Tenso G ad: Di e en iable Tenso -Ne wo k Op imiza ion o G ound S a es and
En anglemen Diagnos ics
Aulia Oc a iani1
1Independen Resea che , Compu a ional Physics, Indonesia
(Da ed: Oc obe 10, 2025)
We p esen Tenso G ad, a di e en iable enso -ne wo k amewo k o a ia ional op imiza ion
o quan um many-body g ound s a es. The me hod combines ini e-di e ence and au odi e en i-
a ion backends o op imize ma ix-p oduc -s a e (MPS) pa ame e s using g adien descen . Ap-
plied o benchma k models such as he T ans e se-Field Ising Model (TFIM) and he Heisenbe g
spin chain, Tenso G ad e icien ly con e ges o low-ene gy con igu a ions using a minimal ansa z.
Fu he mo e, an addi ional wo-body en angle ci cui in oduces con olled quan um co ela ions,
enabling non- i ial educ ions in g ound-s a e ene gy and measu able inc eases in en anglemen
en opy. The amewo k also compu es en anglemen spec a and on Neumann en opy p o iles,
o e ing an accessible pla o m o s udy he in e play be ween a ia ional op imiza ion and en an-
glemen in di e en iable physics.
I. INTRODUCTION
Tenso -ne wo k (TN) me hods such as ma ix p od-
uc s a es (MPS) and p ojec ed en angled-pai s a es
(PEPS) p o ide powe ul ep esen a ions o quan um
many-body sys ems. Thei compac exp essi i y a ises
om en anglemen -limi ed pa ame e iza ions, making
hem ideal o modeling g ound s a es o local Hamil-
onians. Meanwhile, di e en iable p og amming ame-
wo ks ha e e olu ionized machine lea ning, enabling
g adien -based op imiza ion o high-dimensional pa am-
e e spaces. B idging hese ideas, Tenso G ad p o ides
a minimal and di e en iable implemen a ion o TN op-
imiza ion o spin sys ems, illus a ing how a ia ional
quan um simula ions can be ea ed as g adien -based
lea ning asks.
Ou goal is o demons a e ha e en a ligh weigh
di e en iable MPS amewo k can ep oduce physically
meaning ul ene gy landscapes, en anglemen s uc u es,
and con e gence beha io s compa able o adi ional
a ia ional algo i hms, while emaining compu a ionally
simple.
II. METHODS
A. Va ia ional ansa z
We conside a p oduc -s a e MPS o Nspin-1
2si es,
|ψ(θ, ϕ)⟩=
N
O
i=1 cos θi
2|0⟩+eiϕisin θi
2|1⟩.(1)
To in oduce minimal co ela ions, a small wo-body
en angle is applied:
U(γ) = exp iγ
N−1
X
i=1
XiXi+1!,(2)
whe e γcon ols he deg ee o en anglemen . The ull
a ia ional s a e is |Ψ⟩=U(γ)|ψ(θ, ϕ)⟩.
B. Hamil onians
We es Tenso G ad on wo canonical la ice Hamil o-
nians:
1. T ans e se-Field Ising Model (TFIM):
HTFIM =−JX
i
ZiZi+1 −hX
i
Xi,(3)
2. Heisenbe g Model:
HHeis =JxX
i
XiXi+1 +JyX
i
YiYi+1 +JzX
i
ZiZi+1.
(4)
Fini e chains wi h open bounda y condi ions a e used
h oughou (N≤8).
C. Op imiza ion
The objec i e unc ion is he a ia ional ene gy:
E(θ, ϕ) = ⟨Ψ|H|Ψ⟩
⟨Ψ|Ψ⟩.(5)
G adien s a e ob ained ia ini e di e ences o he JAX
au odi backend:
∇ (x)≈ (x+ϵ)− (x−ϵ)
2ϵ,(6)
and pa ame e s a e upda ed using g adien descen :
x +1 =x −η∇ (x ),(7)
whe e ηis he lea ning a e.
III. RESULTS
A. G ound-s a e ene gy op imiza ion
Figu e 1shows con e gence o he a ia ional ene gy
o TFIM wi h N= 6 and N= 8. Tenso G ad quickly
2
minimizes he ene gy o nea g ound-s a e le els wi hin
a ew hund ed s eps.
FIG. 1. Ene gy op imiza ion o he TFIM model using g a-
dien descen . La ge Nyields deepe minima as sys em com-
plexi y g ows.
Fo he Heisenbe g chain wi h en angle γ= 0.02, he
ene gy landscape becomes smoo he and he minimum
ene gy signi ican ly lowe (Fig. 2).
B. En anglemen diagnos ics
We compu e he on Neumann en anglemen en opy,
SA=−T (ρAlog ρA),(8)
o bipa i ions ac oss he chain. Wi hou en angle , he
en opy emains close o ze o (p oduc s a e). When he
en angle is applied, SAinc eases, peaking nea he sys-
em’s cen e (Fig. 3).
The co esponding en anglemen spec um a he cen-
al cu also b oadens (Fig. 4), demons a ing he en an-
glemen ’s non i ial s uc u e.
FIG. 2. Ene gy con e gence o he Heisenbe g chain (N= 6)
wi h en angle γ= 0.02.
FIG. 3. En anglemen en opy p o ile o N= 6 wi h en-
angle γ= 0.02. The inc ease in SAindica es eme gence o
quan um co ela ions.
FIG. 4. En anglemen spec um a mid-cu (N= 6,γ=
0.02).
