Unified Theory of Quantum Chaos and Eigenstate Thermalization in QCA Universe\\ \large ETH, Postulated Chaos and Eigenstate Level Statistics under Unified Time Scale

Unied Theo y o Quan um Chaos and Eigens a e
The maliza ion in QCA Uni e se
ETH, Pos ula ed Chaos and Eigens a e Le el S a is ics unde Unied Time
Scale
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
In he amewo k o Unied Time Scale, Ma ix Uni e se
THE
-
MATRIX
, and
Quan um Disc e e Cellula Au oma on Uni e se
Uqca
, we cons uc a sys ema ic
heo y o Quan um Chaos and he Eigens a e The maliza ion Hypo hesis (ETH).
The Unied Time Scale Mo he Fo mula
κ(ω) = φ′(ω)/π =ρ el(ω) = (2π)−1 Q(ω)
(1)
unies he sca e ing semi-phase de i a i e, ela i e densi y o s a es, and Wigne 
Smi h g oup delay ace in o a single ime scale densi y, he eby p o iding a "Uni-
e se Time Mo he Rule " in he Ma ix Uni e se
Uma
independen o specic
Hamil onian choices. The Uni e se as a QCA objec
Uqca = (Λ,Hcell,Aqloc, α, ω0)
(2)
desc ibes disc e e ime e olu ion ia local uni a y au omo phisms
α
on a coun able
la ice, and econs uc s ela i is ic quan um eld heo y and geome ic s uc u e
in he con inuum limi .
Based on his, we p opose an axioma ic sys em o "Pos ula ed Chaos QCA"
and p o e he ollowing main esul s:
1. On any ni e egion
Ω⊂Λ
, he es ic ed ni e-dimensional uni a y op-
e a o
UΩ
sa ises Disc e e Time ETH o i s quasi-ene gy spec um eigens a es
wi h espec o any local ope a o
OX
(suppo
X⊂Ω
): o he as majo i y o
eigens a es
|ψn⟩
wi hin an ene gy window,
⟨ψn|OX|ψn⟩=⟨OX⟩mic o(εn) + O(e−c|Ω|),
(3)
and he squa ed a e age o o-diagonal ma ix elemen s decays exponen ially wi h
olume, achie ing eigens a e he maliza ion o local obse ables.
2. Unde he axioms o "Pos ula ed Chaos QCA" (ni e p opaga ion adius,
ansla ion symme y, local ga es gene a ing high-o de uni a y designs, no ex a
ex ensi e conse ed quan i ies), he quasi-ene gy le el s a is ics o he ni e egion
UΩ
con e ge a e un olding o he Wigne Dyson dis ibu ion o he CUE andom
1
ma ix class. I s spec al o m ac o exhibi s a ypical " amppla eau" s uc u e,
cons i u ing a s anda d quan um chaos diagnosis a he QCA le el.
3. Rela ing QCAETH o he Unied Time Scale: Unde he unied ime scale
τ
, he " he maliza ion ime scale" and he g ow h a e o local en opy densi y a e
join ly con olled by he ene gy shell a e age o he scale densi y
κ(ω)
and he QCA
ligh cone s uc u e. We p o e a "Unied TimeETHEn opy G ow h Theo em":
o any amily o ni e-densi y local ini ial s a es, he en opy densi y inc eases
mono onically wi h
τ
and app oaches he mic ocanonical en opy densi y in he
long- ime limi .
4. On he global QCA uni e se objec , embedding he abo e local esul s in o he
causal ne wo k and unied ime scale mo he s uc u e shows ha he " he mal ime
a ow" and "mac oscopic i e e sibili y" on he cosmic scale can be unde s ood as
s uc u al in a ian s in he unied Ma ixQCA Uni e se, a he han addi ionally
in oduced dynamical pos ula es.
The appendix p o ides: igo ous deni ions and equi alen o ms o disc e e
ime ETH; exac mapping be ween local andom ci cui s and QCA; p oo de ails
and cons an es ima es o he QCAETH Theo em and CUE le el s a is ics; and a
1D Pos ula ed Chaos QCA model wi h specic p edic ions o nume ical e ica ion.
