alence
Be ween Compu a ional Uni e se and Physical
Uni e se:
Re e sible QCA, Uni ied Time Scale,
and Complexi y Geome ic In a ian s
Haobo Ma1Wenlin Zhang2
1Independen Resea che
2Na ional Uni e si y o Singapo e
No embe 24, 2025
Abs ac
In p e ious wo ks on he “compu a ional uni e se” se ies Ucomp = (X, T,C,I),
we cons uc ed complexi y geome y and in o ma ion geome y a disc e e le el,
ob aining unde uni ied ime scale sca e ing mo he scale con ol mani old (M, G)
and ask in o ma ion mani old (SQ, gQ), such ha complexi y dis ance and in o -
ma ion dis ance a e cha ac e ized espec i ely by geodesic dis ances o Gand gQin
con inuous limi . Howe e , o uly achie e uni ica ion o “compu a ional uni e se
= physical uni e se”, geome ic co espondence alone is insu icien : we need ca -
ego ical equi alence be ween wo “uni e se ca ego ies,” i.e., exis ence o mu ually
in e se unc o s on app op ia e subclasses such ha objec s o physical uni e se
and compu a ional uni e se can co espond one- o-one, wi h geome ic in a ian s
such as complexi y and ime scale p ese ed unde his co espondence.
This pape in oduces a physical uni e se ca ego y PhysUni , whose objec s
a e physical uni e se models sa is ying uni ied ime scale assump ion
Uphys = (M, g, F, κ, S),
whe e (M, g) is space ime mani old wi h me ic (o mo e gene al causal s uc-
u e), Fis ma e ield con en , κ(ω) is uni ied ime scale densi y, S(ω) is sca e ing
da a. Mo phisms a e geome ic mappings p ese ing causal s uc u e, uni ied ime
scale, and sca e ing s uc u e. We simul aneously e iew compu a ional uni e se
ca ego y CompUni , whose objec s a e compu a ional uni e se objec s sa is ying
ini e in o ma ion densi y and locali y axioms, wi h mo phisms being simula ion
mappings wi h complexi y uppe bounds.
On his basis, we selec wo subca ego ies: one is PhysUni QCA consis ing
o physical uni e ses ealizable by e e sible quan um cellula au oma a (QCA),
ano he is CompUni phys consis ing o compu a ional uni e ses physically ealiz-
able unde uni ied ime scale. Re e sible QCA is simul aneously a local disc e e
dynamical sys em and a physical sys em sa is ying uni ied ime scale con ollable
sca e ing s uc u e, hus becoming b idge connec ing wo ca ego ies.
This pape de ines wo co e unc o s:
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1. Func o F:PhysUni QCA →CompUni phys, mapping physical uni e se
Uphys h ough QCA disc e iza ion o compu a ional uni e se Ucomp;
2. Func o G:CompUni phys →PhysUni QCA, econs uc ing om local e-
e sible compu a ional uni e se i s con inuous limi space ime mani old, uni-
ied ime scale, and sca e ing da a.
Unde se o explici echnical axioms (QCA uni e sali y, local e e sibili y, uni-
ied ime scale consis ency, and app op ia e con inuous limi exis ence), we p o e:
Fand Ga e quasi-in e se a objec le el, i.e., o any Uphys ∈PhysUni QCA,
he e exis s na u al isomo phism
ηUphys :Uphys
≃
−−→ G(F(Uphys)),
o any Ucomp ∈CompUni phys, he e exis s na u al isomo phism
ϵUcomp :F(G(Ucomp)) ≃
−−→ Ucomp;
Fand Gp ese e simula ion s uc u e a mo phism le el: geome ic in a i-
an s such as complexi y geome y (me ic Gand geodesic dis ance), uni ied
ime scale densi y κ(ω), and sca e ing phase a e s able unde ca ego ical
equi alence.
Thus ob aining main heo em: on physically ealizable subclass, physical uni-
e se ca ego y and compu a ional uni e se ca ego y a e equi alen in ca ego ical
sense, hey a e me ely di e en p esen a ions o same “uni ied ime scale–complexi y
geome y–sca e ing s uc u e” objec om con inuous and disc e e pe spec i es.
