Qubits: The Basic Unit of Information in Quantum Computers Qubits: The Basic Unit of Information in Quantum Computers Definition C \mathbb{C} C vector space C 2 \mathbb{C}^{2} C 2 [ 1 0 ] \begin{bmatrix} 1 \\ 0 \end{bmatrix} [ 1 0 ] [ 0 1 ] \begin{bmatrix} 0 \\ 1 \end{bmatrix} [ 0 1 ] Dirac notation as follows.
∣ 0 ⟩ = [ 1 0 ] ∣ 1 ⟩ = [ 0 1 ]
\ket{0} = \begin{bmatrix} 1 \\ 0 \end{bmatrix}\qquad \ket{1} = \begin{bmatrix} 0 \\ 1 \end{bmatrix}
∣ 0 ⟩ = [ 1 0 ] ∣ 1 ⟩ = [ 0 1 ]
The element of the set { ∣ 0 ⟩ , ∣ 1 ⟩ } \left\{ \ket{0}, \ket{1} \right\} { ∣ 0 ⟩ , ∣ 1 ⟩ } qubit .
C 2 \mathbb{C}^{2} C 2 n n n tensor product ( C 2 ) ⊗ n = C 2 ⊗ ⋯ ⊗ C 2 ⏞ n \left( \mathbb{C}^{2} \right)^{\otimes n} = \overbrace{\mathbb{C}^{2} \otimes \cdots \otimes \mathbb{C}^{2}}^{n} ( C 2 ) ⊗ n = C 2 ⊗ ⋯ ⊗ C 2 n
{ ∣ 0 ⟩ ⊗ ⋯ ⊗ ∣ 0 ⟩ , … , ∣ 1 ⟩ ⊗ ⋯ ⊗ ∣ 1 ⟩ }
\left\{ \ket{0} \otimes \cdots \otimes \ket{0}, \dots, \ket{1} \otimes \cdots \otimes \ket{1} \right\} { ∣ 0 ⟩ ⊗ ⋯ ⊗ ∣ 0 ⟩ , … , ∣ 1 ⟩ ⊗ ⋯ ⊗ ∣ 1 ⟩ }
is called n n n
Description A qubit is short for qu antum bit . While a bit is the minimum unit of information processing in a classical computer, a qubit performs that role in a quantum computer.
The n n n a = ( a 0 , a 1 , … , a n − 1 ) ∈ { 0 , 1 } n a = (a_{0}, a_{1}, \dots, a_{n-1}) \in \left\{ 0, 1 \right\}^{n} a = ( a 0 , a 1 , … , a n − 1 ) ∈ { 0 , 1 } n n n n
∣ a ⟩ = ∣ a 0 , a 1 , … , a n − 1 ⟩ = ∣ a 0 a 1 … a n − 1 ⟩ = ∣ a 0 ⟩ ⊗ ∣ a 1 ⟩ ⊗ ⋯ ⊗ ∣ a n − 1 ⟩
\begin{align*}
\ket{a}
&= \ket{a_{0}, a_{1}, \dots, a_{n-1}} \\
&= \ket{a_{0} a_{1} \dots a_{n-1}} \\
&= \ket{a_{0}} \otimes \ket{a_{1}} \otimes \cdots \otimes \ket{a_{n-1}}
\end{align*}
