<s>
Quantum	B-Device
error	I-Device
correction	I-Device
(	O
QEC	O
)	O
is	O
used	O
in	O
quantum	B-Architecture
computing	I-Architecture
to	O
protect	O
quantum	O
information	O
from	O
errors	O
due	O
to	O
decoherence	O
and	O
other	O
quantum	O
noise	O
.	O
</s>
<s>
Quantum	B-Device
error	I-Device
correction	I-Device
is	O
theorised	O
as	O
essential	O
to	O
achieve	O
fault	B-General_Concept
tolerant	I-General_Concept
quantum	B-Architecture
computing	I-Architecture
that	O
can	O
reduce	O
the	O
effects	O
of	O
noise	O
on	O
stored	O
quantum	O
information	O
,	O
faulty	O
quantum	B-Application
gates	I-Application
,	O
faulty	O
quantum	O
preparation	O
,	O
and	O
faulty	O
measurements	O
.	O
</s>
<s>
This	O
would	O
allow	O
algorithms	O
of	O
greater	O
circuit	B-Application
depth	O
.	O
</s>
<s>
Classical	O
error	B-Error_Name
correction	I-Error_Name
employs	O
redundancy	B-General_Concept
.	O
</s>
<s>
The	O
simplest	O
albeit	O
inefficient	O
approach	O
is	O
the	O
repetition	B-Error_Name
code	I-Error_Name
.	O
</s>
<s>
Similar	O
to	O
classical	O
error	B-Error_Name
correction	I-Error_Name
,	O
QEC	O
codes	O
do	O
not	O
always	O
correctly	O
decode	O
logical	O
qubits	O
,	O
but	O
their	O
use	O
reduces	O
the	O
effect	O
of	O
noise	O
.	O
</s>
<s>
This	O
theorem	O
seems	O
to	O
present	O
an	O
obstacle	O
to	O
formulating	O
a	O
theory	O
of	O
quantum	B-Device
error	I-Device
correction	I-Device
.	O
</s>
<s>
Peter	O
Shor	O
first	O
discovered	O
this	O
method	O
of	O
formulating	O
a	O
quantum	B-Device
error	I-Device
correcting	I-Device
code	I-Device
by	O
storing	O
the	O
information	O
of	O
one	O
qubit	O
onto	O
a	O
highly	O
entangled	O
state	O
of	O
nine	O
qubits	O
.	O
</s>
<s>
Quantum	B-Device
error	I-Device
correction	I-Device
also	O
employs	O
syndrome	O
measurements	O
.	O
</s>
<s>
The	O
measurement	O
of	O
the	O
syndrome	O
has	O
the	O
projective	B-Algorithm
effect	O
of	O
a	O
quantum	O
measurement	O
,	O
so	O
even	O
if	O
the	O
error	O
due	O
to	O
the	O
noise	O
was	O
arbitrary	O
,	O
it	O
can	O
be	O
expressed	O
as	O
a	O
combination	O
of	O
basis	O
operations	O
called	O
the	O
error	O
basis	O
(	O
which	O
is	O
given	O
by	O
the	O
Pauli	O
matrices	O
and	O
the	O
identity	O
)	O
.	O
</s>
<s>
The	O
syndrome	O
measurement	O
provides	O
information	O
about	O
the	O
error	O
that	O
has	O
happened	O
,	O
but	O
not	O
about	O
the	O
information	O
that	O
is	O
stored	O
in	O
the	O
logical	O
qubit	O
—	O
as	O
otherwise	O
the	O
measurement	O
would	O
destroy	O
any	O
quantum	O
superposition	O
of	O
this	O
logical	O
qubit	O
with	O
other	O
qubits	O
in	O
the	O
quantum	B-Architecture
computer	I-Architecture
,	O
which	O
would	O
prevent	O
it	O
from	O
being	O
used	O
to	O
convey	O
quantum	O
information	O
.	O
</s>
<s>
The	O
repetition	B-Error_Name
code	I-Error_Name
works	O
in	O
a	O
