<s>
Quantum	B-Algorithm
tomography	I-Algorithm
or	O
quantum	B-Algorithm
state	I-Algorithm
tomography	I-Algorithm
is	O
the	O
process	O
by	O
which	O
a	O
quantum	O
state	O
is	O
reconstructed	O
using	O
measurements	O
on	O
an	O
ensemble	O
of	O
identical	O
quantum	O
states	O
.	O
</s>
<s>
The	O
term	O
tomography	B-Algorithm
was	O
first	O
used	O
in	O
the	O
quantum	O
physics	O
literature	O
in	O
a	O
1993	O
paper	O
introducing	O
experimental	O
optical	O
homodyne	O
tomography	B-Algorithm
.	O
</s>
<s>
In	O
quantum	B-Algorithm
process	I-Algorithm
tomography	I-Algorithm
on	O
the	O
other	O
hand	O
,	O
known	O
quantum	O
states	O
are	O
used	O
to	O
probe	O
a	O
quantum	O
process	O
to	O
find	O
out	O
how	O
the	O
process	O
can	O
be	O
described	O
.	O
</s>
<s>
Similarly	O
,	O
quantum	O
measurement	O
tomography	B-Algorithm
works	O
to	O
find	O
out	O
what	O
measurement	O
is	O
being	O
performed	O
.	O
</s>
<s>
Whereas	O
,	O
randomized	B-Device
benchmarking	I-Device
scalably	O
obtains	O
a	O
figure	O
of	O
merit	O
of	O
the	O
overlap	O
between	O
the	O
error	O
prone	O
physical	O
quantum	O
process	O
and	O
its	O
ideal	O
counterpart	O
.	O
</s>
<s>
The	O
general	O
principle	O
behind	O
quantum	B-Algorithm
state	I-Algorithm
tomography	I-Algorithm
is	O
that	O
by	O
repeatedly	O
performing	O
many	O
different	O
measurements	O
on	O
quantum	O
systems	O
described	O
by	O
identical	O
density	O
matrices	O
,	O
frequency	O
counts	O
can	O
be	O
used	O
to	O
infer	O
probabilities	O
,	O
and	O
these	O
probabilities	O
are	O
combined	O
with	O
Born	O
's	O
rule	O
to	O
determine	O
a	O
density	O
matrix	O
which	O
fits	O
the	O
best	O
with	O
the	O
observations	O
.	O
</s>
<s>
The	O
position	O
and	O
momentum	B-Algorithm
of	O
the	O
oscillator	O
at	O
any	O
given	O
point	O
can	O
be	O
measured	O
and	O
therefore	O
the	O
motion	O
can	O
be	O
completely	O
described	O
by	O
the	O
phase	O
space	O
.	O
</s>
<s>
So	O
we	O
have	O
retrieved	O
a	O
function	O
W(x,p )	O
which	O
gives	O
a	O
description	O
of	O
the	O
chance	O
of	O
finding	O
the	O
particle	O
at	O
a	O
given	O
point	O
with	O
a	O
given	O
momentum	B-Algorithm
.	O
</s>
<s>
The	O
only	O
difference	O
is	O
that	O
the	O
Heisenberg	O
's	O
uncertainty	O
principle	O
must	O
n't	O
be	O
violated	O
,	O
meaning	O
that	O
we	O
cannot	O
measure	O
the	O
particle	O
's	O
momentum	B-Algorithm
and	O
position	O
at	O
the	O
same	O
time	O
.	O
</s>
<s>
The	O
particle	O
's	O
momentum	B-Algorithm
and	O
its	O
position	O
are	O
called	O
quadratures	O
(	O
see	O
Optical	O
phase	O
space	O
for	O
more	O
information	O
)	O
in	O
quantum	O
related	O
states	O
.	O
</s>
<s>
In	O
the	O
following	O
text	O
we	O
will	O
see	O
that	O
this	O
probability	O
density	O
is	O
needed	O
to	O