∣ a ⟩ = ∣ a 0 , a 1 , … , a n − 1 ⟩ = ∣ a 0 a 1 … a n − 1 ⟩ = ∣ a 0 ⟩ ⊗ ∣ a 1 ⟩ ⊗ ⋯ ⊗ ∣ a n − 1 ⟩
Example: ( C 2 ) ⊗ 2 (\mathbb{C}^{2}) ^{\otimes 2} ( C 2 ) ⊗ 2 Let’s look at the simplest example, where ( C 2 ) ⊗ 2 = C 2 ⊗ C 2 ≅ C 4 (\mathbb{C}^{2}) ^{\otimes 2} = \mathbb{C}^{2} \otimes \mathbb{C}^{2} \cong \mathbb{C}^{4} ( C 2 ) ⊗ 2 = C 2 ⊗ C 2 ≅ C 4 2 2 2
∣ 00 ⟩ = ∣ 0 , 0 ⟩ = ∣ 0 ⟩ ⊗ ∣ 0 ⟩ , ∣ 01 ⟩ = ∣ 0 , 1 ⟩ = ∣ 0 ⟩ ⊗ ∣ 1 ⟩ ∣ 10 ⟩ = ∣ 1 , 0 ⟩ = ∣ 1 ⟩ ⊗ ∣ 0 ⟩ , ∣ 11 ⟩ = ∣ 1 , 1 ⟩ = ∣ 1 ⟩ ⊗ ∣ 1 ⟩
\ket{00} = \ket{0,0} = \ket{0} \otimes \ket{0},\qquad \ket{01} = \ket{0,1} = \ket{0} \otimes \ket{1} \\
\ket{10} = \ket{1,0} = \ket{1} \otimes \ket{0},\qquad \ket{11} = \ket{1,1} = \ket{1} \otimes \ket{1}
∣ 00 ⟩ = ∣ 0 , 0 ⟩ = ∣ 0 ⟩ ⊗ ∣ 0 ⟩ , ∣ 01 ⟩ = ∣ 0 , 1 ⟩ = ∣ 0 ⟩ ⊗ ∣ 1 ⟩ ∣ 10 ⟩ = ∣ 1 , 0 ⟩ = ∣ 1 ⟩ ⊗ ∣ 0 ⟩ , ∣ 11 ⟩ = ∣ 1 , 1 ⟩ = ∣ 1 ⟩ ⊗ ∣ 1 ⟩
If each of the 2 2 2 Kronecker product , it is as follows.
∣ 00 ⟩ = ∣ 0 ⟩ ⊗ ∣ 0 ⟩ = [ 1 0 ] ⊗ [ 1 0 ] = [ 1 [ 1 0 ] 0 [ 1 0 ] ] = [ 1 0 0 0 ] ∣ 01 ⟩ = ∣ 0 ⟩ ⊗ ∣ 1 ⟩ = [ 1 0 ] ⊗ [ 0 1 ] = [ 1 [ 0 1 ] 0 [ 0 1 ] ] = [ 0 1 0 0 ] ∣ 10 ⟩ = ∣ 1 ⟩ ⊗ ∣ 0 ⟩ = [ 0 1 ] ⊗ [ 1 0 ] = [ 0 [ 1 0 ] 1 [ 1 0 ] ] = [ 0 0 1 0 ] ∣ 11 ⟩ = ∣ 1 ⟩ ⊗ ∣ 1 ⟩ = [ 0 1 ] ⊗ [ 0 1 ] = [ 0 [ 0 1 ] 1 [ 0 1 ] ] = [ 0 0 0 1 ]
\begin{align*}
\ket{00} &= \ket{0} \otimes \ket{0} = \begin{bmatrix} 1 \\ 0 \end{bmatrix} \otimes \begin{bmatrix} 1 \\ 0 \end{bmatrix}
= \begin{bmatrix} 1 \begin{bmatrix} 1 \\ 0 \end{bmatrix} \\[1em] 0 \begin{bmatrix} 1 \\ 0 \end{bmatrix} \end{bmatrix}
= \begin{bmatrix} 1 \\ 0 \\ 0 \\ 0 \end{bmatrix} \\
\ket{01} &= \ket{0} \otimes \ket{1} = \begin{bmatrix} 1 \\ 0 \end{bmatrix} \otimes \begin{bmatrix} 0 \\ 1 \end{bmatrix}