classical	O
channel	O
,	O
because	O
classical	O
bits	O
are	O
easy	O
to	O
measure	O
and	O
to	O
repeat	O
.	O
</s>
<s>
This	O
technique	O
uses	O
entanglement	O
and	O
syndrome	O
measurements	O
and	O
is	O
comparable	O
in	O
performance	O
with	O
the	O
repetition	B-Error_Name
code	I-Error_Name
.	O
</s>
<s>
In	O
the	O
case	O
of	O
the	O
simple	O
three-qubit	O
repetition	B-Error_Name
code	I-Error_Name
,	O
the	O
encoding	O
consists	O
in	O
the	O
mappings	O
and	O
.	O
</s>
<s>
This	O
amounts	O
to	O
performing	O
a	O
measurement	O
with	O
four	O
different	O
outcomes	O
,	O
corresponding	O
to	O
the	O
following	O
four	O
projective	B-Algorithm
measurements:This	O
reveals	O
which	O
qubits	O
are	O
different	O
from	O
the	O
others	O
,	O
without	O
at	O
the	O
same	O
time	O
giving	O
information	O
about	O
the	O
state	O
of	O
the	O
qubits	O
themselves	O
.	O
</s>
<s>
To	O
assess	O
the	O
performance	O
of	O
this	O
error-correcting	B-Error_Name
scheme	O
for	O
a	O
general	O
input	O
we	O
can	O
study	O
the	O
fidelity	O
between	O
the	O
input	O
and	O
the	O
output	O
.	O
</s>
<s>
It	O
follows	O
that	O
This	O
fidelity	O
is	O
to	O
be	O
compared	O
with	O
the	O
corresponding	O
fidelity	O
obtained	O
when	O
no	O
error-correcting	B-Error_Name
protocol	O
is	O
used	O
,	O
which	O
was	O
shown	O
before	O
to	O
equal	O
.	O
</s>
<s>
A	O
little	O
algebra	O
then	O
shows	O
that	O
the	O
fidelity	O
after	O
error	B-Error_Name
correction	I-Error_Name
is	O
greater	O
than	O
the	O
one	O
without	O
for	O
.	O
</s>
<s>
Flipped	O
bits	O
are	O
the	O
only	O
kind	O
of	O
error	O
in	O
classical	O
computer	O
,	O
but	O
there	O
is	O
another	O
possibility	O
of	O
an	O
error	O
with	O
quantum	B-Architecture
computers	I-Architecture
,	O
the	O
sign	O
flip	O
.	O
</s>
<s>
If	O
the	O
three	O
bit	O
flip	O
group	O
(	O
1	O
,	O
2	O
,	O
3	O
)	O
,	O
(	O
4	O
,	O
5	O
,	O
6	O
)	O
,	O
and	O
(	O
7	O
,	O
8	O
,	O
9	O
)	O
are	O
considered	O
as	O
three	O
inputs	O
,	O
then	O
the	O
Shor	O
code	O
circuit	B-Application
can	O
be	O
reduced	O
as	O
a	O
sign	O
flip	O
code	O
.	O
</s>
<s>
One	O
insight	O
offered	O
by	O
these	O
codes	O
is	O
to	O
take	O
advantage	O
of	O
the	O
redundancy	B-General_Concept
within	O
a	O
single	O
system	O
,	O
rather	O
than	O
to	O
duplicate	O
many	O
two-level	O
qubits	O
.	O
</s>
<s>
Schrödinger	O
cat	O
states	O
,	O
superpositions	O
of	O
coherent	O
states	O
,	O
can	O
also	O
be	O
used	O
as	O
logical	O
states	O
for	O
error	B-Error_Name
correction	I-Error_Name
codes	O
.	O
</s>
<s>
where	O
is	O
the	O
orthogonal	O
projection	B-Algorithm
onto	O
.	O
</s>
<s>
Raymond	O
Laflamme	O
and	O
collaborators	O
found	O
a	O
class	O
of	O
5-qubit	O
codes	O
that	O
do	O
the	O
same	O
,	O
which	O
also	O
have	O
the	O
property	O
of	O
being	O
fault-tolerant	B-General_Concept
.	O
</s>
<s>
A	O
5-qubit	B-Device