characterize	O
the	O
particle	O
's	O
quantum	O
state	O
,	O
which	O
is	O
the	O
whole	O
point	O
of	O
quantum	B-Algorithm
tomography	I-Algorithm
.	O
</s>
<s>
Quantum	B-Algorithm
tomography	I-Algorithm
is	O
applied	O
on	O
a	O
source	O
of	O
systems	O
,	O
to	O
determine	O
the	O
quantum	O
state	O
of	O
the	O
output	O
of	O
that	O
source	O
.	O
</s>
<s>
Unlike	O
a	O
measurement	O
on	O
a	O
single	O
system	O
,	O
which	O
determines	O
the	O
system	O
's	O
current	O
state	O
after	O
the	O
measurement	O
(	O
in	O
general	O
,	O
the	O
act	O
of	O
making	O
a	O
measurement	O
alters	O
the	O
quantum	O
state	O
)	O
,	O
quantum	B-Algorithm
tomography	I-Algorithm
works	O
to	O
determine	O
the	O
state(s )	O
prior	O
to	O
the	O
measurements	O
.	O
</s>
<s>
Quantum	B-Algorithm
tomography	I-Algorithm
can	O
be	O
used	O
for	O
characterizing	O
optical	O
signals	O
,	O
including	O
measuring	O
the	O
signal	O
gain	O
and	O
loss	O
of	O
optical	O
devices	O
,	O
as	O
well	O
as	O
in	O
quantum	B-Architecture
computing	I-Architecture
and	O
quantum	O
information	O
theory	O
to	O
reliably	O
determine	O
the	O
actual	O
states	O
of	O
the	O
qubits	O
.	O
</s>
<s>
Not	O
confident	O
with	O
Bob	O
's	O
description	O
of	O
the	O
state	O
,	O
Alice	O
may	O
wish	O
to	O
do	O
quantum	B-Algorithm
tomography	I-Algorithm
to	O
classify	O
the	O
state	O
herself	O
.	O
</s>
<s>
Using	O
Born	O
's	O
rule	O
,	O
one	O
can	O
derive	O
the	O
simplest	O
form	O
of	O
quantum	B-Algorithm
tomography	I-Algorithm
.	O
</s>
<s>
Born	O
's	O
rule	O
states	O
,	O
where	O
is	O
a	O
particular	O
measurement	O
outcome	O
projector	B-Algorithm
and	O
is	O
the	O
density	O
matrix	O
of	O
the	O
system	O
.	O
</s>
<s>
This	O
system	O
is	O
not	O
going	O
to	O
be	O
square	O
in	O
general	O
,	O
as	O
for	O
each	O
measurement	O
being	O
made	O
there	O
will	O
generally	O
be	O
multiple	O
measurement	O
outcome	O
projectors	B-Algorithm
.	O
</s>
<s>
For	O
example	O
,	O
in	O
a	O
2-D	O
Hilbert	O
space	O
with	O
3	O
measurements	O
,	O
each	O
measurement	O
has	O
2	O
outcomes	O
,	O
each	O
of	O
which	O
has	O
a	O
projector	B-Algorithm
Ei	O
,	O
for	O
6	O
projectors	B-Algorithm
,	O
whereas	O
the	O
real	O
dimension	O
of	O
the	O
space	O
of	O
density	O
matrices	O
is	O
(	O
2⋅22	O
)	O
/2	O
=	O
4	O
,	O
leaving	O
to	O
be	O
6	O
x	O
4	O
.	O
</s>
<s>
Now	O
solving	O
for	O
yields	O
the	O
pseudoinverse	B-Algorithm
:	O
</s>
<s>
One	O
notable	O
example	O
is	O
in	O
the	O
tomography	B-Algorithm
of	O
light	O
,	O
known	O
as	O
optical	O
homodyne	O
tomography	B-Algorithm
.	O
</s>
<s>
Using	O
an	O
inverse	O
Radon	B-Algorithm
transformation	I-Algorithm
(	O
the	O
filtered	O
back	O
projection	B-Algorithm
)	O
on	O
leads	O
to	O
the	O
Wigner	O
function	O
,	O
,	O
which	O
can	O
