Universe as Quantum Discrete Cellular Automaton: Axiomatic Characterization

Uni e se as Quan um Disc e e Cellula Au oma on:
Axioma ic Cha ac e iza ion
Haobo Ma
1
Wenlin Zhang
2
1
Independen Resea che
2
Na ional Uni e si y o Singapo e
No embe 19, 2025
Abs ac
Wi hin he amewo ks o Quan um Cellula Au oma a (QCA), quasi-local
C∗
-
algeb as, and causal se s, we cons uc an axioma ic sys em o "Uni e se = Single
Quan um Disc e e Cellula Au oma on". Specically, we use a coun able connec ed
g aph
Λ
as he disc e e space; a ni e-dimensional local Hilbe space
Hcell
and
he quasi-local algeb a
A
on i s inni e enso p oduc o desc ibe local quan um
deg ees o eedom; a
∗
-au omo phism
α:A→A
wi h ni e p opaga ion adius
and i s uni a y implemen a ion
U
o desc ibe disc e e ime e olu ion; and an ini ial
cosmic s a e
ω0
o desc ibe he uni e se a ime
n= 0
. We p o e ha unde hese
axioms, he na u ally induced ela ion on he e en se
E= Λ ×Z
cons i u es a lo-
cally ni e pa ial o de , he eby yielding a disc e e causal se whose local ni eness
s ic ly co esponds o he ni e p opaga ion adius condi ion o he QCA, aligning
wi h he "locally ni e pa ial o de " s uc u e in causal se heo y. Fu he mo e,
we dene he "Uni e se QCA Objec "
UQCA = (Λ,Hcell,A, α, ω0)
, p o ide an equi -
alence heo em o "QCA Locali y
⇐⇒
Local Fini eness o Causal Pa ial O de ",
and cons uc a 1D Di ac- ype QCA in he single-pa icle limi , demons a ing how
o eco e he con inuous Di ac equa ion in app op ia e scaling limi s. Finally, we
discuss he exp ession o obse a ion, en opy, and he a ow o ime wi hin his
amewo k, as well as ela ionships wi h causal se quan um g a i y and disc e iza-
ion schemes o quan um eld heo y.
Keywo ds:
Quan um Cellula Au oma a; Quasi-local
C∗
-Algeb a; Disc e e Uni e se
Model; Causal Se ; Disc e e Time Dynamics; La ice Implemen a ion o Di ac Equa ion
1 In oduc ion & His o ical Con ex
Quan um Cellula Au oma a (QCA) can be iewed as "disc e e- ime, local uni a y dy-
namics on inni e quan um la ices". Thei ep esen a i e axioma iza ion appea ed in he
wo k o Schumache We ne on e e sible QCA: dening QCA as a ansla ion-in a ian
∗
-au omo phism on a quasi-local algeb a wi h a s ic ly ni e p opaga ion adius, en-
su ing ha each e olu ion s ep p opaga es suppo only wi hin a ni e neighbo hood.
Subsequen ly, A ighi, Fa elly, and o he s sys ema ically e iewed s uc u al heo ems,
compu abili y, quan um simula ion, and con inuum limi s o QCA, demons a ing he
1
b oad applicabili y o QCA as a ool o disc e ized quan um eld heo y and opological
phase simula ion. ([a Xi ][1])
On he o he hand, he causal se p og am ep esen ed by So kin and Su ya p oposes
ha space ime on ology can be eplaced by a locally ni e pa ial o de se , whe e he
pa ial o de encodes causal s uc u e and local ni eness encodes disc e eness, sa is ying
he "o de + numbe
∼
geome y" p og am. In his scheme, causal pa ial o de and local
ni eness join ly cha ac e ize a disc e e "p o o-space ime", wi h con inuous Lo en zian
mani olds being me ely limi in e pola ions. ([Li ing Re iews][5])
Exis ing esea ch on QCA mainly ea s i as a "simula ion ool" o "disc e iza ion
scheme": o app oxima e a gi en con inuous quan um eld o condensed ma e sys em,
and con e ge back o con inuous heo y in app op ia e limi s. This pape chooses he
e e se pe spec i e: no ea ing QCA as a nume ical app oxima ion o a con inuous
uni e se, bu dening he "Uni e se On ology" i sel as a QCA objec sa is ying ce ain
axioms; causal s uc u e and "space ime" a e de i ed om he locali y and ime i e a ion
o his objec , a he han being p e-gi en.
