![\"\"](\"https:\/\/learning.oreilly.com\/api\/v2\/epubs\/urn:orm:book:9781482238129\/files\/bga2.png\")
<\/p>\n\n\n\nWe now wish to push the circuit analogy further and explore the possibility<\/p>\n\n\n\n
of universal quantum gates. Let\u2019s start with single-qubit gates. We\u2019ve seen<\/p>\n\n\n\n
that these are 2 \u00d72 unitary matrices, which take a point on the Bloch sphere<\/p>\n\n\n\n
to another. It is easy to see that there are in\ufb01nitely many possible 1-qubit<\/p>\n\n\n\n
gates. These however cannot form a universal set since controlled operations<\/p>\n\n\n\n
cannot be implemented by taking direct products of 1-qubit gates. How do we<\/p>\n\n\n\n
implement controlled gates in general?<\/p>\n\n\n\n
7.4.1 Controlled-U gate<\/p>\n\n\n\n
Working toward a general construction for a controlled U gate for arbitrary<\/p>\n\n\n\n
U makes use of the following representation for U :<\/p>\n\n\n\n
Theorem 7.1. Any unitary 2 \u00d7 2 matrix can be decomposed as<\/p>\n\n\n\n
U = e<\/p>\n\n\n\n
i\u03b8<\/p>\n\n\n\n
A \u03c3<\/p>\n\n\n\n
x<\/p>\n\n\n\n
B \u03c3<\/p>\n\n\n\n
x<\/p>\n\n\n\n
C, s.t A B C = , (7.26)<\/p>\n\n\n\n
where A, B, and C are also unitary.<\/p>\n\n\n\n
Proof. The proof hinges on the fact that any unitary matrix implements a<\/p>\n\n\n\n
rotation on the Bloch sphere, up to an over-all phase factor e<\/p>\n\n\n\n
i\u03b8<\/p>\n\n\n\n
. Suppose V<\/p>\n\n\n\n
is some unitary matrix. The matrix V \u03c3<\/p>\n\n\n\n
x<\/p>\n\n\n\n
V<\/p>\n\n\n\n
\u2020<\/p>\n\n\n\n
is also unitary, so that it can be<\/p>\n\n\n\n
represented (see Equation 7.11) as<\/p>\n\n\n\n
V \u03c3<\/p>\n\n\n\n
x<\/p>\n\n\n\n
V<\/p>\n\n\n\n
\u2020<\/p>\n\n\n\n
= a<\/p>\n\n\n\n
0<\/p>\n\n\n\n
+ ~a \u00b7 ~\u03c3, a<\/p>\n\n\n\n
2<\/p>\n\n\n\n
0<\/p>\n\n\n\n
+ ~a \u00b7~a = 1.<\/p>\n\n\n\n
But V \u03c3<\/p>\n\n\n\n
x<\/p>\n\n\n\n
V<\/p>\n\n\n\n
\u2020<\/p>\n\n\n\n
is a similarity transformation of \u03c3<\/p>\n\n\n\n
x<\/p>\n\n\n\n
. So it must preserve its trace,<\/p>\n\n\n\n
which is zero. Therefore a<\/p>\n\n\n\n
0<\/p>\n\n\n\n
= 0 and<\/p>\n\n\n\n
V \u03c3<\/p>\n\n\n\n
x<\/p>\n\n\n\n
V<\/p>\n\n\n\n
\u2020<\/p>\n\n\n\n
= \u02c6a \u00b7 ~\u03c3 for a real unit \u02c6a.<\/p>\n\n\n\n
Note that \u03c3<\/p>\n\n\n\n
x<\/p>\n\n\n\n
= \u02c6x \u00b7 ~\u03c3. Then V \u03c3<\/p>\n\n\n\n
x<\/p>\n\n\n\n
V<\/p>\n\n\n\n
\u2020<\/p>\n\n\n\n
must be rotating \u02c6x to a new direction \u02c6a.<\/p>\n\n\n\n
Similarly, another unitary W will achieve<\/p>\n\n\n\n
W \u03c3<\/p>\n\n\n\n
x<\/p>\n\n\n\n
W<\/p>\n\n\n\n
\u2020<\/p>\n\n\n\n