code	I-Device
is	O
the	O
smallest	O
possible	O
code	O
that	O
protects	O
a	O
single	O
logical	O
qubit	O
against	O
single-qubit	O
errors	O
.	O
</s>
<s>
A	O
generalisation	O
of	O
the	O
technique	O
used	O
by	O
Steane	O
,	O
to	O
develop	O
the	O
7-qubit	O
code	O
from	O
the	O
classical	B-Error_Name
[	I-Error_Name
7	I-Error_Name
,	I-Error_Name
4 ]	I-Error_Name
Hamming	I-Error_Name
code	I-Error_Name
,	O
led	O
to	O
the	O
construction	O
of	O
an	O
important	O
class	O
of	O
codes	O
called	O
the	O
CSS	O
codes	O
,	O
named	O
for	O
their	O
inventors	O
:	O
Robert	O
Calderbank	O
,	O
Peter	O
Shor	O
and	O
Andrew	O
Steane	O
.	O
</s>
<s>
According	O
to	O
the	O
quantum	O
Hamming	B-Error_Name
bound	O
,	O
encoding	O
a	O
single	O
logical	O
qubit	O
and	O
providing	O
for	O
arbitrary	O
error	B-Error_Name
correction	I-Error_Name
in	O
a	O
single	O
qubit	O
requires	O
a	O
minimum	O
of	O
5	O
physical	O
qubits	O
.	O
</s>
<s>
A	O
more	O
general	O
class	O
of	O
codes	O
(	O
encompassing	O
the	O
former	O
)	O
are	O
the	O
stabilizer	B-Device
codes	I-Device
discovered	O
by	O
Daniel	O
Gottesman	O
,	O
and	O
by	O
Robert	O
Calderbank	O
,	O
Eric	O
Rains	O
,	O
Peter	O
Shor	O
,	O
and	O
N	O
.	O
J	O
.	O
</s>
<s>
Two	O
dimensional	O
Bacon	B-Device
–	I-Device
Shor	I-Device
codes	I-Device
are	O
a	O
family	O
of	O
codes	O
parameterized	O
by	O
integers	O
m	O
and	O
n	O
.	O
There	O
are	O
nm	O
qubits	O
arranged	O
in	O
a	O
square	O
lattice	O
.	O
</s>
<s>
A	O
newer	O
idea	O
is	O
Alexei	O
Kitaev	O
's	O
topological	O
quantum	B-Device
codes	I-Device
and	O
the	O
more	O
general	O
idea	O
of	O
a	O
topological	B-Device
quantum	I-Device
computer	I-Device
.	O
</s>
<s>
Todd	O
Brun	O
,	O
Igor	O
Devetak	O
,	O
and	O
Min-Hsiu	O
Hsieh	O
also	O
constructed	O
the	O
entanglement-assisted	O
stabilizer	B-Device
formalism	I-Device
as	O
an	O
extension	O
of	O
the	O
standard	O
stabilizer	B-Device
formalism	I-Device
that	O
incorporates	O
quantum	O
entanglement	O
shared	O
between	O
a	O
sender	O
and	O
a	O
receiver	O
.	O
</s>
<s>
That	O
these	O
codes	O
allow	O
indeed	O
for	O
quantum	B-Architecture
computations	I-Architecture
of	O
arbitrary	O
length	O
is	O
the	O
content	O
of	O
the	O
quantum	O
threshold	O
theorem	O
,	O
found	O
by	O
Michael	O
Ben-Or	O
and	O
Dorit	O
Aharonov	O
,	O
which	O
asserts	O
that	O
you	O
can	O
correct	O
for	O
all	O
errors	O
if	O
you	O
concatenate	O
quantum	B-Device
codes	I-Device
such	O
as	O
the	O
CSS	O
codes	O
—	O
i.e.	O
</s>
<s>
re-encode	O
each	O
logical	O
qubit	O
by	O
the	O
same	O
code	O
again	O
,	O
and	O
so	O
on	O
,	O
on	O
logarithmically	O
many	O
levels	O
—	O
provided	O
that	O
the	O
error	O
rate	O
of	O
individual	O
quantum	B-Application
gates	I-Application
is	O
below	O
a	O
certain	O
threshold	O
;	O
as	O
otherwise	O
,	O
the	O
attempts	O
to	O
measure	O
the	O
syndrome	O