Keywo ds: Compu a ional uni e se; Physical uni e se; Ca ego y heo y; Ca ego ical
equi alence; Quan um cellula au oma a; Uni ied ime scale; Complexi y geome y; Sca -
e ing heo y
1 In oduc ion
P e ious wo ks ha e sys ema ized he “compu a ional uni e se” idea:
A disc e e le el, a compu a ional uni e se is quad uple Ucomp = (X, T,C,I), whe e
Xis con igu a ion se , Tis one-s ep upda e ela ion, Cis single-s ep cos , Iis in o -
ma ion quali y unc ion. Fi s wo wo ks cons uc ed disc e e complexi y geome y
and disc e e in o ma ion geome y on his basis.
Based on uni ied ime scale sca e ing mo he scale
κ(ω) = φ′(ω)
π=ρ el(ω) = 1
2π Q(ω),
we in oduced physical ime scale in o compu a ional uni e se, making i s single-
s ep cos a unc ion o g oup delay ma ix, p o ing ha in e inemen limi , disc e e
complexi y dis ance con e ges o geodesic dis ance on con ol mani old (M, G).
A ask-awa e in o ma ion geome y le el, we cons uc ed in o ma ion mani old
(SQ, gQ), w i ing on join mani old EQ=M×SQjoin ac ion AQ o ime–
in o ma ion–complexi y, geome izing “op imal algo i hms” as “minimal wo ldlines.”
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Al hough hese esul s achie ed uni ica ion om disc e e compu a ion o con inuous
geome y, “physical uni e se” s ill appea s in ela i ely ex e nal posi ion: i is used o
p o ide uni ied ime scale and sca e ing da a, bu no ye jux aposed wi h compu a ional
uni e se a ca ego ical le el. To uly gi e igo ous meaning o “uni e se is compu a ion,”
we need o in oduce wo “uni e se ca ego ies”:
A ca ego y PhysUni wi h physical heo y objec s as objec s;
A ca ego y CompUni wi h compu a ional uni e se objec s as objec s.
And gi e equi alence on app op ia e subclass
PhysUni QCA ≃CompUni phys.
This pape ’s ask is o cons uc hese wo ca ego ies, ele an subca ego ies, and unc-
o s F, G connec ing hem, p o ing ca ego ical equi alence unde cons ain s o uni ied
ime scale and complexi y geome y.
Sec ion 2 de ines physical uni e se ca ego y and QCA- ealizable subca ego y. Sec ion
3 e iews compu a ional uni e se ca ego y and de ines physically ealizable subca ego y.
Sec ion 4 cons uc s QCA disc e iza ion unc o F om physical uni e se o compu-
a ional uni e se, Sec ion 5 cons uc s con inuous limi unc o G om compu a ional
uni e se o physical uni e se. Sec ion 6 s a es and p o es main heo em o ca ego ical
equi alence, discussing in a iance o complexi y geome y and uni ied ime scale unde
his equi alence. Appendices p o ide de ailed axioms, QCA cons uc ion, and ca ego ical
p oo s.
2 Physical Uni e se Ca ego y and QCA-Realizable
Subca ego y
This sec ion cons uc s physical uni e se ca ego y PhysUni , selec ing wi hin i subca -
ego y PhysUni QCA ealizable by e e sible QCA.
2.1 Physical Uni e se Objec s
Ou wo king physical uni e se objec is mul iple objec wi h space ime geome y, ma e
ields, and uni ied ime scale s uc u e.
De ini ion 2.1 (Physical Uni e se Objec ).A physical uni e se objec is quin uple
Uphys = (M, g, F, κ, S),
whe e:
1. (M, g) is ou -dimensional Lo en zian mani old o mo e gene al causal mani old, M
is space ime mani old, gis me ic o equi alen causal s uc u e;
2. Fis ma e ield con en de ined on (M, g) (such as gauge ields, e mion ields),
ypically solu ion space o ield equa ions o ope a o algeb a;
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3. κ(ω) is uni ied ime scale densi y: o selec ed class o sca e ing p ocesses, i s
sca e ing phase de i a i e, spec al shi unc ion de i a i e, and g oup delay ace
sa is y
κ(ω) = φ′(ω)
π=ρ el(ω) = 1
2π Q(ω);
4. S(ω) is equency- esol ed sca e ing ma ix amily o co esponding sca e ing p o-
cesses, sa is ying s anda d sca e ing heo y axioms (uni a i y, analy ici y, e c.).
