<s>
The	O
time-evolving	B-Algorithm
block	I-Algorithm
decimation	I-Algorithm
(	O
TEBD	B-Algorithm
)	O
algorithm	O
is	O
a	O
numerical	O
scheme	O
used	O
to	O
simulate	O
one-dimensional	O
quantum	O
many-body	O
systems	O
,	O
characterized	O
by	O
at	O
most	O
nearest-neighbour	O
interactions	O
.	O
</s>
<s>
It	O
is	O
dubbed	O
Time-evolving	B-Algorithm
Block	I-Algorithm
Decimation	I-Algorithm
because	O
it	O
dynamically	O
identifies	O
the	O
relevant	O
low-dimensional	O
Hilbert	O
subspaces	O
of	O
an	O
exponentially	O
larger	O
original	O
Hilbert	O
space	O
.	O
</s>
<s>
When	O
the	O
first	O
quantum	B-Architecture
computer	I-Architecture
is	O
plugged	O
in	O
and	O
functioning	O
,	O
the	O
perspectives	O
for	O
the	O
field	O
of	O
computational	O
physics	O
will	O
look	O
rather	O
promising	O
,	O
but	O
until	O
that	O
day	O
one	O
has	O
to	O
restrict	O
oneself	O
to	O
the	O
mundane	O
tools	O
offered	O
by	O
classical	O
computers	O
.	O
</s>
<s>
While	O
experimental	O
physicists	O
are	O
putting	O
a	O
lot	O
of	O
effort	O
in	O
trying	O
to	O
build	O
the	O
first	O
quantum	B-Architecture
computer	I-Architecture
,	O
theoretical	O
physicists	O
are	O
searching	O
,	O
in	O
the	O
field	O
of	O
quantum	O
information	O
theory	O
(	O
QIT	O
)	O
,	O
for	O
genuine	O
quantum	O
algorithms	O
,	O
appropriate	O
for	O
problems	O
that	O
would	O
perform	O
badly	O
when	O
trying	O
to	O
be	O
solved	O
on	O
a	O
classical	O
computer	O
,	O
but	O
pretty	O
fast	O
and	O
successful	O
on	O
a	O
quantum	O
one	O
.	O
</s>
<s>
The	O
search	O
for	O
such	O
algorithms	O
is	O
still	O
going	O
,	O
the	O
best-known	O
(	O
and	O
almost	O
the	O
only	O
ones	O
found	O
)	O
being	O
the	O
Shor	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
,	O
for	O
factoring	O
large	O
numbers	O
,	O
and	O
Grover	B-Algorithm
's	I-Algorithm
search	I-Algorithm
algorithm	I-Algorithm
.	O
</s>
<s>
In	O
the	O
field	O
of	O
QIT	O
one	O
has	O
to	O
identify	O
the	O
primary	O
resources	O
necessary	O
for	O
genuine	O
quantum	B-Architecture
computation	I-Architecture
.	O
</s>
<s>
He	O
asserts	O
that	O
"	O
any	O
quantum	B-Architecture
computation	I-Architecture
with	O
pure	O
states	O
can	O
be	O
efficiently	O
simulated	O
with	O
a	O
classical	O
computer	O
provided	O
the	O
amount	O
of	O
entanglement	O
involved	O
is	O
sufficiently	O
restricted	O
"	O
.	O
</s>
<s>
This	O
happens	O
to	O
be	O
the	O
case	O
with	O
generic	O
Hamiltonians	O
displaying	O
local	O
interactions	O
,	O
as	O
for	O
example	O
,	O
Hubbard-like	O
Hamiltonians	O
.	O
</s>
<s>
The	O
classically	O
(	O
and	O
quantum	O
,	O
as	O
well	O
)	O
feasible	O
simulations	O
are	O
those	O
that	O
involve	O
systems	O
only	O
lightly	O
entangled	O
—	O
the	O
strongly	O
entangled	O
ones	O
being	O
,	O
on	O
the	O
other	O
hand	O
,	O
good	O
candidates	O
only	O
for	O
genuine	O
quantum	B-Architecture
computations	I-Architecture
.	O
</s>
<s>
A	O
useful	O
feature	O
of	O
the	O
TEBD	B-Algorithm
algorithm	O
is	O
that	O
it	O
can	O
be	O
reliably	O
employed	O
for	O
time	O
evolution	O
simulations	O
of	O
time-dependent	O
Hamiltonians	O
,	O
describing	O
systems	O
that	O
can	O
be	O
realized	O
with	O
cold	O
atoms	O
in	O
optical	O
lattices	O
,	O
or	O
in	O
systems	O
far	O
from	O
equilibrium	O
in	O
quantum	O
transport	O
.	O
</s>
<s>
From	O
this	O
point	O
of	O
view	O
,	O
TEBD	B-Algorithm
had	O
a	O
certain	O
ascendance	O
over	O
DMRG	O
,	O
a	O
very	O
powerful	O
technique	O
,	O
but	O
until	O
recently	O
not	O
very	O
well	O
suited	O
for	O
simulating	O
time-evolutions	O
.	O
</s>
<s>
With	O
the	O
Matrix	O
Product	O
States	O
formalism	O
being	O
at	O
the	O
mathematical	O
heart	O
of	O
DMRG	O
,	O
the	O
TEBD	B-Algorithm
scheme	O
was	O
adopted	O
by	O
the	O
DMRG	O
community	O
,	O
thus	O
giving	O
birth	O
to	O
the	O
time	O
dependent	O
DMRG	O