and	O
correct	O
the	O
errors	O
would	O
introduce	O
more	O
new	O
errors	O
than	O
they	O
correct	O
for	O
.	O
</s>
<s>
The	O
error-correction	B-Error_Name
demonstration	O
was	O
performed	O
on	O
Schrodinger-cat	O
states	O
encoded	O
in	O
a	O
superconducting	O
resonator	O
,	O
and	O
employed	O
a	O
quantum	O
controller	O
capable	O
of	O
performing	O
real-time	O
feedback	O
operations	O
including	O
read-out	O
of	O
the	O
quantum	O
information	O
,	O
its	O
analysis	O
,	O
and	O
the	O
correction	O
of	O
its	O
detected	O
errors	O
.	O
</s>
<s>
In	O
2021	O
,	O
an	O
entangling	O
gate	O
between	O
two	O
logical	O
qubits	O
encoded	O
in	O
topological	O
quantum	O
error-correction	B-Error_Name
codes	O
has	O
first	O
been	O
realized	O
using	O
10	O
ions	O
in	O
a	O
trapped-ion	O
quantum	B-Architecture
computer	I-Architecture
.	O
</s>
<s>
2021	O
also	O
saw	O
the	O
first	O
experimental	O
demonstration	O
of	O
fault-tolerant	B-General_Concept
Bacon-Shor	B-Device
code	I-Device
in	O
a	O
single	O
logical	O
qubit	O
of	O
a	O
trapped-ion	O
system	O
,	O
i.e.	O
</s>
<s>
a	O
demonstration	O
for	O
which	O
the	O
addition	O
of	O
error	B-Error_Name
correction	I-Error_Name
is	O
able	O
to	O
suppress	O
more	O
errors	O
than	O
is	O
introduced	O
by	O
the	O
overhead	O
required	O
to	O
implement	O
the	O
error	B-Error_Name
correction	I-Error_Name
as	O
well	O
as	O
fault	B-General_Concept
tolerant	I-General_Concept
Steane	O
code	O
.	O
</s>
<s>
In	O
2022	O
,	O
researchers	O
at	O
the	O
University	O
of	O
Innsbruck	O
have	O
demonstrated	O
a	O
fault-tolerant	B-General_Concept
universal	O
set	O
of	O
gates	O
on	O
two	O
logical	O
qubits	O
in	O
a	O
trapped-ion	O
quantum	B-Architecture
computer	I-Architecture
.	O
</s>
<s>
They	O
have	O
performed	O
a	O
logical	O
two-qubit	O
controlled-NOT	O
gate	O
between	O
two	O
instances	O
of	O
the	O
seven-qubit	O
colour	O
code	O
,	O
and	O
fault-tolerantly	O
prepared	O
a	O
logical	O
magic	B-Device
state	I-Device
.	O
</s>
<s>
In	O
February	O
2023	O
researchers	O
at	O
Google	O
claimed	O
to	O
have	O
decreased	O
quantum	O
errors	O
by	O
increasing	O
the	O
qubit	O
number	O
in	O
experiments	O
,	O
they	O
used	O
a	O
fault	B-General_Concept
tolerant	I-General_Concept
surface	O
code	O
measuring	O
an	O
error	O
rate	O
of	O
3.028	O
%	O
and	O
2.914	O
%	O
for	O
a	O
distance-3	O
qubit	O
array	O
and	O
a	O
distance-5	O
qubit	O
array	O
respectively	O
.	O
</s>
<s>
Also	O
in	O
2022	O
,	O
a	O
research	O
at	O
University	O
of	O
demonstrated	O
error-cancellation	O
by	O
inserting	O
single-qubit	O
Z-axis	O
rotation	O
gates	O
into	O
strategically	O
chosen	O
locations	O
of	O
the	O
superconductor	O
quantum	B-Application
circuits	I-Application
.	O
</s>
<s>
This	O
is	O
a	O
circuit-level	O
calibration	O
scheme	O
that	O
traces	O
deviations	O
(	O
e.g.	O
</s>