=<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
b \u00b7 ~\u03c3 for a real unit<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
b.<\/p>\n\n\n\n
Thus we have<\/p>\n\n\n\n
V \u03c3<\/p>\n\n\n\n
x<\/p>\n\n\n\n
V<\/p>\n\n\n\n
\u2020<\/p>\n\n\n\n
W \u03c3<\/p>\n\n\n\n
x<\/p>\n\n\n\n
W<\/p>\n\n\n\n
\u2020<\/p>\n\n\n\n
= (\u02c6a \u00b7 ~\u03c3) (<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
b \u00b7 ~\u03c3)<\/p>\n\n\n\n
= \u02c6a \u00b7<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
b + i \u02c6a \u00d7<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
b \u00b7 ~\u03c3.<\/p>\n\n\n\n
(Refer to Equation 3.34 you proved in one of the problems of Chapter 3.) We<\/p>\n\n\n\n
can now think of \u02c6a and<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
b as directions with an angle \u03b3 between them so that<\/p>\n\n\n\n
\u02c6a \u00b7<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
b = cos \u03b3, \u02c6a \u00d7<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
b = sin \u03b3 \u02c6n, which is perpendicular to \u02c6a and<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
b<\/p>\n\n\n\n
<\/p>\n\n\n\n
Processing<\/p>\n\n\n\n
Then we can construct<\/p>\n\n\n\n
U = e<\/p>\n\n\n\n
i\u03b8<\/p>\n\n\n\n
V \u03c3<\/p>\n\n\n\n
x<\/p>\n\n\n\n
V<\/p>\n\n\n\n
\u2020<\/p>\n\n\n\n
W \u03c3<\/p>\n\n\n\n
x<\/p>\n\n\n\n
W<\/p>\n\n\n\n
\u2020<\/p>\n\n\n\n
= e<\/p>\n\n\n\n
i\u03b8<\/p>\n\n\n\n
(cos \u03b3 + i sin \u03b3 \u02c6n \u00b7 ~\u03c3)<\/p>\n\n\n\n
= e<\/p>\n\n\n\n
i\u03b8<\/p>\n\n\n\n
e<\/p>\n\n\n\n
i\u03b3 \u02c6n\u00b7~\u03c3<\/p>\n\n\n\n
,<\/p>\n\n\n\n
which is a valid representation for a unitary operator! If we identify<\/p>\n\n\n\n
V = A, V<\/p>\n\n\n\n
\u2020<\/p>\n\n\n\n
W = B and W<\/p>\n\n\n\n
\u2020<\/p>\n\n\n\n
= C,<\/p>\n\n\n\n
then we have the requisite representation for U.<\/p>\n\n\n\n
We can implement C-V \u03c3<\/p>\n\n\n\n
x<\/p>\n\n\n\n
V<\/p>\n\n\n\n
\u2020<\/p>\n\n\n\n
W \u03c3<\/p>\n\n\n\n
x<\/p>\n\n\n\n
W<\/p>\n\n\n\n
\u2020<\/p>\n\n\n\n
by the circuit of Figure (7.16).<\/p>\n\n\n\n
|xi<\/p>\n\n\n\n
\u2022 \u2022<\/p>\n\n\n\n
|yi<\/p>\n\n\n\n
W<\/p>\n\n\n\n
\u2020<\/p>\n\n\n\n
W<\/p>\n\n\n\n
V<\/p>\n\n\n\n
\u2020<\/p>\n\n\n\n
V<\/p>\n\n\n\n
FIGURE 7.16: Circuit to evaluate C-U up to the phase factor<\/p>\n\n\n\n
It is straightforward to see that when x = 0, the output is V V<\/p>\n\n\n\n
\u2020<\/p>\n\n\n\n
W W<\/p>\n\n\n\n
\u2020<\/p>\n\n\n\n
y = y,<\/p>\n\n\n\n
and when x = 1, the output is V \u03c3<\/p>\n\n\n\n
x<\/p>\n\n\n\n
V<\/p>\n\n\n\n
\u2020<\/p>\n\n\n\n
W \u03c3<\/p>\n\n\n\n
x<\/p>\n\n\n\n
