<s>
In	O
gate-based	B-Application
quantum	I-Application
computing	I-Application
,	O
various	O
sets	O
of	O
quantum	O
logic	O
gates	O
are	O
commonly	O
used	O
to	O
express	O
quantum	O
operations	O
.	O
</s>
<s>
Controlled	O
or	O
Hermitian	B-Algorithm
conjugate	I-Algorithm
versions	O
of	O
some	O
of	O
these	O
gates	O
may	O
not	O
be	O
listed	O
.	O
</s>
<s>
It	O
has	O
been	O
described	O
as	O
being	O
a	O
"	O
wait	O
cycle	O
"	O
,	O
and	O
a	O
NOP	B-Language
.	O
</s>
<s>
The	O
rotation	O
operator	O
gates	O
and	O
are	O
the	O
analog	O
rotation	B-Algorithm
matrices	I-Algorithm
in	O
three	O
Cartesian	O
axes	O
of	O
SO(3 )	O
,	O
the	O
axes	O
on	O
the	O
Bloch	O
sphere	O
projection	O
.	O
</s>
<s>
Any	O
unitary	B-Algorithm
matrix	I-Algorithm
in	O
SU(2 )	O
can	O
be	O
written	O
as	O
a	O
product	O
(	O
i.e.	O
</s>
<s>
The	O
rotation	B-Algorithm
matrices	I-Algorithm
are	O
related	O
to	O
the	O
Pauli	O
matrices	O
in	O
the	O
following	O
way	O
:	O
</s>
<s>
The	O
qubit-qubit	O
Ising	O
coupling	O
or	O
Heisenberg	O
interaction	O
gates	O
Rxx	O
,	O
Ryy	O
and	O
Rzz	O
are	O
2-qubit	O
gates	O
that	O
are	O
implemented	O
natively	O
in	O
some	O
trapped-ion	O
quantum	O
computers	O
,	O
using	O
for	O
example	O
the	O
Mølmer	B-Device
–	I-Device
Sørensen	I-Device
gate	I-Device
procedure	I-Device
.	O
</s>
<s>
The	O
gate	O
arises	O
naturally	O
in	O
systems	O
that	O
exploit	O
exchange	B-General_Concept
interaction	I-General_Concept
.	O
</s>
<s>
It	O
has	O
the	O
useful	O
property	O
that	O
the	O
numbers	O
of	O
0s	O
and	O
1s	O
are	O
conserved	O
throughout	O
,	O
which	O
in	O
the	O
billiard	B-Application
ball	I-Application
model	I-Application
means	O
the	O
same	O
number	O
of	O
balls	O
are	O
output	O
as	O
input	O
.	O
</s>