Mo e specically, he co e ques ions o his pape a e:
1. In he con ex o QCA, how o dene "The Uni e se" as a global objec ha
simul aneously con ains space, local deg ees o eedom, dynamics, and ini ial s a e?
2. Can we de i e a locally ni e pa ial o de on he e en se s a ing om QCA
locali y and g aph s uc u e, he eby aligning wi h he undamen al s uc u e o causal
se heo y?
3. In his axioma ic sys em, how o abs ac obse a ion, en opy, and he a ow o
ime, and eco e con inuous ela i is ic eld equa ions, such as he Di ac equa ion, in
app op ia e limi s?
Focusing on hese ques ions, his pape cons uc s he Uni e se QCA Objec
UQCA
,
p o es ha i s induced e en se
(E, ⪯)
is a locally ni e pa ial o de , and p o ides
he con inuous limi cons uc ion o Di ac- ype QCA, demons a ing how o ob ain he
la ice e sion o he s anda d Di ac equa ion and i s limi in he single-pa icle sec o .
2 Model & Assump ions
This sec ion p o ides he basic ma hema ical s uc u es and axioma ic assump ions used
in his pape , including disc e e space, local Hilbe space, quasi-local
C∗
-algeb a, and
he deni ion o QCA, and on his basis denes he "Uni e se QCA Objec ".
2.1 Disc e e Space and G aph S uc u e
Le
Λ
be a coun able se , se ing as he "la ice se " o "disc e e space". Assume
Λ
ca ies an undi ec ed connec ed g aph s uc u e wi h edge se
EΛ⊂Λ×Λ
, sa is ying:
1.
(x, y)∈EΛ⇒(y, x)∈EΛ
;
2. No sel -loops
(x, x)∈EΛ
.
Based on his, dene he g aph dis ance
dis : Λ ×Λ→N∪ {0}
as he numbe o
edges in he sho es pa h connec ing
x
and
y
(o
+∞
i no pa h exis s). The connec i i y
assump ion ensu es ni e dis ance be ween any
x, y ∈Λ
.
Fo any
R∈N
and
x∈Λ
, dene he closed ball
BR(x) := {y∈Λ : dis (x, y)≤R}.
2
Assume each
BR(x)
is a ni e se . This assump ion holds au oma ically on s anda d
la ices
Zd
.
When
Λ = Zd
, dene he ansla ion ac ion
τa: Λ →Λ
as
τa(x) := x+a
, whe e
a∈Zd
. This s uc u e will be used la e when discussing spa ial homogenei y.
2.2 Local Hilbe Space and Quasi-Local
C∗
-Algeb a
Fo each la ice si e
x∈Λ
, associa e a ni e-dimensional Hilbe space
Hx
, and assume
he e exis s a xed ni e-dimensional Hilbe space
Hcell
such ha
Hx≃ Hcell ≃Cd
holds o all
x∈Λ
, whe e
d∈N
is he dimension o local deg ees o eedom o he cell.
Fo any ni e subse
F⋐Λ
, dene he ni e olume Hilbe space
HF:= O
x∈F
Hx,
and he co esponding bounded ope a o algeb a is
AF:= B(HF).
I
F⊂G⋐Λ
, he e is a na u al embedding
ιF,G :AF,→ AG, A 7→ A⊗1G F,
whe e
1G F
is he iden i y ope a o on
Nx∈G FHx
.
Le
Aloc := [
F⋐Λ
AF,
dene he quasi-local
C∗
-algeb a as
A:= Aloc
|·|,
whe e
|·|
is he ope a o no m. The suppo
supp(A)⊂Λ
o an elemen
A∈ Aloc
is
dened as he minimal ni e se
F
such ha
A∈ AF
. Fo gene al
A∈ A
, suppo can
be dened as he closu e o suppo s o local ope a o s app oxima ing
A
.
This cons uc ion is consis en wi h he s anda d quasi-local algeb a o malism o
quan um spin sys ems and quan um la ice sys ems. ([Sp inge Link][7])
2.3 S a es and GNS Rep esen a ion
On he
C∗
-algeb a
A
, a s a e is a posi i e, no malized linea unc ional
ω:A → C, ω(A∗A)≥0, ω(1) = 1.
Physically,
ω(A)
gi es he expec a ion alue o obse able
A
.
By he GNS cons uc ion, o any s a e
ω
, he e exis s a iple
(πω,Hω,Ωω)
, whe e
πω:A→B(Hω)
is a
∗
- ep esen a ion,
Ωω∈ Hω
is a cyclic ec o , such ha
ω(A) = ⟨Ωω, πω(A)Ωω⟩, A ∈ A,
and he se
{πω(A)Ωω:A∈ A}
is dense in
Hω
.