Keywo ds:
Quan um Chaos; Eigens a e The maliza ion Hypo hesis (ETH); Quan um
Cellula Au oma a (QCA); Unied Time Scale; Ma ix Uni e se; Uni a y Design; Spec-
al Fo m Fac o ; Random Ma ix Theo y
1 In oduc ion & His o ical Con ex
1.1 S anda d Pic u e o ETH and Quan um Chaos
In closed many-body quan um sys ems, how pu e s a e uni a y e olu ion gene a es s a-
is ical beha io compa ible wi h he modynamic equilib ium is a co e p oblem in he
ounda ions o quan um s a is ical mechanics. In eg able sys ems ypically e ain a la ge
numbe o conse ed quan i ies, ending owa ds a Gene alized Gibbs Ensemble a e
long- ime e olu ion, whe eas non-in eg able sys ems in high-ene gy densi y egions a e
widely belie ed o sa is y he Eigens a e The maliza ion Hypo hesis (ETH). Deu sch and
S ednicki  s p oposed, h ough andom ma ix heu is ics and eld heo y analysis, ha
in he eigenbasis o a sucien ly chao ic Hamil onian, he eigens a e ma ix elemen s o
a local obse able ope a o
O
can be w i en as
⟨Eα|O|Eβ⟩=O(¯
E)δαβ + e−S(¯
E)/2 O(¯
E, ω)Rαβ,
(4)
whe e
¯
E= (Eα+Eβ)/2
,
ω=Eα−Eβ
,
S(¯
E)
is he mic oscopic en opy, and
Rαβ
a e
quasi-Gaussian andom numbe s wi h ze o mean and uni a iance. The diagonal e m
gi es he ene gy-dependen he mal equilib ium alue, while he o-diagonal e ms a e
exponen ially small in sys em olume, ensu ing ha ime-a e aged obse ables app oxi-
ma e he mic ocanonical ensemble a e age and ime uc ua ions a e supp essed.
Subsequen ly, ex ensi e nume ical and heo e ical wo k has e ied ETH in con ex s
such as spin chains, Bose/Fe mi la ice models, and Floque many-body sys ems, sys-
ema ically analyzing i s scope and ailu e mechanisms (e.g., many-body localiza ion).
Pa allel o his, esea ch on andom ma ix heo y and quan um chaos p o ides ano he
diagnos ic ou e: i le el s a is ics exhibi a Wigne Dyson dis ibu ion a e app op ia e
un olding, and he spec al o m ac o shows a " amppla eau", he sys em is conside ed
o be in a quan um chaos phase.
2
1.2 Disc e e Time Sys ems and Random Quan um Ci cui s
In Floque sys ems and andom quan um ci cui s, ime e olu ion is desc ibed by a single
ime-s ep uni a y ope a o
U
, and he quasi-ene gy spec um is dened by
U|ψn⟩= e−iεn∆ |ψn⟩.
(5)
ETH can be ew i en as a s a emen abou quasi-ene gy eigens a es
|ψn⟩
, whe e he mal
equilib ium co esponds o a mic ocanonical dis ibu ion on a xed quasi-ene gy shell.
Local andom quan um ci cui s ha e been p o en o achie e high-o de uni a y designs
a polynomial dep h, wi h hei eigens a es and spec al s a is ics s ic ly app oaching
he ypical p ope ies o Haa andom uni a y ma ices. Recen wo k has u he im-
p o ed he ade-o be ween design o de and ci cui dep h, p o iding ne es ima es o
" andomness" and "sc ambling speed".
These esul s indica e ha in disc e e- ime many-body sys ems wi h locali y con-
s ain s, quan um chaos and ETH possess he uni e sali y p edic ed by andom ma ix
heo y and can be s ic ly con olled ia he language o local andom ci cui s and uni a y
designs.
1.3 S uc u e and De elopmen o Quan um Cellula Au oma a
(QCA)
Quan um Cellula Au oma a (QCA) a e ano he class o uni a y dynamical models dis-
c e e in bo h ime and space, encoding s uc u es like "locali y, ansla ion symme y,
ni e p opaga ion speed" unde igo ous ma hema ical axioms. Schumache and We ne
p o ided he deni ion and s uc u e heo ems o e e sible QCA, emphasizing ha
QCA a e ansla ion-co a ian dynamics wi h ni e p opaga ion adius on inni e la ice
sys ems; he same local ule can gene a e global ime s eps unde ni e pe iodic bounda y
condi ions. In he 1D case, G oss, Nesme, Vog s, and We ne de eloped index heo y o
QCA and quan um walks, p o iding opological indices on K- heo y o classi y 1D QCA.
O he wo ks cha ac e ize he s uc u e and simulabili y o local QCA om a quan um
in o ma ion pe spec i e.
Compa ed o local andom ci cui s, QCA a e close o he ideal abs ac ion o "cos-
mic dynamics": hei deni ion does no include ex e nal noise o measu emen , elying
only on disc e e ime-s ep uni a y upda es and spa ial locali y. The e o e, i he cosmic
on ology is cha ac e ized as a ce ain QCA, hen ETH, quan um chaos, i e e sibili y,
and he he mal ime a ow should all nd explana ions wi hin he QCA amewo k.