W<\/p>\n\n\n\n
\u2020<\/p>\n\n\n\n
y = Uy up to the phase. So<\/p>\n\n\n\n
this gives C-U up to the phase factor. We need to additionally implement the<\/p>\n\n\n\n
controlled phase C-\u0398 where<\/p>\n\n\n\n
\u0398 =<\/p>\n\n\n\n
“<\/p>\n\n\n\n
e<\/p>\n\n\n\n
i\u03b8<\/p>\n\n\n\n
0<\/p>\n\n\n\n
0 e<\/p>\n\n\n\n
i\u03b8<\/p>\n\n\n\n
.<\/p>\n\n\n\n
#<\/p>\n\n\n\n
Now check that<\/p>\n\n\n\n
C\u2212\u0398 =<\/p>\n\n\n\n
“<\/p>\n\n\n\n
0<\/p>\n\n\n\n
0 \u0398<\/p>\n\n\n\n
#<\/p>\n\n\n\n
=<\/p>\n\n\n\n
“<\/p>\n\n\n\n
1 0<\/p>\n\n\n\n
0 e<\/p>\n\n\n\n
i\u03b8<\/p>\n\n\n\n
#<\/p>\n\n\n\n
\u2297 .<\/p>\n\n\n\n
We then have the implementation of the full C-U illustrated in Fig-<\/p>\n\n\n\n
ure (7.17), that uses only CNOT gates and single-qubit gates.<\/p>\n\n\n\n
|xi<\/p>\n\n\n\n
\u2022 \u2022<\/p>\n\n\n\n
“<\/p>\n\n\n\n
1 0<\/p>\n\n\n\n
0 e<\/p>\n\n\n\n
i\u03b8<\/p>\n\n\n\n
#<\/p>\n\n\n\n
|yi<\/p>\n\n\n\n
W<\/p>\n\n\n\n
V W<\/p>\n\n\n\n
\u2020<\/p>\n\n\n\n
W<\/p>\n\n\n\n
FIGURE 7.17: Implementation of controlled U gate.<\/p>\n\n\n\n
Here, V and W are arbitrary unitaries. We are now half-way through in our<\/p>\n\n\n\n
quest for universal quantum gates, of which one set is given in the following<\/p>\n\n\n\n
theorem:<\/p>\n\n\n\n
<\/p>\n\n\n\n
Quantum Gates and Circuits 137<\/p>\n\n\n\n
Theorem 7.2. Universal Quantum Gates: the CNOT gate along with<\/p>\n\n\n\n
single-qubit gates is universal.<\/p>\n\n\n\n
How do we prove this? Now classically, the To\ufb00oli gate, which is a C-<\/p>\n\n\n\n
C-NOT gate, is universal. We\u2019ll now show that given our construction for<\/p>\n\n\n\n
C-U gates, we can build doubly controlled C-C-U gates as follows. Consider<\/p>\n\n\n\n
a unitary Q such that Q<\/p>\n\n\n\n
2<\/p>\n\n\n\n
= U . Then we can build a C-C-U by the circuit<\/p>\n\n\n\n
in Figure 7.18. Let\u2019s work through this circuit algebraically to show that it<\/p>\n\n\n\n
|xi<\/p>\n\n\n\n
\u2022 \u2022 \u2022<\/p>\n\n\n\n
|yi<\/p>\n\n\n\n
\u2022 \u2022<\/p>\n\n\n\n
|zi<\/p>\n\n\n\n
Q<\/p>\n\n\n\n
Q<\/p>\n\n\n\n
\u2020<\/p>\n\n\n\n
Q<\/p>\n\n\n\n
FIGURE 7.18: Implementation of C-C-U gate.<\/p>\n\n\n\n
works as expected:<\/p>\n\n\n\n
|xi|yi|zi \u2192 |xi|yi Q<\/p>\n\n\n\n
x<\/p>\n\n\n\n
|zi<\/p>\n\n\n\n
\u2192 |xi|x \u2295 yi Q<\/p>\n\n\n\n
x<\/p>\n\n\n\n
|zi<\/p>\n\n\n\n
\u2192 |xi|x \u2295 yi (Q<\/p>\n\n\n\n
\u2020<\/p>\n\n\n\n
)<\/p>\n\n\n\n
x\u2295y<\/p>\n\n\n\n
Q<\/p>\n\n\n\n
x<\/p>\n\n\n\n
|zi<\/p>\n\n\n\n
\u2192 |xi|yi (Q<\/p>\n\n\n\n
\u22121<\/p>\n\n\n\n
)<\/p>\n\n\n\n
x\u2295y<\/p>\n\n\n\n
Q<\/p>\n\n\n\n
x<\/p>\n\n\n\n