3
2.4 Heisenbe g Pic u e Deni ion o QCA
Adop ing he algeb aic deni ion om Schumache We ne and subsequen wo ks, QCA
is iewed as a
∗
-au omo phism on a quasi-local algeb a wi h ni e p opaga ion adius
and commu ing wi h ansla ions. ([a Xi ][1])
Deni ion 2.1 (Quan um Cellula Au oma on)
Le
R∈N
. A map
α:A → A
is called a Quan um Cellula Au oma on wi h adius a mos
R
, i i sa ises:
1.
α
is a
∗
-au omo phism o he
C∗
-algeb a, i.e., o any
A, B ∈ A
and
λ∈C
,
α(AB) = α(A)α(B), α(A∗) = α(A)∗, α(1) = 1,
and
α
is bijec i e and con inuous.
2. Locali y: Fo any ni e
F⋐Λ
and any
AF∈ AF
, he e exis s a ni e se
G⊂Λ
such ha
α(AF)∈ AG
, and sa is ying
G⊂BR(F) := [
x∈F
BR(x).
3. I
Λ
ca ies ansla ion ac ion
τa
, he e exis s a co esponding ansla ion au omo -
phism
θa:A→A
, such ha o all
a
and
A∈ A
,
α◦θa=θa◦α.
I he e exis s some ni e
R
such ha he abo e condi ions hold,
α
is called a local
QCA.
F om
α
, in ege i e a ions can be dened
αn:= α◦ · · · ◦ α
| {z }
n
imes
, n ∈Z,
whe e
α0= id
,
α−n:= (α−1)n
.
2.5 Sch ödinge Pic u e and Uni a y Implemen a ion
In he Sch ödinge pic u e, s a es e ol e wi h ime. Gi en QCA
α
, o any s a e
ω
, dene
he s a e a e s ep
n
as
ωn:= ω◦α−n, n ∈Z.
In an app op ia e GNS ep esen a ion, QCA can be implemen ed by a uni a y ope a o .
Le
ω
be a ai h ul and
α
-in a ian s a e, i.e.,
ω◦α=ω
, and
ω(A∗A) = 0 ⇒A= 0
.
Then in he GNS ep esen a ion
(πω,Hω,Ωω)
, he e exis s a unique uni a y ope a o
U:Hω→ Hω
such ha
πω(α(A)) = Uπω(A)U†, A ∈ A,
and
UΩω= Ωω
. This esul is a s anda d conclusion o GNS o malism applica ion; p oo
is gi en in Appendix B.
In specic models, a uni a y ope a o
U
is o en di ec ly specied on a gi en Hilbe
space
H
, se ing
α(A) := U†AU
, and hen e i ying i sa ises locali y and ansla ion
symme y condi ions.
4
2.6 Uni e se QCA Objec
Based on he abo e s uc u es, dene he "Uni e se QCA Objec ", summa izing space,
local deg ees o eedom, dynamics, and ini ial s a e in o a quin uple.
Deni ion 2.2 (Uni e se QCA Objec )
A se o da a
UQCA = (Λ,Hcell,A, α, ω0)
is called a Uni e se QCA Objec i i sa ises:
1.
Λ
is he e ex se o a coun able inni e connec ed g aph wi h g aph dis ance
dis
,
and o any
x∈Λ, R ∈N
, he ball
BR(x)
is ni e.
2.
Hx≃ Hcell
is a ni e-dimensional Hilbe space, and
A
is i s quasi-local
C∗
-algeb a.
3.
α:A→A
is a QCA wi h some ni e adius
R
.
4. I
Λ
ca ies ansla ion ac ion,
α
commu es wi h he co esponding ansla ion
au omo phism.
5.
ω0:A → C
is a no malized s a e, called he ini ial cosmic s a e; o
n∈Z
, dene
ωn:= ω0◦α−n.
In his deni ion,
Λ
and
Hcell
x he " ype o disc e e space and local deg ees o ee-
dom o he uni e se",
α
xes he "dynamical laws", and
ω0
xes he "ini ial condi ions".
3 Main Resul s (Theo ems and alignmen s)
This sec ion p esen s he main esul s: he causal s uc u e de i ed om he Uni e se
QCA Objec , he equi alen cha ac e iza ion o i s local ni eness and QCA locali y, and
he con inuous limi o Di ac- ype QCA.