be	O
converted	O
by	O
an	O
inverse	O
Fourier	O
transform	O
into	O
the	O
density	O
matrix	O
for	O
the	O
state	O
in	O
any	O
basis	O
.	O
</s>
<s>
A	O
similar	O
technique	O
is	O
often	O
used	O
in	O
medical	B-Algorithm
tomography	I-Algorithm
.	O
</s>
<s>
The	O
single-qubit	O
state	O
tomography	B-Algorithm
can	O
be	O
performed	O
by	O
means	O
of	O
single-qubit	O
Pauli	O
measurements	O
:	O
</s>
<s>
This	O
algorithm	O
is	O
the	O
foundation	O
for	O
qubit	O
tomography	B-Algorithm
and	O
is	O
used	O
in	O
some	O
quantum	B-Device
programming	I-Device
routines	O
,	O
like	O
that	O
of	O
Qiskit	B-Device
.	O
</s>
<s>
Balanced	O
homodyne	O
tomography	B-Algorithm
is	O
a	O
reliable	O
technique	O
of	O
reconstructing	O
quantum	O
states	O
in	O
the	O
optical	O
domain	O
.	O
</s>
<s>
This	O
technique	O
combines	O
the	O
advantages	O
of	O
the	O
high	O
efficiencies	O
of	O
photodiodes	O
in	O
measuring	O
the	O
intensity	O
or	O
photon	O
number	O
of	O
light	O
,	O
together	O
with	O
measuring	O
the	O
quantum	O
features	O
of	O
light	O
by	O
a	O
clever	O
set-up	O
called	O
the	O
homodyne	O
tomography	B-Algorithm
detector	O
.	O
</s>
<s>
Quantum	O
homodyne	O
tomography	B-Algorithm
is	O
understood	O
by	O
the	O
following	O
example	O
.	O
</s>
<s>
One	O
of	O
the	O
photons	O
is	O
used	O
as	O
a	O
trigger	O
signal	O
used	O
to	O
trigger	O
(	O
start	O
)	O
the	O
readout	O
event	O
of	O
the	O
homodyne	O
tomography	B-Algorithm
detector	O
.	O
</s>
<s>
The	O
other	O
photon	O
is	O
directed	O
into	O
the	O
homodyne	O
tomography	B-Algorithm
detector	O
,	O
in	O
order	O
to	O
reconstruct	O
its	O
quantum	O
state	O
.	O
</s>
<s>
Now	O
consider	O
the	O
homodyne	O
tomography	B-Algorithm
detector	O
as	O
depicted	O
in	O
figure	O
4	O
(	O
figure	O
missing	O
)	O
.	O
</s>
<s>
The	O
homodyne	O
tomography	B-Algorithm
system	O
functions	O
as	O
an	O
amplifier	O
.	O
</s>
<s>
The	O
quantum	B-Architecture
computations	I-Architecture
for	O
retrieving	O
the	O
quadrature	O
component	O
from	O
the	O
current	O
difference	O
are	O
performed	O
as	O
follows	O
.	O
</s>
<s>
Bayesian	B-General_Concept
mean	O
estimation	O
(	O
BME	O
)	O
is	O
a	O
relatively	O
new	O
approach	O
which	O
addresses	O
the	O
problems	O
of	O
maximum	O
likelihood	O
estimation	O
.	O
</s>
<s>
realize	O
quantum	B-Algorithm
tomography	I-Algorithm
with	O
fewer	O
measurements	O
.	O
</s>
<s>
Also	O
known	O
as	O
"	O
quantum	O
estimation	O
"	O
,	O
tomography	B-Algorithm
techniques	O
are	O
increasingly	O
important	O
including	O
those	O
for	O
quantum	O
measurement	O
tomography	B-Algorithm
and	O
the	O
very	O
similar	O
quantum	B-Algorithm
state	I-Algorithm
tomography	I-Algorithm
.	O
</s>
<s>
Not	O
surprisingly	O
,	O
this	O
suffers	O
from	O
the	O
