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线性代数的基本对象是向量，而量子计算的基本单位是量子比特，量子比特用向量来描述。
注：两能级系统和量子比特经典计算机：用逻辑电路中的高低电平表示 0、1；量子计算机：用粒子所处的两种能级基态 ∣ g > \left| g \right> ∣ g ⟩ 和激发态 ∣ e > \left| e \right> ∣ e ⟩ 构成基本计算单元，用向量的方式表示为：∣ g > = [ 0 1 ] , ∣ e > = [ 1 0 ]
\left| g \right> =\left[ \begin{array}{c}
0\\
1\\
\end{array} \right] ,\left| e \right> =\left[ \begin{array}{c}
1\\
0\\
\end{array} \right]
∣ g ⟩ = [ 0 1 ] , ∣ e ⟩ = [ 1 0 ] 和经典的比特类比，常将 ∣ g > \left| g \right> ∣ g ⟩ ∣ 1 > \left| 1 \right> ∣ 1 ⟩ ∣ e > \left| e \right> ∣ e ⟩ ∣ 0 > \left| 0 \right> ∣ 0 ⟩ 量子比特（quantum bits）。
[ z 1 ⋮ z n ]
\left[ \begin{array}{c}
z_1\\
\vdots\\
z_n\\
\end{array} \right]
⎣ ⎢ ⎡ z 1 ⋮ z n ⎦ ⎥ ⎤
加法：[ z 1 ⋮ z n ] + [ z 1 ′ ⋮ z n ′ ] = [ z 1 + z 1 ′ ⋮ z n + z n ′ ]
\left[ \begin{array}{c}
z_1\\
\vdots\\
z_n\\
\end{array} \right] +\left[ \begin{array}{c}
z_{1}^{'}\\
\vdots\\
z_{n}^{'}\\
\end{array} \right] =\left[ \begin{array}{c}
z_1+z_{1}^{'}\\
\vdots\\
z_n+z_{n}^{'}\\
\end{array} \right]
⎣ ⎢ ⎡ z 1 ⋮ z n ⎦ ⎥ ⎤ + ⎣ ⎢ ⎡ z 1 ′ ⋮ z n ′ ⎦ ⎥ ⎤ = ⎣ ⎢ ⎡ z 1 + z 1 ′ ⋮ z n + z n ′ ⎦ ⎥ ⎤
标量乘：z [ z 1 ⋮ z n ] = [ z z 1 ⋮ z z n ]
z\left[ \begin{array}{c}
z_1\\
\vdots\\
z_n\\
\end{array} \right] =\left[ \begin{array}{c}
zz_1\\
\vdots\\
zz_n\\
\end{array} \right]
z ⎣ ⎢ ⎡ z 1 ⋮ z n ⎦ ⎥ ⎤ = ⎣ ⎢ ⎡ z z 1 ⋮ z z n ⎦ ⎥ ⎤
新的符号：狄拉克（Dirac）符号∣ ψ > \left| \psi \right> ∣ ψ ⟩
在量子理论中，描述量子态的向量称为态矢，态矢分为左矢和右矢。右失（ k e t ）： ∣ ψ > = [ c 1 c 2 ⋯ c n ] T = [ c 1 c 2 ⋮ c n ] 左失（ b r a ）： < ψ ∣ = [ c 1 ∗ c 2 ∗ ⋯ c n ∗ ]
\text{右失（}ket\text{）：}\left| \psi \right> =\left[ \begin{matrix}
c_1& c_2& \cdots& c_n\\
\end{matrix} \right] ^{\mathrm{T}}=\left[ \begin{array}{c}
c_1\\
c_2\\
\vdots\\
c_n\\
\end{array} \right]
\\
\text{左失（}bra\text{）：}\left< \psi \right|=\left[ \begin{matrix}
c_{1}^{*}& c_{2}^{*}& \cdots& c_{n}^{*}\\
\end{matrix} \right]
右失（ k e t ）： ∣ ψ ⟩ = [ c 1 c 2 ⋯ c n ] T = ⎣ ⎢ ⎢ ⎢ ⎡ c 1 c 2 ⋮ c n ⎦ ⎥ ⎥ ⎥ ⎤ 左失（ b r a ）： ⟨ ψ ∣ = [ c 1 ∗ c 2 ∗ ⋯ c n ∗ ]
对于一组向量 ∣ u 1 > , ⋯ , ∣ u n > \left| u_1 \right> ,\cdots ,\left| u_n \right> ∣ u 1 ⟩ , ⋯ , ∣ u n ⟩ n n n C n C^n C n ∣ v > = ∑ i x i ∣ u i > = [ x 1 x 2 ⋮ x n ]
\left| v \right> =\sum_i{x_i\left| u_i \right>}=\left[ \begin{array}{c}
x_1\\
x_2\\
\vdots\\
x_n\\
\end{array} \right]
∣ v ⟩ = i ∑ x i ∣ u i ⟩ = ⎣ ⎢ ⎢ ⎢ ⎡ x 1 x 2 ⋮ x n ⎦ ⎥ ⎥ ⎥ ⎤ { ∣ u 1 > , ⋯ , ∣ u n > } \left\{ \left| u_1 \right> ,\cdots ,\left| u_n \right> \right\} { ∣ u 1 ⟩ , ⋯ , ∣ u n ⟩ } C n C^n C n
在 C n C^n C n ( ( a 1 , ⋯ a n ) , ( b 1 , ⋯ , b n ) ) = ∑ i = 1 n a i ∗ b i
\left( \left( a_1,\cdots a_n \right) ,\left( b_1,\cdots ,b_n \right) \right) =\sum_{i=1}^n{a_{i}^{*}b_i}
( ( a 1 , ⋯ a n ) , ( b 1 , ⋯ , b n ) ) = i = 1 ∑ n a i ∗ b i < w ∣ v > = ( ∣ w > , ∣ v > )
\left< w \mid v \right> =\left( \left| w \right> ,\left| v \right> \right)
⟨ w ∣ v ⟩ = ( ∣ w ⟩ , ∣ v ⟩ )