3.1 E en Se and Causal Reachabili y Rela ion
In he Uni e se QCA Objec , he na u al e en se is
E:= Λ ×Z,
whe e
(x, n)
deno es he e en a la ice si e
x
a ime s ep
n
. Fo
e= (x, n)∈E
, deno e
spa ial coo dina e as
sp(e) = x
, and ime as
m(e) = n
.
The ni e p opaga ion adius o QCA de e mines he "disc e e ligh cone". Le he
p opaga ion adius o
α
be
R
. Fo any
n < m
and local ope a o
By
suppo ed on a
single poin
{y}
, by locali y we ha e
suppαm−n(By)⊂BR(m−n)(y).
Dene he geome ic ela ion
(x, n)≤geo (y, m)⇐⇒ m≥n, dis (x, y)≤R(m−n).
To ob ain causal ela ions, we need o link s a is ical co ela ion wi h geome ic ligh
cones. Dene:
Deni ion 3.1 (Causal Reachabili y Rela ion)
Fo
(x, n),(y, m)∈E
, we say
(x, n)⪯(y, m)
5

i : 1.
m≥n
;
2. The e exis s a local ope a o
Ax∈ A{x}
suppo ed on
{x}
, and
By∈ A{y}
, and
some s a e
ω
, such ha
ωαn(Ax)αm(By)=ωαn(Ax)ωαm(By).
Tha is, a local pe u ba ion a
x
a ime
n
can p oduce a s a is ical inuence a
y
a
ime
m
.
3.2 Theo em 1: QCA Causal S uc u e is a Locally Fini e Pa ial
O de
Theo em 3.2 (Pa ial O de and Local Fini eness)
In any Uni e se QCA Objec
UQCA
, he ela ion
⪯
induced by Deni ion 3.1 is equi alen o he geome ic ela ion
≤geo
, i.e.,
(x, n)⪯(y, m)⇐⇒ (x, n)≤geo (y, m).
Thus:
1.
(E, ⪯)
is a pa ial o de se ( eexi e, an isymme ic, ansi i e);
2.
(E, ⪯)
is locally ni e, i.e., o any
e1, e2∈E
, he causal in e al
I(e1, e2) := {e∈E:e1⪯e⪯e2}
is a ni e se .
The e o e,
(E, ⪯)
sha es he same s uc u al ype as "locally ni e pa ial o de " in
causal se heo y.
The p oo idea elies on wo poin s: Fi s , i
(x, n)
is wi hin he geome ic ligh cone o
(y, m)
, he e exis local ope a o s and s a es c ea ing non- i ial s a is ical co ela ions;
Second, i he dis ance condi ion is no me , ope a o suppo s can be decomposed in o
disjoin egions, leading o ac o iza ion o expec a ion alues and no causal inuence.
Local ni eness is gi en by he ni eness o each ball
BR(x)
in he g aph s uc u e and
he ni e die ence in ime s eps. De ailed p oo in Appendix A.
3.3 Theo em 2: Equi alen Cha ac e iza ion o QCA Locali y
and Locally Fini e Pa ial O de
Theo em 3.2 gi es he cons uc ion om QCA o causal se s. The e e se s a emen is:
Gi en a disc e e ime dynamics and a locally ni e pa ial o de on he e en se , he
p opaga ion adius o QCA can be econs uc ed om he "ligh cone in e ace" o his
pa ial o de .
Theo em 3.3 (Equi alen Cha ac e iza ion)
Le
α:A→A
be a
C∗
-algeb a
au omo phism,
Λ
be a connec ed g aph,
E= Λ ×Z
. The ollowing wo a e equi alen :
1. The e exis s
R < ∞
, such ha o any ni e
F⋐Λ
and
AF∈ AF
,
supp(α(AF)) ⊂
BR(F)
;
2. The e exis s a ela ion
⪯
such ha
(E, ⪯)
is a locally ni e pa ial o de , and
sa ises: i
(x, n)⪯(y, m)
and
m=n+ 1
, hen
dis (x, y)≤R
, and o all
(x, n)
, i s
one-s ep u u e eachable poin se is con ained in
BR(x)× {n+ 1}
.
In o he wo ds, he ni e p opaga ion adius condi ion o QCA is equi alen o he
locally ni e pa ial o de s uc u e on he e en se whe e "causal links pe ime s ep
6
only connec ni e neighbo hoods". This equi alen cha ac e iza ion es a es he QCA
deni ion o Schumache We ne and subsequen wo ks in he language o causal se s as
"disc e e ime + locally ni e pa ial o de + au omo phism on quasi-local algeb a".