1.4 Unied Time Scale and Ma ix Uni e se
In sca e ing and spec al heo y, he Wigne Smi h ime delay ma ix
Q(ω) = −iS(ω)†∂ωS(ω)
(6)
ela es he equency de i a i e o he mul i-channel sca e ing ma ix
S(ω)
o "g oup
delay", wi h i s ace gi ing he o al delay ime. On he o he hand, he Bi manK e
in
o mula and Li shi sK e
in ace o mula ela e he spec al shi unc ion
ξ(ω)
o he
sca e ing de e minan
de S(ω)
, showing ha
ξ(ω) = −1
2πilog de S(ω), ρ el(ω) = −ξ′(ω)
(7)
3
a e well-dened o e y gene al ope a o pai s
(H0, H)
.
The Unied Time Scale Mo he Fo mula
κ(ω) = φ′(ω)/π =ρ el(ω) = (2π)−1 Q(ω)
(8)
is in e p e ed in his con ex as a unied densi y o " ela i e densi y o s a es  g oup
delay  sca e ing phase de i a i e", dening a "Time Mo he Rule " independen o
specic coo dina es and local Hamil onian choices.
The Ma ix Uni e se
THE
-
MATRIX
iews he uni e se as a amily o gian sca e ing
ma ices
S(ω)
decomposed by ene gy, and hei Wigne Smi h ma ices
Q(ω)
, wi h block
spa se s uc u e encoding causal pa ial o de and spec al da a ealizing he unied ime
scale. In geome icbounda y ime s uc u e, he unied ime scale co esponds o objec s
like modula ow and GHY bounda y ime ansla ion, he eby downg ading " ime" o
a unc ion o sca e ing phase and spec al shi .
1.5 Objec i es and Wo k O e iew
The goal o his pape is o es ablish an axioma ic and heo em-based heo y o "The
alidi y o Quan um Chaos and ETH in QCA Uni e se and Le el S a is ics" unde he
dual amewo k o Unied Time Scale and Ma ix Uni e seQCA Uni e se.
The co e ideas a e:
1. Model he uni e se as a e e sible QCA objec
Uqca
sa is ying he Schumache 
We ne axioms, ob aining ni e-dimensional uni a y ope a o s
UΩ
on ni e egions.
2. In oduce he axioma ic sys em o "Pos ula ed Chaos QCA", such ha
UΩ
is
equi alen o a amily o local andom ci cui s a ni e dep h. These ci cui s cons i u e
high-o de uni a y designs wi hin polynomial dep h, he eby app oxima ing Haa andom
uni a ies in local obse ables and spec al s a is ics.
3. U ilize he ETH ypicali y o Haa andom uni a ies and andom ma ix heo y o
es ablish QCAETH heo ems and CUE- ype spec al s a is ics heo ems o he eigen-
s a e ma ix elemen s, le el spacings, and spec al o m ac o o
UΩ
.
4. Rela e he Unied Time Scale Mo he Fo mula
κ(ω)
o he disc e e ime s ep o
QCA, p o ing he Unied TimeETHEn opy G ow h Theo em, and es a ing he he -
mal ime a ow and mac oscopic i e e sibili y a he le el o he cosmic causal ne wo k.
Following he gi en s uc u e, he subsequen sec ions p esen he model and pos-
ula es, main heo ems, p oo s, model applica ions, enginee ing p oposals, discussion,
conclusion, and de ailed p oo s in he appendices.
2 Model Assump ions
2.1 Unied Time Scale Mo he Fo mula and Sca e ing S uc u e
Le
H
be a sepa able Hilbe space, and
H0, H
be a pai o sel -adjoin ope a o s sa is ying
app op ia e ace-class pe u ba ion condi ions such ha wa e ope a o s exis and a e
comple e, and he sca e ing ope a o
S=W†
+W−
(9)
can be w i en unde spec al decomposi ion as
S=Z⊕
S(ω) dµ(ω),
(10)
4
whe e
ω
ep esen s ene gy o equency pa ame e . Fo almos e e y
ω
,
S(ω)
is a uni a y
ope a o on he be Hilbe space.
The Bi manK e
in o mula asse s he exis ence o a spec al shi unc ion
ξ(ω)
such
ha
ξ(ω) = −1
2πilog de S(ω), ρ el(ω) = −ξ′(ω)
(11)
gi ing he ela i e densi y o s a es in b oad cases. On he o he hand, he Wigne Smi h
ime delay ma ix is dened as
Q(ω) = −iS(ω)†∂ωS(ω),
(12)
whose ace cha ac e izes he o al g oup delay ime.