|zi<\/p>\n\n\n\n
\u2192 |xi|yi Q<\/p>\n\n\n\n
y<\/p>\n\n\n\n
Q<\/p>\n\n\n\n
\u2212x\u2295y<\/p>\n\n\n\n
Q<\/p>\n\n\n\n
x<\/p>\n\n\n\n
|zi<\/p>\n\n\n\n
The power of Q that acts on |zi in the end is<\/p>\n\n\n\n
y \u2212 (x \u2295 y) + x = y \u2212(x + y \u2212 2xy) + x = 2xy.<\/p>\n\n\n\n
So the e\ufb00ect of this circuit is<\/p>\n\n\n\n
|zi \u2192 Q<\/p>\n\n\n\n
2xy<\/p>\n\n\n\n
|zi = U<\/p>\n\n\n\n
xy<\/p>\n\n\n\n
|zi,<\/p>\n\n\n\n
which is exactly what we want. We can for instance construct a quantum<\/p>\n\n\n\n
To\ufb00oli gate by using Q<\/p>\n\n\n\n
2<\/p>\n\n\n\n
= X. One such \u201csquare root of NOT\u201d gate is<\/p>\n\n\n\n
\u221a<\/p>\n\n\n\n
X =<\/p>\n\n\n\n
1<\/p>\n\n\n\n
2<\/p>\n\n\n\n
“<\/p>\n\n\n\n
1 + i 1 \u2212 i<\/p>\n\n\n\n
1 \u2212 i 1 + i<\/p>\n\n\n\n
#<\/p>\n\n\n\n
. (7.27)<\/p>\n\n\n\n
Example 7.4.1. A useful question to ask in designing circuits is how to min-<\/p>\n\n\n\n
imize the number of basic gates required for a given implementation. In our<\/p>\n\n\n\n
construction for the C-C-U gate given above, we require 2 CNOTs plus 2<\/p>\n\n\n\n
CNOTs for each C-Q gate, that is a total of 8 CNOTs. Can we be more<\/p>\n\n\n\n
frugal? Here is an example from Mermin [48] of a construction for a To\ufb00oli<\/p>\n\n\n\n
![\"\"](\"https:\/\/learning.oreilly.com\/api\/v2\/epubs\/urn:orm:book:9781482238129\/files\/bga3.png\")
<\/p>\n\n\n\n
138 Introduction to Quantum Physics and Information Processing<\/p>\n\n\n\n
gate using only 4 CNOT gates. Consider two unitaries A and B such that<\/p>\n\n\n\n
A<\/p>\n\n\n\n
2<\/p>\n\n\n\n
= = B<\/p>\n\n\n\n
2<\/p>\n\n\n\n
. This means that<\/p>\n\n\n\n
A = V<\/p>\n\n\n\n
\u2020<\/p>\n\n\n\n
XV, B = W<\/p>\n\n\n\n
\u2020<\/p>\n\n\n\n
XW.<\/p>\n\n\n\n
Thus each C-A and C-B gate requires only one CNOT gate and two single-<\/p>\n\n\n\n
qubit gates.<\/p>\n\n\n\n
You should be able to work out that the circuit of Figure 7.19 implements<\/p>\n\n\n\n
a doubly controlled (BA)<\/p>\n\n\n\n
2<\/p>\n\n\n\n
gate, up to a phase \u03b1.<\/p>\n\n\n\n
\u2022 \u2022<\/p>\n\n\n\n
“<\/p>\n\n\n\n
1 0<\/p>\n\n\n\n
0 e<\/p>\n\n\n\n
i\u03b1<\/p>\n\n\n\n
#<\/p>\n\n\n\n
\u2022 \u2022<\/p>\n\n\n\n
B A B A<\/p>\n\n\n\n
FIGURE 7.19: E\ufb03cient implementation of a To\ufb00oli gate.<\/p>\n\n\n\n
Now<\/p>\n\n\n\n
AB = V<\/p>\n\n\n\n
\u2020<\/p>\n\n\n\n
XV W<\/p>\n\n\n\n
\u2020<\/p>\n\n\n\n
XW = (\u02c6a \u00b7 ~\u03c3)(<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
b \u00b7 ~\u03c3)<\/p>\n\n\n\n
= \u02c6a \u00b7<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
b + i(\u02c6a \u00d7<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