We only conside uni e se objec s sa is ying “dis inguishable uni ied ime scale,” i.e.,
he e exis s a leas one amily o sca e ing p ocesses such ha abo e mo he o mula
holds and κ(ω) is non-degene a e.
2.2 Physical Uni e se Mo phisms
In ca ego y PhysUni , mo phisms should be mappings p ese ing causal s uc u e, uni-
ied ime scale, and sca e ing ea u es.
De ini ion 2.2 (Physical Uni e se Mo phism).Gi en wo physical uni e se objec s
Uphys = (M, g, F, κ, S), U′
phys = (M′, g′,F′, κ′,S′),
a mo phism :Uphys →U′
phys consis s o ollowing da a:
1. A smoo h mapping M:M→M′, locally bijec i e in causal sense (p ese ing
imelike causal o de );
2. Push o wa d F:F → F′be ween ield con en s, co a ian wi h M;
3. P ese a ion o uni ied ime scale and sca e ing da a: he e exis s equency ans-
o ma ion ω: Ω →Ω′, such ha
κ′(ω′) = κ(ω),S′(ω′)≃S(ω) when ω′= ω(ω),
whe e “≃” deno es equi alence unde gauge ans o ma ion and isomo phism.
Wi h physical uni e se objec s as objec s and physical uni e se mo phisms as mo -
phisms, we o m ca ego y PhysUni .
2.3 QCA-Realizable Subca ego y PhysUni QCA
We ca e abou hose physical uni e se objec s ha can be ealized o app oxima ed by
e e sible quan um cellula au oma a (QCA).
De ini ion 2.3 (QCA-Realizable Physical Uni e se).A physical uni e se objec Uphys is
called QCA- ealizable i he e exis :
1. A la ice se Λ ⊂Mand i s embedding i: Λ ,→M, o ming uni o m co e ing o M
a la ge scales;
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2. Fini e-dimensional local Hilbe space Hxa each la ice poin and global Hilbe
space
H=O
x∈Λ
Hx;
3. A QCA e olu ion ope a o U:H → H sa is ying locali y and e e sibili y, such
ha :
I s Lieb–Robinson ligh cone app oxima es causal s uc u e o (M, g) a la ge
scales;
I s sca e ing ma ix amily SQCA(ω) app oxima es S(ω) in app op ia e limi ,
uni ied ime scale densi y κQCA(ω) consis en wi h κ(ω).
All QCA- ealizable physical uni e se objec s and mo phisms induced by QCA simu-
la ion o m subca ego y deno ed PhysUni QCA ⊂PhysUni .
3 Compu a ional Uni e se Ca ego y and Physically
Realizable Subca ego y
This sec ion e iews de ini ion o compu a ional uni e se ca ego y, selec ing wi hin i
physically ealizable subca ego y.
3.1 Compu a ional Uni e se Ca ego y CompUni
A compu a ional uni e se objec is
Ucomp = (X, T,C,I),
sa is ying axioms o ini e in o ma ion densi y, local upda e, (gene alized) e e sibili y,
and cos addi i i y.
Mo phisms a e simula ion mappings:
De ini ion 3.1 (Simula ion Mapping Recalled).I he e exis :X→X′and cons an s
α, β > 0, mono one unc ion Φ, such ha
1. S ep p ese a ion: (x, y)∈T⇒( (x), (y)) ∈T′;
2. Cos con ol: o any pa h γ:x→y, he e exis s γ′: (x)→ (y) such ha
C′(γ′)≤αC(γ) + β;
3. In o ma ion ideli y: I(x)≤Φ(I′( (x)));
hen is a simula ion mapping om Ucomp o U′
comp.
Wi h compu a ional uni e se objec s as objec s and simula ion mappings as mo -
phisms, we o m ca ego y CompUni .
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3.2 Physically Realizable Subca ego y CompUni phys
We need o selec hose compu a ional uni e se objec s ealizable unde uni ied ime scale
and QCA amewo k.