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IV. DISCUSSION
The esul s ob ained wi h Tenso G ad demons a e
ha e en a compac , di e en iable implemen a ion o
ma ix p oduc s a es (MPS) can e ec i ely ep oduce
he quali a i e physics o quan um g ound s a es. The
success ul con e gence o he a ia ional ene gy in bo h
he TFIM and Heisenbe g models con i ms ha g adien -
based me hods, when applied o enso -ne wo k an-
sä ze, can se e as an al e na i e o adi ional op imiza-
ion echniques such as he densi y ma ix eno maliza-
ion g oup (DMRG) o a ia ional quan um eigensol e s
(VQE). While Tenso G ad ope a es a a small sys em
size, i s beha io e lec s many o he cha ac e is ic ea-
u es o hese la ge amewo ks.
The amewo k p o ides a b idge be ween physics-
based a ia ional p inciples and mode n di e en iable
p og amming pa adigms. In his sense, i se es as a
concep ual labo a o y o explo ing how au oma ic di -
e en ia ion, loss-based aining, and pa ame e -space ge-
ome y can be used o op imize quan um s a es di ec ly
wi hin enso -ne wo k mani olds.
The inclusion o a simple XX en angle ci cui plays
a c ucial ole. This ope a o ac s as a minimal gene a-
o o local co ela ions, analogous o wo-body ga es in
a ia ional quan um ci cui s o local p ojec o s in PEPS.
I s e ec —lowe ing he a ia ional ene gy and inc easing
he on Neumann en anglemen en opy—illus a es how
con olled en anglemen can sys ema ically imp o e a i-
a ional exp essi i y. This obse a ion is consis en wi h
p e ious indings in enso -ne wo k heo y, whe e in o-
ducing sho - ange en anglemen is known o enhance ap-
p oxima ion accu acy.
F om a me hodological s andpoin , Tenso G ad high-
ligh s ha di e en iable op imiza ion does no necessa -
ily equi e la ge-scale au oma ic di e en ia ion ame-
wo ks; ini e-di e ence g adien s, when combined wi h
clea enso abs ac ions, a e su icien o explo e a i-
a ional landscapes and isualize op imiza ion dynam-
ics. This balance be ween simplici y and physical in e -
p e abili y makes Tenso G ad a use ul pedagogical and
esea ch ool o s udying di e en iable enso ne wo ks.
Finally, he amewo k opens a pa h owa d in eg a -
ing di e en iable op imiza ion in o enso -ne wo k algo-
i hms o mo e complex sys ems—such as chi al spin
liquids, opologically o de ed PEPS, o models wi h long-
ange in e ac ions—whe e analy ic g adien s and en an-
glemen diagnos ics may e eal new s uc u al insigh s
in o he na u e o quan um co ela ions.
V. CONCLUSION
We in oduced Tenso G ad, a compac and di e en-
iable enso -ne wo k amewo k ha success ully ep o-
duces essen ial ea u es o quan um many-body g ound-
s a e op imiza ion. By combining nume ical s abili y
wi h concep ual anspa ency, Tenso G ad demons a es
ha g adien -based lea ning pa adigms can be di ec ly
applied o physical a ia ional p oblems wi hou equi -
ing la ge-scale symbolic algeb a o high-pe o mance con-
ac ion lib a ies.
I s modula design encompassing MPS-based p oduc
s a es, di e en iable en angle s, and au oma ic compu-
a ion o obse ables such as ene gy and en anglemen
en opy p o ides a e sa ile educa ional and esea ch
es bed. Beyond a simple pedagogical ool, Tenso G ad
also es ablishes a concep ual link be ween di e en iable
p og amming and enso -ne wo k a ia ional p inciples.
The obse ed lowe ing o he Heisenbe g g ound-s a e
ene gy h ough a small XX en angle and he eme gence
o non-ze o en anglemen en opy show ha e en mini-
mal di e en iable ci cui s can mimic he physics o co e-
la ed quan um sys ems. This highligh s he po en ial o
hyb id physics-in o med di e en iable amewo ks o ex-
plo ing quan um co ela ions in low-dimensional la ice
models.
Fu u e Wo k. Building upon his ounda ion, se e al
esea ch di ec ions na u ally eme ge:
•Implemen ing highe -dimensional enso ne wo ks
such as PEPS, including co ne - ans e and
bounda y-MPS eno maliza ion.
•In eg a ing JAX o PyTo ch backends o ull au-
oma ic di e en ia ion o enso con ac ions and
ene gy unc ionals.
•Explo ing he connec ion be ween di e en iable en-
angle s and uni a y pa ame iza ions o opologi-
cal o de and chi al spin liquids.
•Ex ending he amewo k owa d di e en iable
quan um simula ion pipelines, whe e loss unc ions
a e physically mo i a ed quan i ies such as ideli y
o mu ual in o ma ion.
•Benchma king Tenso G ad agains es ablished
DMRG and VQE algo i hms o assess con e gence
scaling and a ia ional exp essi i y.
O e all, Tenso G ad b idges nume ical op imiza ion
and heo e ical quan um in o ma ion concep s, o e ing
a small ye powe ul p o o ype o he nex gene a ion
o di e en iable enso -ne wo k esea ch. I s simplici y
makes i accessible o s uden s, while i s ex ensibili y
enables in eg a ion in o mode n machine-lea ning-based
quan um simula ion en i onmen s.
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