The Unied Time Scale Mo he Fo mula is dened as
κ(ω) = φ′(ω)
π=ρ el(ω) = 1
2π Q(ω),
(13)
whe e
φ(ω)
is he sca e ing semi-phase (phase o he ela i e sca e ing de e minan ).
In he unied amewo k,
κ(ω)
is ega ded as he "Uni e se Time Densi y". Any physical
ime pa ame e , i ob ained h ough an obse able measu emen p ocess, belongs o he
equi alence class o
τsca (ω) = Zω
κ(˜ω) d˜ω
(14)
on la ge scales.
2.2 Ma ix Uni e se
THE
-
MATRIX
The Ma ix Uni e se objec
Uma
is dened as
Uma =Hchan, S(ω), Q(ω), κ, A∂, ω∂,
(15)
whe e: 1.
Hchan =L ∈VH
is he di ec sum o channel Hilbe spaces, each
co e-
sponding o a mac oscopic "po " o bounda y egion; 2.
S(ω)∈ B(Hchan)
is a amily
o equency-dependen sca e ing ma ices, whose block spa se s uc u e on
V×V
en-
codes causal pa ial o de and in e ac ion s uc u e; 3.
Q(ω) = −iS(ω)†∂ωS(ω)
is he
Wigne Smi h g oup delay ma ix, and he unied ime scale densi y is gi en by
κ(ω) = (2π)−1 Q(ω);
(16)
4.
A∂
is he bounda y obse able algeb a, and
ω∂
is he bounda y quan um s a e, con-
nec ed o sca e ing da a ia bounda y ime geome y and modula ow.
In his objec , quan um chaos and ETH co espond o: sca e ing phases obeying
andom ma ix heo y p edic ions s a is ically o e ene gy windows, and local p ojec-
ions unde channel decomposi ion exhibi ing only exponen ially small de ia ions be ween
eigens a e a e ages and co esponding mic ocanonical a e ages.
2.3 QCA Uni e se
Uqca
The QCA Uni e se objec
Uqca
is dened as
Uqca = (Λ,Hcell,Aqloc, α, ω0),
(17)
5

sa is ying he ollowing axioms: 1.
Λ
is a coun able connec ed g aph ( ypically
Zd
o i s
ni e me ic de o ma ion); 2. Each la ice si e
x∈Λ
ca ies a ni e-dimensional Hilbe
space
Hx∼
=Hcell
; 3. The quasi-local algeb a
Aqloc
is he no m closu e o all ni ely
suppo ed ope a o s; 4.
α:Aqloc → Aqloc
is a
∗
-au omo phism, wi h a p opaga ion
adius
R < ∞
, such ha o any ni e egion
X⊂Λ
, an ope a o
OX
suppo ed on
X
is
mapped o an ope a o suppo ed on
X+R={y∈Λ|dis (y, X)≤R};
(18)
5.
α
is implemen ed by a global uni a y ope a o
U
, i.e.,
α(O) = U†OU
; 6.
α
is ansla ion
co a ian , i.e., commu es wi h he la ice ansla ion g oup ac ion; 7. The ini ial s a e
ω0
is a s a e on
Aqloc
, gi ing he quan um s a e o he uni e se a ime s ep
n= 0
.
Fo any ni e egion
Ω⊂Λ
, dene
HΩ=Nx∈ΩHx
. Res ic ing
U
yields
UΩ
(unde
app op ia e bounda y condi ions), wi h spec um
UΩ|ψn⟩= e−iεn∆ |ψn⟩.
(19)
These ni e-dimensional uni a y ope a o s a e he undamen al objec s o discussing
QCAETH and le el s a is ics.
2.4 Pos ula ed Chaos QCA
To make QCA exhibi s a is ical p ope ies simila o local andom ci cui s on ni e
egions, we in oduce he ollowing deni ion.
Deni ion 2.1
(Pos ula ed Chaos QCA)
.