3.4 Theo em 3: Con inuous Limi o Di ac-Type QCA
QCA is widely used as a disc e iza ion amewo k o con inuous quan um eld heo y.
Below we p esen he con inuous limi esul o 1D Di ac- ype QCA, demons a ing how
he s anda d Di ac equa ion is ob ained wi hin he Uni e se QCA Objec .
Conside
Λ = Z
, each si e ca ying
Hx≃C2
, basis ec o s deno ed
|x, ↑⟩,|x, ↓⟩
. Dene
he ime s ep e olu ion ope a o
U:= S◦R,
whe e
1. Local spin o a ion
Rx= e−iθσy= cos θ1−i sin θ σy
ac s independen ly on each si e;
2. Condi ional ansla ion
S|x, ↑⟩ =|x+ 1,↑⟩, S|x, ↓⟩ =|x−1,↓⟩.
This
U
is a uni a y ope a o wi h p opaga ion adius
R= 1
, co esponding o 1D
Di ac- ype QCA. Deno e he single-pa icle s a e a ime s ep
n
as
|ψn⟩=X
x∈Zψ↑
n(x)|x, ↑⟩ +ψ↓
n(x)|x, ↓⟩,
e olu ion equa ion is
|ψn+1⟩=U|ψn⟩.
Le la ice spacing
ε > 0
, dene con inuous coo dina es
X=εx
,
T=εn
. Take o a ion
angle scaling as
θ=εm
, whe e
m > 0
is cons an . I assuming he wa e unc ion is
smoo h in he limi
ε→0
, sa is ying
ψ↑
n(x)≈ψ↑(X, T), ψ↓
n(x)≈ψ↓(X, T),
hen e aining up o  s o de e ms in Taylo expansion, we ob ain he ollowing con-
inuous equa ions:
∂Tψ↑=−∂Xψ↑−m ψ↓, ∂Tψ↓=∂Xψ↓+m ψ↑.
Deno ing
Ψ=(ψ↑, ψ↓)T
as a wo-componen spino , he abo e can be w i en as
i∂TΨ = −iσz∂X+mσyΨ,
which is a s anda d o m o he 1D Di ac equa ion. This esul shows ha in app op ia e
scaling limi s, he Di ac- ype QCA in he Uni e se QCA Objec p oduces dynamics
consis en wi h he con inuous Di ac equa ion in he single-pa icle sec o . De ailed
de i a ion in Appendix C.
7
4 P oo s
This sec ion p o ides he p oo amewo k o he main heo ems; ull de ails a e in
appendices.
4.1 O e iew o P oo o Theo em 3.2
We need o p o e wo poin s:
1. Causal ela ion
⪯
is equi alen o geome ic ela ion
≤geo
;
2. On his basis,
(E, ⪯)
is a locally ni e pa ial o de .
Causal Reachabili y wi hin Geome ic Ligh Cone
I
(x, n)≤geo (y, m)
, hen
m≥n
and
dis (x, y)≤R(m−n)
. By locali y, o any
By
suppo ed on
{y}
,
suppαm−n(By)⊂BR(m−n)(y),
so his suppo con ains deg ees o eedom nea
x
. Choose
Ax
and
By
such ha
αm−n(By)
does no commu e wi h
Ax
nea
x
, and pick a s a e
ω
capable o esol ing his non-
commu a i i y, hen
ωαn(Ax)αm(By)−ωαn(Ax)ωαm(By)
can be cons uc ed o be non-ze o, hus
(x, n)⪯(y, m)
. This cons uc ion u ilizes he
ni e dimensionali y o local Hilbe spaces and eedom o choose Pauli- ype ope a o s.
Non-Causali y Ou side Geome ic Ligh Cone
I
m≥n
and
dis (x, y)> R(m−
n)
, hen he suppo o
αm−n(By)
is comple ely con ained in some ni e se
G⊂Λ
, and
x /∈G
. Selec a ni e se
F⋐Λ
con aining
x
and disjoin om
G
, hen
Ax∈ AF, αm−n(By)∈ AG, F ∩G=∅.
Unde enso decomposi ion
AF∪G∼
=AF⊗ AG
, hey ac on die en ac o s. Fo any
p oduc s a e
ωF⊗ωG
,
ωαn(Ax)αm(By)=ωF(. . . )ωG(. . . ),
gene al s a es can be cons uc ed by limi s o p oduc s a es, yielding ac o iza ion. Thus
(x, n)⪯ (y, m)
.