= \begin{bmatrix} 1 \begin{bmatrix} 0 \\ 1 \end{bmatrix} \\[1em] 0 \begin{bmatrix} 0 \\ 1 \end{bmatrix} \end{bmatrix}
= \begin{bmatrix} 0 \\ 1 \\ 0 \\ 0 \end{bmatrix} \\
\ket{10} &= \ket{1} \otimes \ket{0} = \begin{bmatrix} 0 \\ 1 \end{bmatrix} \otimes \begin{bmatrix} 1 \\ 0 \end{bmatrix}
= \begin{bmatrix} 0 \begin{bmatrix} 1 \\ 0 \end{bmatrix} \\[1em] 1 \begin{bmatrix} 1 \\ 0 \end{bmatrix} \end{bmatrix}
= \begin{bmatrix} 0 \\ 0 \\ 1 \\ 0 \end{bmatrix} \\
\ket{11} &= \ket{1} \otimes \ket{1} = \begin{bmatrix} 0 \\ 1 \end{bmatrix} \otimes \begin{bmatrix} 0 \\ 1 \end{bmatrix}
= \begin{bmatrix} 0 \begin{bmatrix} 0 \\ 1 \end{bmatrix} \\[1em] 1 \begin{bmatrix} 0 \\ 1 \end{bmatrix} \end{bmatrix}
= \begin{bmatrix} 0 \\ 0 \\ 0 \\ 1 \end{bmatrix}
\end{align*}
∣ 00 ⟩ ∣ 01 ⟩ ∣ 10 ⟩ ∣ 11 ⟩ = ∣ 0 ⟩ ⊗ ∣ 0 ⟩ = [ 1 0 ] ⊗ [ 1 0 ] = 1 [ 1 0 ] 0 [ 1 0 ] = 1 0 0 0 = ∣ 0 ⟩ ⊗ ∣ 1 ⟩ = [ 1 0 ] ⊗ [ 0 1 ] = 1 [ 0 1 ] 0 [ 0 1 ] = 0 1 0 0 = ∣ 1 ⟩ ⊗ ∣ 0 ⟩ = [ 0 1 ] ⊗ [ 1 0 ] = 0 [ 1 0 ] 1 [ 1 0 ] = 0 0 1 0 = ∣ 1 ⟩ ⊗ ∣ 1 ⟩ = [ 0 1 ] ⊗ [ 0 1 ] = 0 [ 0 1 ] 1 [ 0 1 ] = 0 0 0 1
Therefore, ⟨ i k ∣ j l ⟩ = δ i j δ k l \braket{ik | jl} = \delta_{ij}\delta_{kl} ⟨ ik ∣ j l ⟩ = δ ij δ k l δ \delta δ Kronecker delta . Any element of ( C 2 ) ⊗ 2 (\mathbb{C}^{2}) ^{\otimes 2} ( C 2 ) ⊗ 2
( α 0 ∣ 0 ⟩ + α 1 ∣ 1 ⟩ ) ⊗ ( β 0 ∣ 0 ⟩ + β 1 ∣ 1 ⟩ ) = α 0 β 0 ∣ 0 ⟩ ⊗ ∣ 0 ⟩ + α 0 β 1 ∣ 0 ⟩ ⊗ ∣ 1 ⟩ + α 1 β 0 ∣ 1 ⟩ ⊗ ∣ 0 ⟩ + α 1 β 1 ∣ 1 ⟩ ⊗ ∣ 1 ⟩ = α 0 β 0 ∣ 00 ⟩ + α 0 β 1 ∣ 01 ⟩ + α 1 β 0 ∣ 10 ⟩ + α 1 β 1 ∣ 11 ⟩ = α 00 ∣ 00 ⟩ + α 01 ∣ 01 ⟩ + α 10 ∣ 10 ⟩ + α 11 ∣ 11 ⟩
\begin{align*}
& (\alpha_{0}\ket{0} + \alpha_{1}\ket{1}) \otimes (\beta_{0}\ket{0} + \beta_{1}\ket{1})\\