A ansla ion-in a ian QCA
U
is called a
Pos ula ed Chaos QCA i i sa ises: 1. **Fini e P opaga ion Radius and Locali y:**
The e exis s an in ege
R
such ha o any ni e
X⊂Λ
,
α(AX)⊂ AX+R
; 2. **Local
Ci cui Rep esen a ion:** On any ni e egion
Ω
,
UΩ
can be w i en in a sui able basis
as a ni e-dep h local quan um ci cui
UΩ=
D
Y
ℓ=1
Uℓ, Uℓ=O
j
Uℓ,j,
(20)
whe e each ga e
Uℓ,j
ac s on a ni e subse
Xℓ,j ⊂Ω
and commu es wi h all ga es
sepa a ed by mo e han a ni e dis ance; 3. **App oxima e Uni a y Design:** The e
exis s
0∈N
and a unc ion
ϵ (|Ω|)
(decaying exponen ially wi h
|Ω|
), such ha o any
≤ 0
, he uni a y amily gene a ed by
UΩ
cons i u es an
ϵ
-app oxima e uni a y design
in he
- h momen , i.e., o any polynomial
P(U, U†)
(deg ee no exceeding
),


EUΩ[P(UΩ)] −EU∼Haa [P(U)]

≤ϵ (|Ω|);
(21)
whe e
U∼Haa
deno es Haa andom uni a y on
U(dim HΩ)
. 4. **No Ex a Ex ensi e
Conse ed Quan i ies:** Excep o possibly a ew global quan um numbe s (e.g., o al
pa icle numbe , spin), he e a e no independen ex ensi e local conse ed quan i ies
in he sys em; 5. **The maliza ion Ene gy Window:** The e exis s an ene gy window
I⊂(−π/∆ , π/∆ ]
, wi hin which he numbe o eigens a es g ows exponen ially wi h
|Ω|
, and ene gy le el degene acy p oduces only ni ely many symme y mul iplici ies.
These condi ions encapsula e ma u e esul s o local andom ci cui s ealizing high-
o de uni a y designs and sa is ying ETH, embedding hem in o he language o QCA.
6
2.5 Deni ion o Disc e e Time ETH
Conside a ni e egion
Ω
and i s e olu ion ope a o
UΩ
, wi h spec al decomposi ion as
be o e. Fo a gi en ene gy window cen e
ε
and wid h
δ > 0
, dene he quasi-ene gy
shell subspace
HΩ(ε, δ) = span{|ψn⟩ | εn∈(ε−δ, ε +δ)},
(22)
wi h dimension deno ed by
Dε,δ
. Dene he mic ocanonical a e age
⟨OX⟩mic o(ε) = D−1
ε,δ X
εn∈(ε−δ,ε+δ)
⟨ψn|OX|ψn⟩
(23)
( aking
δ
scaling polynomially wi h
|Ω|
).
Deni ion 2.2
(Disc e e Time ETH)
.
UΩ
is said o sa is y Disc e e Time ETH o a
amily o local ope a o s
{OX}
in ene gy window
I
, i he e exis cons an
c > 0
and
smoo h unc ions
OX(ε)
,
σX(ε)
, such ha o any
X⊂Ω
and he as majo i y o
n
(
εn∈I
): 1. **Diagonal ETH:**
⟨ψn|OX|ψn⟩=OX(εn) + O(e−c|Ω|);
(24)
2. **O-Diagonal ETH:** Fo almos all
m=n
and
¯ε= (εm+εn)/2∈I
,
|⟨ψm|OX|ψn⟩| ≤ e−S(¯ε)/2σX(¯ε),
(25)
whe e
S(¯ε)∼s(¯ε)|Ω|
is he mic ocanonical en opy o he ene gy shell.
I his holds o all amilies o local ope a o s, he QCA is said o sa is y ETH in ha
egion.
3 Main Resul s (Theo ems and alignmen s)
Fo b e i y, deno e
dim HΩ=D∼es|Ω|
.
3.1 QCAETH Main Theo em
Theo em 3.1
(QCAETH Theo em)
.
Le
U
be a Pos ula ed Chaos QCA,
Ω⋐Λ
a
sucien ly la ge ni e egion,
UΩ
he uni a y ope a o es ic ed o
Ω
, and
{|ψn⟩, εn}
i s
quasi-ene gy eigenpai s. Then he e exis an ene gy window
I
and a cons an
c > 0
such
ha o any local ope a o
OX
wi h ni e suppo
X⊂Ω
, he e exis s a smoo h unc ion
OX(ε)
sa is ying:
1. **Diagonal ETH:** Fo he as majo i y o
n
in he ene gy window (
εn∈I
),
⟨ψn|OX|ψn⟩=OX(εn) + O(e−c|Ω|);
(26)
2. **O-Diagonal ETH:** The second momen sa ises
E|⟨ψm|OX|ψn⟩|2≤e−S(¯ε)gO(¯ε, ω),
(27)
whe e
¯ε= (εm+εn)/2
,
S(¯ε)∼s(¯ε)|Ω|
, and
gO
is bounded;
3. I he ini ial s a e
|ψ0⟩
has a na ow ene gy dis ibu ion wi hin window
I
, i.e.,
|cn|2
is signican only o
εn∈I
, hen i s ime-a e aged local obse a ion sa ises
⟨OX⟩=⟨OX⟩mic o(ε) + O(e−c|Ω|),
(28)
and ime uc ua ions a e exponen ially supp essed by he o-diagonal ETH index.
