在量子資訊科學 中,量子位元 (英語:quantum bit ),又稱Q位元 (qubit [ 1] )是量子信息的計量單位 。傳統電腦 使用的是0和1,量子電腦 雖然也是使用0跟1,但不同的是,量子電腦 的0與1可以同時計算。在古典系统中,一个位元在同一时间,只有0或1,只存在一種狀態,但量子位元可以同時是1和0,兩種狀態同時存在,這種效果叫量子疊加 。這是量子電腦計算目前獨有的特性。
4個量子位元的IBM實驗晶片,但最後並無實用價值。
若設定
|
0
⟩
{\displaystyle |0\rangle }
、
|
1
⟩
{\displaystyle |1\rangle }
順沿直角坐標系 的z方向,則有諸多表示法。可採上述向量 形式如狄拉克標記 的右括向量,亦可將之表為行矩陣;另外有密度矩陣 形式,可表為右括向量乘以左括向量,或表為方块矩阵 ,可見如下:
向量:
z
+
=
|
0
⟩
=
(
1
0
)
,
z
−
=
|
1
⟩
=
(
0
1
)
{\displaystyle z_{+}=|0\rangle ={\begin{pmatrix}1\\0\end{pmatrix}},\quad z_{-}=|1\rangle ={\begin{pmatrix}0\\1\end{pmatrix}}}
密度矩陣:
z
+
=
|
0
⟩
⟨
0
|
=
(
1
0
)
∗
(
1
0
)
=
(
1
0
0
0
)
,
{\displaystyle z_{+}=|0\rangle \langle 0|={\begin{pmatrix}1\\0\end{pmatrix}}*{\begin{pmatrix}1&0\end{pmatrix}}={\begin{pmatrix}1&0\\0&0\end{pmatrix}},}
z
−
=
|
1
⟩
⟨
1
|
=
(
0
1
)
∗
(
0
1
)
=
(
0
0
0
1
)
{\displaystyle z_{-}=|1\rangle \langle 1|={\begin{pmatrix}0\\1\end{pmatrix}}*{\begin{pmatrix}0&1\end{pmatrix}}={\begin{pmatrix}0&0\\0&1\end{pmatrix}}}
向量:
x
+
=
|
x
+
⟩
=
(
1
2
1
2
)
,
x
−
=
|
x
−
⟩
=
(
1
2
−
1
2
)
{\displaystyle x_{+}=|x_{+}\rangle ={\begin{pmatrix}{\frac {1}{\sqrt {2}}}\\{\frac {1}{\sqrt {2}}}\end{pmatrix}},\quad x_{-}=|x_{-}\rangle ={\begin{pmatrix}{\frac {1}{\sqrt {2}}}\\-{\frac {1}{\sqrt {2}}}\end{pmatrix}}}
密度矩陣:
x
+
=
|
x
+
⟩
⟨
x
+
|
=
(
1
2
1
2
)
∗
(
1
2
1
2
)
=
(
1
2
1
2
1
2
1
2
)
,
{\displaystyle x_{+}=|x_{+}\rangle \langle x_{+}|={\begin{pmatrix}{\frac {1}{\sqrt {2}}}\\{\frac {1}{\sqrt {2}}}\end{pmatrix}}*{\begin{pmatrix}{\frac {1}{\sqrt {2}}}&{\frac {1}{\sqrt {2}}}\end{pmatrix}}={\begin{pmatrix}{\frac {1}{2}}&{\frac {1}{2}}\\{\frac {1}{2}}&{\frac {1}{2}}\end{pmatrix}},}
x
−
=
|
x
−
⟩
⟨
x
−
|
=
(
1
2
−
1
2
)
∗
(
1
2
−
1
2
)
=
(
1
2
−
1
2
−
1
2
1
2
)
{\displaystyle x_{-}=|x_{-}\rangle \langle x_{-}|={\begin{pmatrix}{\frac {1}{\sqrt {2}}}\\-{\frac {1}{\sqrt {2}}}\end{pmatrix}}*{\begin{pmatrix}{\frac {1}{\sqrt {2}}}&-{\frac {1}{\sqrt {2}}}\end{pmatrix}}={\begin{pmatrix}{\frac {1}{2}}&-{\frac {1}{2}}\\-{\frac {1}{2}}&{\frac {1}{2}}\end{pmatrix}}}
向量:
y
+
=
|
y
+
⟩
=
(
1
2
i
2
)
,
y
−
=
|
y
−
⟩
=
(
1
2
−
i
2
)
{\displaystyle y_{+}=|y_{+}\rangle ={\begin{pmatrix}{\frac {1}{\sqrt {2}}}\\{\frac {i}{\sqrt {2}}}\end{pmatrix}},\quad y_{-}=|y_{-}\rangle ={\begin{pmatrix}{\frac {1}{\sqrt {2}}}\\-{\frac {i}{\sqrt {2}}}\end{pmatrix}}}
密度矩陣:
y
+
=
|
y
+
⟩
⟨
y
+
|
=
(
1
2
i
2
)
∗
(
1
2
−
i
2
)
=
(
1
2
−
i
2
i
2
1
2
)
,
{\displaystyle y_{+}=|y_{+}\rangle \langle y_{+}|={\begin{pmatrix}{\frac {1}{\sqrt {2}}}\\{\frac {i}{\sqrt {2}}}\end{pmatrix}}*{\begin{pmatrix}{\frac {1}{\sqrt {2}}}&-{\frac {i}{\sqrt {2}}}\end{pmatrix}}={\begin{pmatrix}{\frac {1}{2}}&-{\frac {i}{2}}\\{\frac {i}{2}}&{\frac {1}{2}}\end{pmatrix}},}
y
−
=
|
y
−
⟩
⟨
y
−
|
=
(
1
2
−
i
2
)
∗
(
1
2
i
2
)
=
(
1
2
i
2
−
i
2
1
2
)
{\displaystyle y_{-}=|y_{-}\rangle \langle y_{-}|={\begin{pmatrix}{\frac {1}{\sqrt {2}}}\\-{\frac {i}{\sqrt {2}}}\end{pmatrix}}*{\begin{pmatrix}{\frac {1}{\sqrt {2}}}&{\frac {i}{\sqrt {2}}}\end{pmatrix}}={\begin{pmatrix}{\frac {1}{2}}&{\frac {i}{2}}\\-{\frac {i}{2}}&{\frac {1}{2}}\end{pmatrix}}}
MA Nielsen, IL Chuang. Quantum Computation and Quantum Information , Cambridge University Press, Cambridge (2000).
Michael A. Nielsen, Isaac L. Chuang: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge 2000, ISBN 0-521-63503-9 .
Oliver Morsch: Quantum bits and quantum secrets - how quantum physics is revolutionizing codes and computers. Wiley-VCH, Weinheim 2008, ISBN 978-3-527-40710-1 .
Anthony J. Leggett: Quantum computing and quantum bits in mesoscopic systems. Kluwer Academic, New York 2004, ISBN 0-306-47904-4 .