Combining he abo e wo poin s gi es equi alence o
⪯
and
≤geo
. Pa ial o de p op-
e ies ollow di ec ly om eexi i y, an isymme y, and ansi i i y o
≤geo
.
Local ni eness: Fo
e1= (x, n)⪯e2= (y, m)
, i
m < n
he in e al is emp y. I
m≥n
, any
e= (z, k)∈I(e1, e2)
sa ises
n≤k≤m, dis (x, z)≤R(k−n),dis (z, y)≤R(m−k),
hus
dis (x, z)≤R(m−n),dis (y, z)≤R(m−n),
i.e.,
z∈BR(m−n)(x)∩BR(m−n)(y)
, which is ni e; and
k
can only ake ni ely many
in ege alues, so
I(e1, e2)
is a ni e se .
Full echnical de ails in Appendix A.
8
4.2 O e iew o P oo o Theo em 3.3
The cons uc ion om ni e p opaga ion condi ion o QCA o locally ni e pa ial o de
is done in Theo em 3.2. Re e se di ec ion: Assume
(E, ⪯)
is a locally ni e pa ial
o de , and he e exis s in ege
R
such ha o any
(x, n)
, i s one-s ep u u e
{(y, n +
1) : (x, n)⪯(y, n + 1)}
sa ises
dis (x, y)≤R
. Using his condi ion, one can dene
"neighbo hood p opaga ion bound" o each ime s ep, cha ac e izing he p opaga ion
adius o au omo phism
α
as no mo e han
R
.
Specically, examine he se o all la ice poin s ha may ha e co ela ions a e one
s ep o e olu ion on any ni e egion
F
, and use local ni eness o ensu e his se emains
ni e. Fu he p o e ha o any
AF∈ AF
, suppo o
α(AF)
is con ained in
BR(F)
,
ob aining he ni e p opaga ion adius condi ion. Comple e p oo equi es o malizing he
ela ionship be ween "s a is ical co ela ion" and "suppo con ainmen ", see Appendix
A.
4.3 O e iew o P oo o Theo em 3.4
P oo o con inuous limi o Di ac- ype QCA is based on s anda d scaling limi analysis.
Key s eps a e:
1. Scale disc e e space and ime coo dina es as
X=εx, T =εn
, and le o a ion
angle
θ=εm
scale wi h
ε
;
2. Pe o m  s -o de Taylo expansion o
cos θ, sin θ
and wa e unc ion
ψ↑,↓
n(x±1)
;
3. Subs i u e expansions in o disc e e e olu ion equa ions
ψ↑
n+1(x) = cos θ ψ↑
n(x−1) −sin θ ψ↓
n(x−1),
ψ↓
n+1(x) = sin θ ψ↑
n(x+ 1) + cos θ ψ↓
n(x+ 1)
eplacing e ms wi h con inuous unc ions and de i a i es, igno ing
ε2
and highe e ms;
4. Ob ain sys em o  s -o de pa ial die en ial equa ions, ea ange in o spino
o m o ge 1D Di ac equa ion.
This cons uc ion aligns wi h ypical me hods in quan um walks and QCA simula ion
o Di ac equa ions. Full expansion calcula ion in Appendix C.
5 Model Apply
This sec ion demons a es he applica ion o Uni e se QCA Objec in specic models,
ocusing on he Di ac- ype QCA example and i s embedding in he Uni e se QCA ame-
wo k.
5.1 1D Di ac-Type Uni e se Sec ion
Le
Λ = Z
,
Hcell ≃C2
,
A
be he co esponding quasi-local algeb a. Choose Di ac- ype
QCA e olu ion
α(A) = U†AU
, whe e
U=S◦R
as desc ibed be o e. Take a ansla ion-
in a ian g ound s a e
ω0
as ini ial s a e, e.g., spin-up lled s a e, KMS s a e, o Gaussian
s a e.
Then he quin uple
UDi ac
QCA = (Z,Hcell,A, α, ω0)
9
B Appendix B: P oo o QCA Uni a y Implemen a ion
Theo em
This appendix p o es ha QCA can be implemen ed by a unique uni a y ope a o in he
GNS ep esen a ion o an
α
-in a ian ai h ul s a e.