7
3.2 CUE- ype Con e gence o Quasi-Ene gy S a is ics
Le
θn=εn∆ ∈(−π, π]
, so ed in ascending o de and un olded o a iables
sn
wi h
a e age spacing 1.
Theo em 3.2
(CUE Beha io o QCA Le el S a is ics)
.
Unde he assump ions o The-
o em 3.1, he un olded nea es -neighbo spacing dis ibu ion
P(s)
con e ges in he limi
|Ω|→∞
o he Wigne Dyson dis ibu ion o he CUE andom ma ix ensemble:
PCUE(s)∼32
π2s2e−4s2/π.
(29)
Simul aneously, he no malized spec al o m ac o
K( ) = D−1| U
Ω|2
(30)
exhibi s a " amppla eau" s uc u e a e app op ia e escaling, consis en wi h he uni-
e sal spec al uc ua ions o CUE.
3.3 Unied TimeETHEn opy G ow h Theo em
In he QCA Uni e se, in oduce an ane ela ion be ween unied ime scale
τ
and disc e e
ime s ep
n
:
τ=an∆ +b, a > 0.
(31)
Conside a amily o "low en opy" ini ial s a es
{ρ0}
wi hin ene gy window
I
, whose on
Neumann en opy densi y
s0=S(ρ0)/|Ω|
is less han he mic ocanonical en opy densi y
smc(ε)
.
Theo em 3.3
(Unied TimeETHEn opy G ow h)
.
Unde Pos ula ed Chaos QCA and
he assump ions o Theo em 3.1, he e exis s a unc ion
en (ε)>0
and a cons an
c′>0
such ha o any ni e
X⊂Ω
, in he unied ime scale in e al
τ∈[0, τ h]
, he en opy
densi y o he educed s a e
ρX(τ) = Ω Xρ(τ)
,
sX(τ) = |X|−1S(ρX(τ)),
(32)
sa ises
sX(τ)≥s0+ en (ε)τ
ℓe
− O(e−c′|Ω|),
(33)
and app oaches
smc(ε)
a e
τ≳τ h
. He e
ℓe
is de e mined by he p opaga ion adius o
he QCA and LiebRobinson ype ligh cone eloci y, and
en (ε)
can be w i en as a unc-
ion o he a e age o he unied scale densi y
κ(ω)
o e window
I
and local in e ac ion
s eng h.
3.4 Cosmic Scale ETH and The mal Time A ow
View he QCA Uni e se
Uqca
as he di ec limi o a amily o ni e egions
{ΩL}
,
ΩL↗Λ
.
Fo each
L
, es ic ing
U
yields
UΩL
.
P oposi ion 3.4
(ETH on Cosmic Causal Ne wo k)
.
I
U
is a Pos ula ed Chaos QCA,
hen o any ni e causal diamond (de e mined by some obse e 's wo ldline), he e exis s
L
such ha his diamond is con ained in some sucien ly la ge egion
ΩL
. Consequen ly,
8
unde he unied ime scale, all local obse ables wi hin his diamond end o mic ocanon-
ical equilib ium a e a sucien ly long ime, and he en opy densi y g ows mono onically
wi h
τ
un il sa u a ion.
The e o e, he he mal ime a ow and mac oscopic i e e sibili y on he cosmic scale
can be iewed as s uc u al esul s join ly de e mined by QCAETH and he Unied Time
Scale.
4 P oo s
This sec ion p o ides p oo s o he main heo ems. Technical de i a ions o ope a o
in eg als, spec al un olding, and concen a ion inequali ies a e placed in he appendices.
4.1 Local Random Ci cui s and App oxima e Uni a y Designs
Recall he esul s o local andom quan um ci cui s ealizing app oxima e uni a y designs.
B andão, Ha ow, and Ho odecki p o ed ha on a 1D chain, local andom ci cui s com-
posed o nea es -neighbo wo-body ga es achie e
- h o de app oxima e uni a y design
wi hin dep h
O( 10n2)
. Ha ow e al. imp o ed he dep h es ima e o highe dimensions
and mo e gene al la ice s uc u es o he op imal scaling o
poly( )n1/D
. Subsequen
wo k cons uc ed explici amilies o local designs unde symme y cons ain s and pa -
icle numbe conse a ion.
