<s>
Tensor	B-Application
networks	I-Application
or	O
tensor	B-Application
network	I-Application
states	I-Application
are	O
a	O
class	O
of	O
variational	O
wave	O
functions	O
used	O
in	O
the	O
study	O
of	O
many-body	B-Algorithm
quantum	O
systems	O
.	O
</s>
<s>
Tensor	B-Application
networks	I-Application
extend	O
one-dimensional	O
matrix	O
product	O
states	O
to	O
higher	O
dimensions	O
while	O
preserving	O
some	O
of	O
their	O
useful	O
mathematical	O
properties	O
.	O
</s>
<s>
The	O
wave	O
function	O
is	O
encoded	O
as	O
a	O
tensor	B-Device
contraction	O
of	O
a	O
network	O
of	O
individual	O
tensors	B-Device
.	O
</s>
<s>
The	O
structure	O
of	O
the	O
individual	O
tensors	B-Device
can	O
impose	O
global	O
symmetries	O
on	O
the	O
wave	O
function	O
(	O
such	O
as	O
antisymmetry	B-Algorithm
under	I-Algorithm
exchange	I-Algorithm
of	O
fermions	O
)	O
or	O
restrict	O
the	O
wave	O
function	O
to	O
specific	O
quantum	O
numbers	O
,	O
like	O
total	O
charge	O
,	O
angular	O
momentum	O
,	O
or	O
spin	O
.	O
</s>
<s>
It	O
is	O
also	O
possible	O
to	O
derive	O
strict	O
bounds	O
on	O
quantities	O
like	O
entanglement	O
and	O
correlation	O
length	O
using	O
the	O
mathematical	O
structure	O
of	O
the	O
tensor	B-Application
network	I-Application
.	O
</s>
<s>
This	O
has	O
made	O
tensor	B-Application
networks	I-Application
useful	O
in	O
theoretical	O
studies	O
of	O
quantum	O
information	O
in	O
many-body	B-Algorithm
systems	I-Algorithm
.	O
</s>
<s>
They	O
have	O
also	O
proved	O
useful	O
in	O
variational	O
studies	O
of	O
ground	O
states	O
,	O
excited	O
states	O
,	O
and	O
dynamics	O
of	O
strongly	O
correlated	O
many-body	B-Algorithm
systems	I-Algorithm
.	O
</s>
<s>
In	O
general	O
,	O
a	O
tensor	B-Application
network	I-Application
diagram	O
(	O
Penrose	O
diagram	O
)	O
can	O
be	O
viewed	O
as	O
a	O
graph	O
where	O
nodes	O
(	O
or	O
vertices	O
)	O
represent	O
individual	O
tensors	B-Device
,	O
while	O
edges	O
represent	O
summation	O
over	O
an	O
index	O
.	O
</s>
<s>
For	O
instance	O
,	O
one	O
can	O
use	O
trapezoids	O
for	O
unitary	O
matrices	O
or	O
tensors	B-Device
with	O
similar	O
behaviour	O
.	O
</s>
<s>
Tensor	B-Application
networks	I-Application
have	O
been	O
adapted	O
for	O
supervised	B-General_Concept
learning	I-General_Concept
,	O
taking	O
advantage	O
of	O
similar	O
mathematical	O
structure	O
in	O
variational	O
studies	O
in	O
quantum	O
mechanics	O
and	O
large-scale	O
machine	O
learning	O
.	O
</s>
<s>
This	O
crossover	O
has	O
spurred	O
collaboration	O
between	O
researchers	O
in	O
artificial	B-Application
intelligence	I-Application
and	O
quantum	O
information	O
science	O
.	O
</s>
<s>
In	O
June	O
2019	O
,	O
Google	B-Application
,	O
the	O
Perimeter	O
Institute	O
for	O
Theoretical	O
Physics	O
,	O
and	O
X	O
(	O
company	O
)	O
,	O
released	O
TensorNetwork	O
,	O
an	O
open-source	O
library	O
for	O
efficient	O
tensor	B-Device
calculations	O
.	O
</s>
<s>
The	O
main	O
interest	O
in	O
tensor	B-Application
networks	I-Application
and	O
their	O
study	O
from	O
the	O
perspective	O
of	O
machine	O
learning	O
is	O
to	O
reduce	O
the	O
number	O
of	O
trainable	O
parameters	O
(	O
in	O
a	O
layer	O
)	O
by	O
approximating	O
a	O
high-order	O
tensor	B-Device
with	O
a	O
network	O
of	O
lower-order	O
ones	O
.	O
</s>
<s>
Using	O
the	O
so-called	O
tensor	B-Device
train	O
technique	O
(	O
TT	O
)	O
,	O
one	O
can	O
reduce	O
an	O
N-order	O
tensor	B-Device
(	O
containing	O
exponentially	O
many	O
trainable	O
parameters	O
)	O
to	O
a	O
chain	O
of	O
N	O
tensors	B-Device
of	O
order	O
2	O
or	O
3	O
,	O
which	O
gives	O
us	O
a	O
polynomial	O
number	O
of	O
parameters	O
.	O
</s>