Le
α:A→A
be a QCA,
ω
be a ai h ul and
α
-in a ian s a e, i.e.,
ω◦α=ω
, and
ω(A∗A) = 0 ⇒A= 0
. Le
(πω,Hω,Ωω)
be i s GNS ep esen a ion.
B.1 B.1 Cons uc ion and Isome y o Ope a o
U
Dene linea ope a o on dense subspace
D0:= {πω(A)Ωω:A∈ A}
U0:D0→ Hω, U0πω(A)Ωω:= πω(α(A))Ωω.
Fo any
A, B ∈ A
, we ha e
⟨U0πω(A)Ωω, U0πω(B)Ωω⟩=⟨πω(α(A))Ωω, πω(α(B))Ωω⟩=ωα(A)∗α(B).
Using
α
is a
∗
-au omo phism and
ω◦α=ω
,
ωα(A)∗α(B)=ωA∗B=⟨πω(A)Ωω, πω(B)Ωω⟩.
Thus
U0
p ese es inne p oduc on
D0
, is an isome ic linea ope a o . Since
D0
is dense
in
Hω
,
U0
uniquely ex ends o bounded ope a o
U:Hω→ Hω
, and
U
is isome ic, i.e.,
⟨Uϕ, Uψ⟩=⟨ϕ, ψ⟩, ϕ, ψ ∈ Hω.
B.2 B.2 Su jec i i y and Uni a i y
Need o p o e
U
is su jec i e. Fo any
B∈ A
, since
α
is bijec i e, he e exis s
A∈ A
such ha
B=α(A)
. Thus
πω(B)Ωω=πω(α(A))Ωω=U0πω(A)Ωω∈Ran(U0).
The e o e
Ran(U0)
con ains
{πω(B)Ωω:B∈ A}
, which is dense in
Hω
. Since
U
is
con inuous ex ension o
U0
,
Ran(U)
is closu e o
Ran(U0)
, bo h closed and dense, hus
Ran(U) = Hω
,
U
is su jec i e isome y, i.e., uni a y ope a o .
B.3 B.3 Conjuga e Ac ion Implemen s
α
Fo any
A, B ∈ A
,
Uπω(A)U†πω(B)Ωω=Uπω(A)U†πω(B)Ωω.
No e ha
U†πω(B)Ωω
16

is some ec o , and by deni ion can be w i en as linea combina ion o GNS ec o s.
Mo e con enien is o calcula e di ec ly on
D0
:
Uπω(A)U†πω(B)Ωω=Uπω(A)U†πω(B)Ωω
=Uπω(A)πω(α−1(B))Ωω
=U0πω(Aα−1(B))Ωω
=πω(α(Aα−1(B)))Ωω
=πω(α(A)B)Ωω
=πω(α(A))πω(B)Ωω
Since
{πω(B)Ωω:B∈ A}
is dense in
Hω
, abo e shows
Uπω(A)U†=πω(α(A))
holds on en i e Hilbe space.
Finally,
UΩω=Uπω(1)Ωω=πω(α(1))Ωω=πω(1)Ωω= Ωω,
showing GNS ec o is
U
-in a ian . This comple es p oo o uni a y implemen a ion
heo em.
C Appendix C: 1D Di ac-Type QCA and Con inuous
Limi o Di ac Equa ion
This appendix gi es specic cons uc ion and con inuous limi de i a ion o 1D Di ac- ype
QCA.
C.1 C.1 Model Deni ion and Disc e e E olu ion Equa ion
Take
Λ = Z
, each si e ca ies
Hx≃C2
, basis ec o s
|x, ↑⟩,|x, ↓⟩
. O e all Hilbe space
o mally
H=O
x∈Z
Hx.
Dene local spin o a ion ope a o
Rx= e−iθσy= cos θ1−i sin θ σy,
global o a ion
R:= O
x∈Z
Rx.
Dene condi ional ansla ion ope a o
S
ac ion on single pa icle basis as
S|x, ↑⟩ =|x+ 1,↑⟩, S|x, ↓⟩ =|x−1,↓⟩.
Time s ep e olu ion ope a o is
U:= S◦R.
U
is uni a y wi h p opaga ion adius
R= 1
.
17
In single pa icle sec o , w i e
|ψn⟩=X
x∈Zψ↑
n(x)|x, ↑⟩ +ψ↓
n(x)|x, ↓⟩,
disc e e e olu ion is
|ψn+1⟩=U|ψn⟩.