De ini ion 3.2 (Physically Realizable Compu a ional Uni e se).Compu a ional uni-
e se objec Ucomp = (X, T,C,I) is called physically ealizable i he e exis :
1. A QCA sys em (Λ,Hx, U), and con igu a ion encoding mapping e:X→ H o
no malized basis ec o subse ;
2. A amily o con ol pa ame e s θ∈ M and sca e ing ma ix amily S(ω;θ), such
ha one-s ep e olu ion o Uco esponds o some con ol s ep size;
3. Single-s ep cos C(x, y) can be w i en as disc e e in eg al o uni ied ime scale
densi y:
C(x, y) = ZΩx,y
κ(ω;θ) dµx,y(ω);
4. Complexi y geome y in e inemen limi is app oxima ed by geodesic dis ance o
some con ol mani old (M, G), as s a ed in p e ious heo em on Riemannian limi .
All physically ealizable compu a ional uni e se objec s and mo phisms induced by
physically ealizable simula ion o m subca ego y CompUni phys ⊂CompUni .
4 Func o F om Physical Uni e se o Compu a-
ional Uni e se: QCA Disc e iza ion
This sec ion cons uc s unc o
F:PhysUni QCA →CompUni phys,
mapping each QCA- ealizable physical uni e se objec o compu a ional uni e se ob-
jec .
4.1 Objec Le el: QCA Disc e iza ion Cons uc ion
Gi en Uphys = (M, g, F, κ, S)∈PhysUni QCA, by de ini ion he e exis la ice Λ ⊂M
and QCA sys em (Λ,Hx, U).
1. Con igu a ion se X: Selec se o no malized basis ec o s Bx o each Hx, le
X=Y
x∈Λ
Bx,
i.e., se o all basis enso p oduc labels. Any x∈Xco esponds o basis ec o
|x⟩∈H.
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2. One-s ep upda e ela ion T: De ine
T={(x, y)∈X×X:⟨y|U|x⟩ = 0}.
I Udecomposes in o undamen al ga e sequence wi h ime s ep ∆ , we can de ine
Tby his s ep size as “one physical ime s ep” upda e ela ion.
3. Single-s ep cos C: Using uni ied ime scale densi y κ(ω) and co esponding
sca e ing ma ix S(ω), assign cos o each (x, y)∈T
C(x, y) = ZΩx,y
κ(ω) dµx,y(ω),
whe e Ωx,y and spec al measu e µx,y gi en by local sca e ing s uc u e o QCA.
Fo non-adjacen pai s (x, y)/∈Tle C(x, y) = ∞.
4. In o ma ion quali y unc ion I: Acco ding o ask, choose app op ia e obse -
a ion ope a o amily, ansla ing ask in o ma ion on physical ield con en F o
I:X→Ron con igu a ion space X. Fo example, o ou pu dis ibu ion o gi en
sca e ing expe imen , de ine I(x) as nega i e ela i e en opy o likelihood ela i e
o some a ge dis ibu ion.
F om locali y and e e sibili y o QCA we can e i y: Ucomp = (X, T,C,I) sa is ies
compu a ional uni e se axioms and is physically ealizable.
De ini ion 4.1 (Objec Mapping).Le
F(Uphys)=(X, T,C,I).
4.2 Mo phism Le el: Disc e e Simula ion o Physical Uni e se
Mo phisms
Gi en physical uni e se mo phism
:Uphys →U′
phys,
om i s ealiza ion a QCA le el, we ge uni a y mapping o isome ic embedding
H:H → H′be ween Hilbe spaces, he eby inducing basis ec o le el mapping X:
X→X′.
We equi e X o sa is y:
1. S ep p ese a ion: i (x, y)∈Tand ⟨y|U|x⟩ = 0, hen ( X(x), X(y)) ∈T′, co e-
sponding o ⟨ X(y)|U′| X(x)⟩ = 0;
2. Cos con ol: he e exis α, β > 0, such ha o any pa h γi s image X(γ) has
cos sa is ying
C′( X(γ)) ≤αC(γ) + β;
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3. In o ma ion ideli y: in luence o physical mo phism on sca e ing ou pu con olled
by F, he eby inducing mono one unc ion Φ on ask in o ma ion, such ha
I(x)≤Φ(I′( X(x))).
This is p ecisely he condi ion o simula ion mapping.
De ini ion 4.2 (Mo phism Mapping).Le
F( ) = X:F(Uphys)⇝F(U′
phys).
4.3 Func o iali y
P oposi ion 4.3. Abo e de ini ion gi es co a ian unc o
F:PhysUni QCA →CompUni phys.