These heo ems can be summa ized as: on a ni e egion
Ω
, o a amily o local ga es
sa is ying ce ain gene ici y and non-degene acy condi ions, he dis ibu ion induced by
sucien ly deep andom ci cui s on he uni a y g oup is close o Haa in he
- h momen
sense.
In he deni ion o Pos ula ed Chaos QCA, he p ope y o app oxima e uni a y
design is embedded ia he local ci cui ep esen a ion o
UΩ
. Since QCA e olu ion
is de e minis ic a he han andom, he "se o mul iple ime s eps
Un
Ω
" needs o be
iewed as a ci cui amily: when
n
a ies wi hin an app op ia e ime window,
{Un
Ω}
o ms a amily o o bi s in he local ga e pa ame e space. In some QCAs, his amily
o o bi s is sucien o ealize design p ope ies; mo e gene ally, ni e supe -pe iods o
spa ial ansla ions can be in oduced in dening he cosmic QCA o achie e "eec i e
andomiza ion". In he pos ula es o his pape , hese de ails a e abs ac ed as " he
uni a y amily gene a ed by
UΩ
wi hin polynomial ime s eps is an app oxima e uni a y
design".
4.2 ETH Typicali y o Haa Random Uni a ies
Fo a Haa andom uni a y
U∈U(D)
, i s eigen ec o s a e uni o mly dis ibu ed on
he complex sphe e o he Hilbe space. Fo any xed local ope a o
OX
, eigens a e
ma ix elemen s a is ics can be calcula ed using Haa in eg a ion o mulas. The ol-
lowing conclusions nd sys ema ic p oo s in andom ma ix heo y and high-dimensional
geome y.
Lemma 4.1
(Diagonal S a is ics o Haa Random Eigenbasis)
.
Le
U
be Haa andom,
{|ψn⟩}
be i s eigenbasis, and
OX
be a local ope a o suppo ed on
|X| ≪ |Ω|
. Then: 1.
E[⟨ψn|OX|ψn⟩] = (OX)/D
is independen o le el
n
; 2.
Va [⟨ψn|OX|ψn⟩]∼ O(D−1)
,
9
and ini ial s a e
|ψ0⟩=Pncn|ψn⟩
. The ime a e age o a local obse a ion is
⟨OX⟩= lim
N→∞
1
N
N−1
X
k=0
⟨ψ0|U†k
ΩOXUk
Ω|ψ0⟩=X
n
|cn|2⟨ψn|OX|ψn⟩,
(49)
assuming non-degene a e ene gy le els.
I Diagonal ETH holds, i.e.,
⟨ψn|OX|ψn⟩=OX(εn) + δn,|δn| ≤ e−c|Ω|,
(50)
and he ini ial s a e ene gy dis ibu ion is concen a ed in a na ow window, hen
⟨OX⟩=X
n
|cn|2OX(εn) + O(e−c|Ω|)≈OX(ε)≈ ⟨OX⟩mic o(ε).
(51)
The e o e, Diagonal ETH is equi alen o he maliza ion o ime-a e aged local obse a-
ions (in he sense o exponen ially small e o ).
A.2 A.2 O-Diagonal ETH and Time Fluc ua ions
Time uc ua ions can be exp essed as
δOX(k) = ⟨OX⟩(k)−⟨OX⟩=X
m=n
c∗
mcnei(εm−εn)k∆ ⟨ψm|OX|ψn⟩.
(52)
An uppe bound o i s a iance is
|δOX|2≤X
m=n
|cm|2|cn|2|⟨ψm|OX|ψn⟩|2.
(53)
I O-Diagonal ETH gi es
E|⟨ψm|OX|ψn⟩|2≤e−S(¯ε)gO(¯ε, ω),
(54)
hen gi en he numbe o eigens a es in he ene gy shell
Dε,δ ∼eS(ε)
, he uc ua ion
a iance is
O(e−S(ε))
, decaying exponen ially wi h olume.
A.3 A.3 Rela ion be ween Floque ETH and Hamil onian ETH
When an eec i e con inuum limi exis s o he QCA, an eec i e Hamil onian can be
dened
He =i
∆ log U,
(55)
whose spec um ela es o quasi-ene gy spec um as
En≈εn
. I
∆
is sucien ly small
and he b anch cu o
log U
is chosen p ope ly, Floque ETH and Hamil onian ETH a e
equi alen in he same ene gy window, and mic ocanonical ensembles and quasi-ene gy
shells can be in e changed.