Expanding gi es ecu ence ela ion:
ψ↑
n+1(x) = cos θ ψ↑
n(x−1) −sin θ ψ↓
n(x−1),
ψ↓
n+1(x) = sin θ ψ↑
n(x+ 1) + cos θ ψ↓
n(x+ 1).
C.2 C.2 Con inuous Limi and Taylo Expansion
In oduce la ice spacing
ε > 0
, dene con inuous a iables
X=εx, T =εn.
Assume smoo h unc ions
ψ↑,↓(X, T)
exis such ha
ψ↑
n(x)≈ψ↑(X, T), ψ↓
n(x)≈ψ↓(X, T)
a
X=εx, T =εn
. Le o a ion angle scale wi h
ε
as
θ=εm,
whe e
m > 0
is cons an .
Fi s o de expansion o
cos θ
and
sin θ
:
cos θ= cos(εm)≈1−1
2ε2m2,
sin θ= sin(εm)≈εm.
Fi s o de expansion o spa ial ansla ion and ime s ep:
ψ↑
n(x±1) ≈ψ↑(X±ε, T)≈ψ↑(X, T)±ε∂Xψ↑(X, T),
ψ↓
n(x±1) ≈ψ↓(X±ε, T)≈ψ↓(X, T)±ε∂Xψ↓(X, T),
ψ↑
n+1(x)≈ψ↑(X, T +ε)≈ψ↑(X, T) + ε∂Tψ↑(X, T),
ψ↓
n+1(x)≈ψ↓(X, T +ε)≈ψ↓(X, T) + ε∂Tψ↓(X, T).
Subs i u e in o disc e e e olu ion equa ion, keep  s o de e ms in
ε
.
Fo spin-up componen :
ψ↑(X, T) + ε∂Tψ↑(X, T)≈1−1
2ε2m2ψ↑(X−ε, T)−εm ψ↓(X−ε, T).
RHS expands o
1−1
2ε2m2ψ↑(X, T)−ε∂Xψ↑(X, T)−εmψ↓(X, T)−ε∂Xψ↓(X, T).
18
Igno ing
ε2
e ms, ge
ψ↑(X, T) + ε∂Tψ↑(X, T)≈ψ↑(X, T)−ε∂Xψ↑(X, T)−εm ψ↓(X, T).
Sub ac
ψ↑(X, T)
om bo h sides, ea ange o
∂Tψ↑(X, T) = −∂Xψ↑(X, T)−m ψ↓(X, T).
Fo spin-down componen :
ψ↓(X, T) + ε∂Tψ↓(X, T)≈εm ψ↑(X+ε, T) + 1−1
2ε2m2ψ↓(X+ε, T).
RHS expands o
εmψ↑(X, T) + ε∂Xψ↑(X, T)+ψ↓(X, T) + ε∂Xψ↓(X, T),
Igno ing
ε2
e ms, ge
ψ↓(X, T) + ε∂Tψ↓(X, T)≈ψ↓(X, T) + ε∂Xψ↓(X, T) + εm ψ↑(X, T).
Sub ac
ψ↓(X, T)
om bo h sides, ea ange o
∂Tψ↓(X, T) = ∂Xψ↓(X, T) + m ψ↑(X, T).
W i e in spino o m. Dene
Ψ(X, T) := ψ↑(X, T)
ψ↓(X, T),
hen
∂TΨ(X, T) = −σz∂XΨ(X, T)−m σxΨ(X, T).
Mul iply by
i
and ea ange
i∂TΨ(X, T) = −iσz∂X+mσyΨ(X, T),
whe e
−mσx
and
mσy
can be in e changed by basis edeni ion. This is a s anda d o m
o 1D Di ac equa ion.
C.3 C.3 Rela ion o Uni e se QCA Objec
In Uni e se QCA amewo k, 1D Di ac- ype model co esponds o
UDi ac
QCA = (Z,Hcell,A, α, ω0),
whe e
α(A) = U†AU
. This objec induces e en se
E=Z×Z
and causal pa ial
o de
⪯
, QCA p opaga ion adius
R= 1
co esponds o "maximum p opaga ion speed",
and Di ac equa ion in con inuous limi is he eec i e desc ip ion o his uni e se a low
ene gy and long wa eleng h scales.
This example shows: unde he p emise ha he uni e se is dened as a QCA objec ,
h ough app op ia e scaling and coa se-g aining, con inuous ela i is ic eld heo y can
be ep oduced wi hin he in e nal cons uc ion, p o iding conc e e ma hema ical suppo
o a ionali y o "Uni e se as Quan um Disc e e Cellula Au oma on".
19