same	O
pitfalls	O
as	O
in	O
quantum	B-Algorithm
state	I-Algorithm
tomography	I-Algorithm
:	O
namely	O
,	O
non-physical	O
results	O
,	O
in	O
particular	O
negative	O
probabilities	O
.	O
</s>
<s>
Bayesian	B-General_Concept
methods	O
as	O
well	O
as	O
Maximum	O
likelihood	O
estimation	O
of	O
the	O
density	O
matrix	O
can	O
be	O
used	O
to	O
restrict	O
the	O
operators	O
to	O
valid	O
physical	O
results	O
.	O
</s>
<s>
Quantum	B-Algorithm
process	I-Algorithm
tomography	I-Algorithm
(	O
QPT	O
)	O
deals	O
with	O
identifying	O
an	O
unknown	O
quantum	O
dynamical	O
process	O
.	O
</s>
<s>
The	O
first	O
approach	O
,	O
introduced	O
in	O
1996	O
and	O
sometimes	O
known	O
as	O
standard	O
quantum	B-Algorithm
process	I-Algorithm
tomography	I-Algorithm
(	O
SQPT	O
)	O
involves	O
preparing	O
an	O
ensemble	O
of	O
quantum	O
states	O
and	O
sending	O
them	O
through	O
the	O
process	O
,	O
then	O
using	O
quantum	B-Algorithm
state	I-Algorithm
tomography	I-Algorithm
to	O
identify	O
the	O
resultant	O
states	O
.	O
</s>
<s>
Other	O
techniques	O
include	O
ancilla-assisted	O
process	O
tomography	B-Algorithm
(	O
AAPT	O
)	O
and	O
entanglement-assisted	O
process	O
tomography	B-Algorithm
(	O
EAPT	O
)	O
which	O
require	O
an	O
extra	O
copy	O
of	O
the	O
system	O
.	O
</s>
<s>
Each	O
of	O
the	O
techniques	O
listed	O
above	O
are	O
known	O
as	O
indirect	O
methods	O
for	O
characterization	O
of	O
quantum	O
dynamics	O
,	O
since	O
they	O
require	O
the	O
use	O
of	O
quantum	B-Algorithm
state	I-Algorithm
tomography	I-Algorithm
to	O
reconstruct	O
the	O
process	O
.	O
</s>
<s>
In	O
contrast	O
,	O
there	O
are	O
direct	O
methods	O
such	O
as	O
direct	O
characterization	O
of	O
quantum	O
dynamics	O
(	O
DCQD	O
)	O
which	O
provide	O
a	O
full	O
characterization	O
of	O
quantum	O
systems	O
without	O
any	O
state	O
tomography	B-Algorithm
.	O
</s>
<s>
The	O
number	O
of	O
experimental	O
configurations	O
(	O
state	O
preparations	O
and	O
measurements	O
)	O
required	O
for	O
full	O
quantum	B-Algorithm
process	I-Algorithm
tomography	I-Algorithm
grows	O
exponentially	O
with	O
the	O
number	O
of	O
constituent	O
particles	O
of	O
a	O
system	O
.	O
</s>
<s>
The	O
method	O
of	O
compressed	O
quantum	B-Algorithm
process	I-Algorithm
tomography	I-Algorithm
(	O
CQPT	O
)	O
uses	O
the	O
compressed	O
sensing	O
technique	O
and	O
applies	O
the	O
sparsity	O
assumption	O
to	O
reconstruct	O
a	O
quantum	O
dynamical	O
map	O
from	O
an	O
incomplete	O
set	O
of	O
measurements	O
or	O
test	O
state	O
preparations	O
.	O
</s>
<s>
By	O
sending	O
each	O
through	O
many	O
times	O
,	O
quantum	B-Algorithm
state	I-Algorithm
tomography	I-Algorithm
can	O
be	O
used	O
to	O
determine	O
the	O
coefficients	O
experimentally	O
.	O
</s>