&= \alpha_{0}\beta_{0} \ket{0} \otimes \ket{0} + \alpha_{0}\beta_{1} \ket{0} \otimes \ket{1} + \alpha_{1}\beta_{0} \ket{1} \otimes \ket{0} + \alpha_{1}\beta_{1} \ket{1} \otimes \ket{1} \\
&= \alpha_{0}\beta_{0} \ket{00} + \alpha_{0}\beta_{1} \ket{01} + \alpha_{1}\beta_{0} \ket{10} + \alpha_{1}\beta_{1} \ket{11} \\
&= \alpha_{00}\ket{00} + \alpha_{01}\ket{01} + \alpha_{10}\ket{10} + \alpha_{11}\ket{11} \\
\end{align*}
( α 0 ∣ 0 ⟩ + α 1 ∣ 1 ⟩ ) ⊗ ( β 0 ∣ 0 ⟩ + β 1 ∣ 1 ⟩ ) = α 0 β 0 ∣ 0 ⟩ ⊗ ∣ 0 ⟩ + α 0 β 1 ∣ 0 ⟩ ⊗ ∣ 1 ⟩ + α 1 β 0 ∣ 1 ⟩ ⊗ ∣ 0 ⟩ + α 1 β 1 ∣ 1 ⟩ ⊗ ∣ 1 ⟩ = α 0 β 0 ∣ 00 ⟩ + α 0 β 1 ∣ 01 ⟩ + α 1 β 0 ∣ 10 ⟩ + α 1 β 1 ∣ 11 ⟩ = α 00 ∣ 00 ⟩ + α 01 ∣ 01 ⟩ + α 10 ∣ 10 ⟩ + α 11 ∣ 11 ⟩
In particular, if { ∣ a ⟩ } a ∈ { 0 , 1 } 2 \left\{ \ket{a} \right\}_{a \in \left\{ 0, 1 \right\}^{2}} { ∣ a ⟩ } a ∈ { 0 , 1 } 2 ( C 2 ) ⊗ 2 (\mathbb{C}^{2}) ^{\otimes 2} ( C 2 ) ⊗ 2 ∣ ψ ⟩ ∈ ( C 2 ) ⊗ 2 \ket{\psi} \in (\mathbb{C}^{2}) ^{\otimes 2} ∣ ψ ⟩ ∈ ( C 2 ) ⊗ 2
∣ ψ ⟩ = ∑ a ∈ { 0 , 1 } 2 ⟨ a ∣ ψ ⟩ ∣ a ⟩ = ∑ a ∈ { 0 , 1 } 2 ψ a ∣ a ⟩
\ket{\psi} = \sum\limits_{a \in \left\{ 0, 1 \right\}^{2}} \braket{a | \psi} \ket {a} = \sum\limits_{a \in \left\{ 0, 1 \right\}^{2}} \psi_{a} \ket {a}
∣ ψ ⟩ = a ∈ { 0 , 1 } 2 ∑ ⟨ a ∣ ψ ⟩ ∣ a ⟩ = a ∈ { 0 , 1 } 2 ∑ ψ a ∣ a ⟩
The inner product of ∣ ψ ⟩ , ∣ ξ ⟩ ∈ ( C 2 ) ⊗ 2 \ket{\psi}, \ket{\xi} \in (\mathbb{C}^{2}) ^{\otimes 2} ∣ ψ ⟩ , ∣ ξ ⟩ ∈ ( C 2 ) ⊗ 2
⟨ ψ ∣ ξ ⟩ = ∑ a ∈ { 0 , 1 } 2 ψ a ‾ ξ a
\braket{\psi | \xi} = \sum\limits_{a \in \left\{ 0, 1 \right\}^{2}} \overline{\psi_{a}} \xi_{a}
⟨ ψ ∣ ξ ⟩ = a ∈ { 0 , 1 } 2 ∑ ψ a ξ a
Here, ψ a ‾ \overline{\psi_{a}} ψ a conjugate complex of ψ a \psi_{a} ψ a
See Also 