b) \u00b7 ~\u03c3.<\/p>\n\n\n\n
If we choose the angle between \u02c6a and<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
b to be \u03c0\/4, and also let \u02c6a \u00d7<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
b point<\/p>\n\n\n\n
along \u02c6x, then we have<\/p>\n\n\n\n
AB = cos<\/p>\n\n\n\n
\u03c0<\/p>\n\n\n\n
4<\/p>\n\n\n\n
+ i sin<\/p>\n\n\n\n
\u03c0<\/p>\n\n\n\n
4<\/p>\n\n\n\n
\u03c3<\/p>\n\n\n\n
x<\/p>\n\n\n\n
=<\/p>\n\n\n\n
1<\/p>\n\n\n\n
\u221a<\/p>\n\n\n\n
2<\/p>\n\n\n\n
( + iX)<\/p>\n\n\n\n
(AB)<\/p>\n\n\n\n
2<\/p>\n\n\n\n
=<\/p>\n\n\n\n
1<\/p>\n\n\n\n
2<\/p>\n\n\n\n
( + 2iX + (\u2212X)<\/p>\n\n\n\n
2<\/p>\n\n\n\n
) = iX.<\/p>\n\n\n\n
Thus we can regard AB as the square-root of X up to a phase of i. This phase<\/p>\n\n\n\n
can be cancelled if we choose \u03b1 = \u2212\u03c0\/2 and this circuit implements a To\ufb00oli<\/p>\n\n\n\n
gate with just 4 CNOTs and single-qubit gates.<\/p>\n\n\n\n
One can construct multiply controlled U gates, a C<\/p>\n\n\n\n
n<\/p>\n\n\n\n
-U gate, by a cascad-<\/p>\n\n\n\n
ing circuit using n control bits, To\ufb00oli gates and n \u2212 1 auxiliary bits, as in<\/p>\n\n\n\n
Figure 7.20.<\/p>\n\n\n\n
Verify that this works! The use of the To\ufb00oli gates performs an \u201cAND\u201d of<\/p>\n\n\n\n
all the control bits, which \ufb01nally controls the U gate. Also note that all the<\/p>\n\n\n\n
auxiliaries can be returned to their original state of |0i by adding the reverse<\/p>\n\n\n\n
of each of the actions after obtaining C<\/p>\n\n\n\n
n<\/p>\n\n\n\n
-U.<\/p>\n\n\n\n
![\"\"](\"https:\/\/learning.oreilly.com\/api\/v2\/epubs\/urn:orm:book:9781482238129\/files\/bga4.png\")
<\/p>\n\n\n\n
Quantum Gates and Circuits 139<\/p>\n\n\n\n
x<\/p>\n\n\n\n
n\u22121<\/p>\n\n\n\n
\u2022<\/p>\n\n\n\n
x<\/p>\n\n\n\n
n\u22122<\/p>\n\n\n\n
\u2022<\/p>\n\n\n\n
x<\/p>\n\n\n\n
n\u22123<\/p>\n\n\n\n
\u2022<\/p>\n\n\n\n
n controls<\/p>\n\n\n\n
.<\/p>\n\n\n\n
.<\/p>\n\n\n\n
.<\/p>\n\n\n\n
.<\/p>\n\n\n\n
.<\/p>\n\n\n\n
.<\/p>\n\n\n\n
.<\/p>\n\n\n\n
.<\/p>\n\n\n\n
.<\/p>\n\n\n\n
.<\/p>\n\n\n\n
.<\/p>\n\n\n\n
.<\/p>\n\n\n\n
.<\/p>\n\n\n\n
.<\/p>\n\n\n\n
.<\/p>\n\n\n\n
x<\/p>\n\n\n\n
0<\/p>\n\n\n\n
\u2022<\/p>\n\n\n\n
\uf8f1<\/p>\n\n\n\n
\uf8f4<\/p>\n\n\n\n
\uf8f4<\/p>\n\n\n\n
\uf8f4<\/p>\n\n\n\n
\uf8f4<\/p>\n\n\n\n
\uf8f4<\/p>\n\n\n\n
\uf8f4<\/p>\n\n\n\n
\uf8f4<\/p>\n\n\n\n
\uf8f4<\/p>\n\n\n\n
\uf8f4<\/p>\n\n\n\n
\uf8f4<\/p>\n\n\n\n
\uf8f2<\/p>\n\n\n\n
\uf8f4<\/p>\n\n\n\n
\uf8f4<\/p>\n\n\n\n
\uf8f4<\/p>\n\n\n\n
\uf8f4<\/p>\n\n\n\n
\uf8f4<\/p>\n\n\n\n
\uf8f4<\/p>\n\n\n\n