16

B Appendix B: Technical De ails o QCAETH Theo-
em
B.1 B.1 Quan i a i e Bounds o App oxima e Uni a y Designs
Fo a 1D chain o
n q
-dimensional qubi s, he design heo em by B andãoHa ow
Ho odecki can be w i en as: he e exis cons an s
C, c > 0
such ha local andom
ci cui s o nea es -neighbo ga es wi h dep h
L≥C 10n2
(56)
cons i u e an
ϵ
-app oxima e
-design, whe e
ϵ≤e−cn
.
Ha ow e al. u he p o ed ha on a
D
-dimensional la ice, he dep h scaling can
be imp o ed o
poly( )n1/D
. Combined wi h ou pos ula es, we can ake
0
as a xed
cons an , and
ϵ 0(|Ω|)≤e−c|Ω|
.
B.2 B.2 Haa In eg a ion and Eigens a e Ma ix Elemen s
Haa in eg a ion o mulas gi e
ZU(D)
Ui1j1· · · UikjkUi′
1j′
1· · · Ui′
kj′
kdµHaa (U)
(57)
exp essible using Weinga en unc ions on he symme ic g oup
Sk
. Fo
k≤ 0
, expec a-
ions can be exp essed as ni e sums. This leads o he mean and a iance es ima es in
Lemma 4.1 and Lemma 4.2.
In he QCA scena io, since
UΩ
is an app oxima e uni a y design a o de
0
, he
die ence be ween he abo e expec a ions and a iances unde he dis ibu ion induced
by
UΩ
and Haa measu e is also con olled by
ϵ 0
.
B.3 B.3 Concen a ion Inequali ies
Lipschi z unc ions on he complex sphe e o Hilbe space sa is y Le y's concen a ion
inequali y: o an
L
-Lipschi z unc ion
,
P| −E |> ϵ≤2 exp(−cDϵ2/L2).
(58)
Taking
as unc ions o
⟨ψ|OX|ψ⟩
o
|⟨ψm|OX|ψn⟩|2
yields p obabili y bounds o de i-
a ions o eigens a e ma ix elemen s om mean alues.
C Appendix C: Spec al Fo m Fac o and Wigne Dyson
Dis ibu ion
C.1 C.1 Spec al Fo m Fac o o CUE
The spec al o m ac o o a CUE andom ma ix
U∈U(D)
is dened as
KCUE( ) = D−1E| U |2.
(59)
Random ma ix heo y gi es an explici exp ession in he limi
D→ ∞
: unde escaled
ime
τ= /D
,
KCUE(τ)
exhibi s a linea " amp" and sa u a ion "pla eau".
17
C.2 C.2 Spec al Fo m Fac o in QCA Models
In Pos ula ed Chaos QCA,
UΩ
is app oxima ely Haa andom on ni e-o de ace poly-
nomials, so he s a is ics o
K( )
wi hin a ni e ime window
| | ≤ max ∼poly(|Ω|)
ma ch
KCUE( )
o CUE, wi h de ia ion
O(ϵ 0)
.
Th ough Fou ie ans o m,
K( )
can be ela ed o le el co ela ion unc ions, he eby
ob aining he Wigne Dyson o m o nea es -neighbo spacing dis ibu ion and highe -
o de spacing dis ibu ions.
D Appendix D: Nume ical Ve ica ion F amewo k o
1D Pos ula ed Chaos QCA
D.1 D.1 Model Pa ame e Selec ion
In 1D b ick-wall QCA, wo-body ga es can be chosen as
Uga e = exp−i(Jxσx⊗σx+Jyσy⊗σy+Jzσz⊗σz+hx(σx⊗I+I⊗σx))∆ ,
(60)
wi h pa ame e s
(Jx, Jy, Jz, hx)
subjec ed o quasi- andom pe u ba ions o e die en
ime pe iods o b eak in eg abili y and ex a symme ies.
D.2 D.2 Nume ical S eps
1. Cons uc
UΩ
o a chain o leng h
L
and pe o m exac diagonaliza ion ( easible o
L≤16
); 2. Calcula e he dis ibu ion o ma ix elemen s o local ope a o s (e.g., single-
body Pauli ma ices o wo-body in e ac ion e ms) on eigens a es, e i ying Diagonal and
O-Diagonal ETH; 3. Calcula e un olded spacing dis ibu ion and spec al o m ac o ,
compa ing wi h CUE esul s; 4. Simula e ime e olu ion o die en ini ial s a e amilies,
e i ying he maliza ion o local obse ables and g ow h o en opy densi y, compa ing
wi h p edic ions o Theo em 3.3.
This amewo k p o ides a conc e e ealizable nume ical and expe imen al pa h o
e i ying Pos ula ed Chaos QCA axioms and he heo ems o his pape .
18