,	O
t-DMRG	O
for	O
short	O
.	O
</s>
<s>
Those	O
last	O
approaches	O
actually	O
provide	O
a	O
formalism	O
that	O
is	O
more	O
general	O
than	O
the	O
original	O
TEBD	B-Algorithm
approach	O
,	O
as	O
it	O
also	O
allows	O
to	O
deal	O
with	O
evolutions	O
with	O
matrix	O
product	O
operators	O
;	O
this	O
enables	O
the	O
simulation	O
of	O
nontrivial	O
non-infinitesimal	O
evolutions	O
as	O
opposed	O
to	O
the	O
TEBD	B-Algorithm
case	O
,	O
and	O
is	O
a	O
crucial	O
ingredient	O
to	O
deal	O
with	O
higher-dimensional	O
analogues	O
of	O
matrix	O
product	O
states	O
.	O
</s>
<s>
The	O
trick	O
of	O
TEBD	B-Algorithm
is	O
to	O
re-write	O
the	O
coefficients	O
:	O
</s>
<s>
One	O
can	O
proceed	O
now	O
to	O
investigate	O
the	O
behaviour	O
of	O
the	O
decomposition	O
D	O
when	O
acted	O
upon	O
with	O
one-qubit	O
gates	B-Application
(	O
OQG	O
)	O
and	O
two-qubit	O
gates	B-Application
(	O
TQG	O
)	O
acting	O
on	O
neighbouring	O
qubits	O
.	O
</s>
<s>
The	O
OQGs	O
are	O
affecting	O
only	O
the	O
qubit	O
they	O
are	O
acting	O
upon	O
,	O
the	O
update	O
of	O
the	O
state	O
after	O
a	O
unitary	B-Algorithm
operator	I-Algorithm
at	O
qubit	O
k	O
does	O
not	O
modify	O
the	O
Schmidt	O
eigenvalues	O
or	O
vectors	O
on	O
the	O
left	O
,	O
consequently	O
the	O
'	O
s	O
,	O
or	O
on	O
the	O
right	O
,	O
hence	O
the	O
'	O
s	O
.	O
</s>
<s>
The	O
changes	O
required	O
to	O
update	O
the	O
'	O
s	O
and	O
the	O
'	O
s	O
,	O
following	O
a	O
unitary	B-Algorithm
operation	I-Algorithm
V	O
on	O
qubits	O
k	O
,	O
k+1	O
,	O
concern	O
only	O
,	O
and	O
.	O
</s>
<s>
To	O
find	O
out	O
the	O
new	O
decomposition	O
,	O
the	O
new	O
'	O
s	O
at	O
the	O
bond	O
k	O
and	O
their	O
corresponding	O
Schmidt	O
eigenvectors	O
must	O
be	O
computed	O
and	O
expressed	O
in	O
terms	O
of	O
the	O
'	O
s	O
of	O
the	O
decomposition	O
D	O
.	O
The	O
reduced	O
density	O
matrix	O
is	O
therefore	O
diagonalized	B-Algorithm
:	O
</s>
<s>
Expressing	O
the	O
eigenvectors	O
of	O
the	O
diagonalized	B-Algorithm
matrix	I-Algorithm
in	O
the	O
basis	O
:	O
the	O
'	O
s	O
are	O
obtained	O
as	O
well	O
:	O
</s>
<s>
The	O
dimension	O
of	O
the	O
largest	O
tensors	B-Device
in	O
D	O
is	O
of	O
the	O
order	O
;	O
when	O
constructing	O
the	O
one	O
makes	O
the	O
summation	O
over	O
,	O
and	O
for	O
each	O
,	O
adding	O
up	O
to	O
a	O
total	O
of	O
operations	O
.	O
</s>
<s>
For	O
simulations	O
of	O
quantum	O
dynamics	O
it	O
is	O
useful	O
to	O
use	O
operators	O
that	O
are	O
unitary	B-Algorithm
,	O
conserving	O
the	O
norm	O
(	O
unlike	O
power	O
series	O
expansions	O
)	O
,	O
and	O
there	O
's	O
where	O
the	O
Trotter-Suzuki	O
expansion	O
comes	O
in	O
.	O
</s>
<s>
The	O
trick	O
of	O
the	O
ST2	O
is	O
to	O
write	O
the	O
unitary	B-Algorithm
operators	I-Algorithm
as	O
:	O
</s>
<s>
Mind	O
this	O
would	O
imply	O
diagonalizing	B-Algorithm
somewhat	O
generous	O
reduced	O
density	O
matrices	O
,	O
which	O
,	O
depending	O
on	O
the	O
system	O
one	O
has	O
to	O
simulate	O
,	O
might	O
be	O
a	O
task	O
beyond	O
our	O
reach	O
and	O
patience	O
.	O
</s>
<s>
Hence	O
,	O
at	O
the	O
first	O
bond	O
,	O
instead	O
of	O
futilely	O
diagonalizing	B-Algorithm
,	O
let	O
us	O
say	O
,	O
10	O
by	O
10	O
or	O
20	O
by	O
20	O
matrices	O
,	O
we	O
can	O
just	O
restrict	O
ourselves	O
to	O
ordinary	O
2	O
by	O
2	O
ones	O
,	O
thus	O
making	O
the	O
algorithm	O
generally	O
faster	O
.	O
</s>
<s>
TEBD	B-Algorithm
also	O
offers	O
the	O
possibility	O
of	O
straightforward	O
parallelization	O
due	O
to	O
the	O
factorization	O
of	O
the	O
exponential	O
time-evolution	O
operator	O
using	O
the	O
Suzuki	O
–	O
Trotter	O
expansion	O
.	O
</s>
<s>
A	O
parallel-TEBD	O
has	O
the	O
same	O
mathematics	O
as	O
its	O
non-parallelized	O
counterpart	O
,	O
the	O
only	O
difference	O
is	O
in	O
the	O
numerical	O
implementation	O
.	O
</s>