\uf8f4<\/p>\n\n\n\n
\uf8f4<\/p>\n\n\n\n
\uf8f4<\/p>\n\n\n\n
\uf8f4<\/p>\n\n\n\n
\uf8f3<\/p>\n\n\n\n
0<\/p>\n\n\n\n
\u2022<\/p>\n\n\n\n
0<\/p>\n\n\n\n
n \u2212 1<\/p>\n\n\n\n
.<\/p>\n\n\n\n
.<\/p>\n\n\n\n
.<\/p>\n\n\n\n
.<\/p>\n\n\n\n
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.<\/p>\n\n\n\n
.<\/p>\n\n\n\n
auxiliaries<\/p>\n\n\n\n
0<\/p>\n\n\n\n
\u2022<\/p>\n\n\n\n
\uf8f1<\/p>\n\n\n\n
\uf8f4<\/p>\n\n\n\n
\uf8f4<\/p>\n\n\n\n
\uf8f4<\/p>\n\n\n\n
\uf8f4<\/p>\n\n\n\n
\uf8f4<\/p>\n\n\n\n
\uf8f4<\/p>\n\n\n\n
\uf8f4<\/p>\n\n\n\n
\uf8f4<\/p>\n\n\n\n
\uf8f4<\/p>\n\n\n\n
\uf8f4<\/p>\n\n\n\n
\uf8f2<\/p>\n\n\n\n
\uf8f4<\/p>\n\n\n\n
\uf8f4<\/p>\n\n\n\n
\uf8f4<\/p>\n\n\n\n
\uf8f4<\/p>\n\n\n\n
\uf8f4<\/p>\n\n\n\n
\uf8f4<\/p>\n\n\n\n
\uf8f4<\/p>\n\n\n\n
\uf8f4<\/p>\n\n\n\n
\uf8f4<\/p>\n\n\n\n
\uf8f4<\/p>\n\n\n\n
\uf8f3<\/p>\n\n\n\n
0<\/p>\n\n\n\n
\u2022<\/p>\n\n\n\n
y<\/p>\n\n\n\n
U<\/p>\n\n\n\n
U<\/p>\n\n\n\n
x<\/p>\n\n\n\n
0<\/p>\n\n\n\n
x<\/p>\n\n\n\n
1<\/p>\n\n\n\n
…x<\/p>\n\n\n\n
n\u22121<\/p>\n\n\n\n
y<\/p>\n\n\n\n
FIGURE 7.20: Implementation of C<\/p>\n\n\n\n
n<\/p>\n\n\n\n
-U gate<\/p>\n","protected":false},"excerpt":{"rendered":"
We now wish to push the circuit analogy further and explore the possibility of universal quantum gates. Let\u2019s start with single-qubit gates. We\u2019ve seen that these are 2 \u00d72 unitary matrices, which take a point on the Bloch sphere to another. It is easy to see that there are in\ufb01nitely many possible 1-qubit gates. These […]<\/p>\n","protected":false},"author":1,"featured_media":4041,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[496],"tags":[],"class_list":["post-4074","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-quantum-gates-and-circuits"],"jetpack_featured_media_url":"https:\/\/workhouse.sweetdishy.com\/wp-content\/uploads\/2024\/09\/quantum-2.png","_links":{"self":[{"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/4074","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/comments?post=4074"}],"version-history":[{"count":1,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/4074\/revisions"}],"predecessor-version":[{"id":4075,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/4074\/revisions\/4075"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/media\/4041"}],"wp:attachment":[{"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/media?parent=4074"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/categories?post=4074"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/tags?post=4074"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}