P oo in Appendix A.1. Key is e i ying: iden i y mo phism maps o iden i y sim-
ula ion mapping, mo phism composi ion a disc e e le el co esponds o composi ion o
simula ion mappings, and complexi y and in o ma ion con ol pa ame e s sa is y simula-
ion condi ions a e composi ion.
5 Func o G om Compu a ional Uni e se o Phys-
ical Uni e se: Con inuous Limi Recons uc ion
This sec ion cons uc s unc o
G:CompUni phys →PhysUni QCA,
s a ing om physically ealizable compu a ional uni e se, econs uc ing con inuous
physical uni e se objec unde cons ain s o uni ied ime scale and complexi y geome y.
5.1 F om Disc e e Con ol o Space ime Mani old
Gi en Ucomp = (X, T,C,I)∈CompUni phys, by assump ion he e exis con ol mani old
(M, G) and QCA ealiza ion. We i s cons uc space ime mani old (M, g) om con ol–
complexi y geome y.
1. Con ol mani old Mis pa ame e space o “space + in e nal deg ees o eedom,”
al eady equipped wi h Riemannian me ic G.
2. Using uni ied ime scale densi y κ(ω;θ), we can cons uc e ec i e space ime me ic
o causal s uc u e (M, g) on R× M, e.g., in simpli ied case le
M=R × M, g =−c2(θ)d 2+Gab(θ)dθadθb,
whe e c(θ) ela ed o κ, ensu ing ligh cone s uc u e consis en wi h Lieb–Robinson
ligh cone o QCA.
Mo e gene ally, we can join ly de ine Lo en z- ype me ic using causal s uc u e (up-
da e di ec ion) o complexi y g aph and uni ied ime scale, such ha “ eachabili y ela-
ion” and “nonze o p opaga ion eloci y” co espond o causal s uc u e o g.
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5.2 Recons uc ion o Sca e ing and Uni ied Time Scale
By physically ealizable assump ion, compu a ional uni e se has QCA ealiza ion U,
whose sca e ing ma ix amily SQCA(ω;θ) ealizes equency domain cha ac e is ics o
compu a ional upda e. Using p e ious uni ied ime scale mo he o mula, we de ine
κ(ω;θ) = 1
2π QQCA(ω;θ), QQCA(ω;θ) = −iS†
QCA∂ωSQCA.
Taking his as uni ied ime scale densi y o physical uni e se, de ine sca e ing da a
S(ω;θ) = SQCA(ω;θ),
he eby sa is ying sca e ing and ime scale s uc u e axioms o physical uni e se
objec .
5.3 Embedding o Field Con en and In o ma ion Quali y
Con igu a ion in o ma ion Xand ask in o ma ion Io compu a ional uni e se can be
embedded in o physical ield con en F h ough local ope a o s o QCA, e.g., iewing
Xas se o expec a ion alues o ce ain local ope a o s o bounda y condi ion se .
Mo e p ecise app oach is cons uc ing local ope a o algeb a ne A(O)⊂ B(H), ea ing
con igu a ion in o ma ion and ask in o ma ion as unc ions o hese ope a o s unde
QCA e olu ion.
F om ca ego ical s uc u e pe spec i e, we only need o ensu e exis ence o embedding
om (X, I) o F, such ha compa ison ela ion o in o ma ion quali y is p ese ed unde
his embedding (mono one homomo phism).
5.4 Objec and Mo phism Mappings
De ini ion 5.1 (Objec Mapping).Gi en Ucomp ∈CompUni phys, le
G(Ucomp)=(M, g, F, κ, S),
whe e (M, g) and (κ, S) cons uc ed espec i ely om con ol–complexi y geome y
and QCA sca e ing s uc u e, Fis ield con en gene a ed by QCA local ope a o algeb a.
De ini ion 5.2 (Mo phism Mapping).Gi en simula ion mapping
:Ucomp ⇝U′
comp,
i s co esponding QCA ealiza ion induces mappings be ween con ol mani olds, space-
ime mani olds, and sca e ing da a
M:M→M′, F:F → F′, ω: Ω →Ω′,
de ine
G( ) = ( M, F, ω) : G(Ucomp)→G(U′
comp).
P oposi ion 5.3. Abo e de ini ion gi es co a ian unc o
G:CompUni phys →PhysUni QCA.
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