两个向量 ∣ v > , ∣ w > \left| v \right> ,\left| w \right> ∣ v ⟩ , ∣ w ⟩ < w ∣ v > = 0 \left< w \mid v \right> =0 ⟨ w ∣ v ⟩ = 0 ∣ i > \left| i \right> ∣ i ⟩ i i i < j ∣ i > = δ i j , δ i j = { 0 , i ≠ j 1 , i = j
\left< j \mid i \right> =\delta _{ij},\delta _{ij}=\begin{cases}
0, i\ne j\\
1, i=j\\
\end{cases}
⟨ j ∣ i ⟩ = δ i j , δ i j = { 0 , i = j 1 , i = j ∑ i = 1 n ∣ i > < i ∣ = I
\sum_{i=1}^n{\left| i \right> \left< i \right|=I}
i = 1 ∑ n ∣ i ⟩ ⟨ i ∣ = I A = [ a 1 b 1 c 1 a 2 b 2 c 2 a 3 b 3 c 3 ]
A=\left[ \begin{matrix}
a_1& b_1& c_1\\
a_2& b_2& c_2\\
a_3& b_3& c_3\\
\end{matrix} \right]
A = ⎣ ⎡ a 1 a 2 a 3 b 1 b 2 b 3 c 1 c 2 c 3 ⎦ ⎤
定义：∣ α > < β ∣ = [ a i b j ∗ ] m × n \left| \alpha \right> \left< \beta \right|=\left[ a_ib_{j}^{*} \right] _{m\times n} ∣ α ⟩ ⟨ β ∣ = [ a i b j ∗ ] m × n ∣ φ > < ψ ∣ \left| \varphi \right> \left< \psi \right| ∣ φ ⟩ ⟨ ψ ∣ ( ∣ φ > < ψ ∣ ) ∣ ψ ′ > = ∣ φ > ( < ψ ∣ ∣ ψ ′ > ) = ∣ φ > < ψ ∣ ψ ′ > = < ψ ∣ ψ ′ > ∣ φ > \left( \left| \varphi \right> \left< \psi \right| \right) \left| \psi ' \right> =\left| \varphi \right> \left( \left< \psi \right|\left| \psi ' \right> \right) =\left| \varphi \right> \left< \psi \mid \psi ' \right> =\left< \psi \mid \psi ' \right> \left| \varphi \right> ( ∣ φ ⟩ ⟨ ψ ∣ ) ∣ ψ ′ ⟩ = ∣ φ ⟩ ( ⟨ ψ ∣ ∣ ψ ′ ⟩ ) = ∣ φ ⟩ ⟨ ψ ∣ ψ ′ ⟩ = ⟨ ψ ∣ ψ ′ ⟩ ∣ φ ⟩
给定一个线性算子：A A A A ( ∑ i a i ∣ v i > ) = ∑ i a i A ( ∣ v i > ) A\left( \sum_i{a_i\left| v_i \right>} \right) =\sum_i{a_iA(\left| v_i \right>)} A ( ∑ i a i ∣ v i ⟩ ) = ∑ i a i A ( ∣ v i ⟩ ) A A A A ∣ v > = A ( ∣ v > ) A\left| v \right> =A\left( \left| v \right> \right) A ∣ v ⟩ = A ( ∣ v ⟩ ) B A ∣ v > = B ( A ( ∣ v > ) ) BA\left| v \right> =B\left( A\left( \left| v \right> \right) \right) B A ∣ v ⟩ = B ( A ( ∣ v ⟩ ) )
泡利矩阵（泡利算子）σ 0 = I = ( 1 0 0 1 ) \sigma _0=I=\left( \begin{matrix}
1& 0\\
0& 1\\
\end{matrix} \right) σ 0 = I = ( 1 0 0 1 ) I I I σ 1 = σ x = X = ( 0 1 1 0 ) \sigma _1=\sigma _x=X=\left( \begin{matrix}
0& 1\\
1& 0\\
\end{matrix} \right) σ 1 = σ x = X = ( 0 1 1 0 ) X X X σ 2 = σ y = Y = ( 0 − i i 0 ) \sigma _2=\sigma _y=Y=\left( \begin{matrix}
0& -i\\
i& 0\\
\end{matrix} \right) σ 2 = σ y = Y = ( 0 i − i 0 ) σ 3 = σ z = Z = ( 1 0 0 − 1 ) \sigma _3=\sigma _z=Z=\left( \begin{matrix}
1& 0\\
0& -1\\
\end{matrix} \right) σ 3 = σ z = Z = ( 1 0 0 − 1 )
厄米算子（自伴算子）A A A A † A^{\dagger} A † A = A † A=A^{\dagger} A = A † B B B ( A B ) † = B † A † \left( AB \right) ^{\dagger}=B^{\dagger}A^{\dagger} ( A B ) † = B † A † ( A B ) † = ( A ∗ B ∗ ) T = B ∗ T A ∗ T = B † A † \left( AB \right) ^{\dagger}=\left( A^*B^* \right) ^{\mathrm{T}}=B^{*\mathrm{T}}A^{*\mathrm{T}}=B^{\dagger}A^{\dagger} ( A B ) † = ( A ∗ B ∗ ) T = B ∗ T A ∗ T = B † A † ∣ v > † = < v ∣ , ( ∣ v > < w ∣ ) † = ∣ w > < v ∣ \left| v \right> ^{\dagger}=\left< v \right|, \left( \left| v \right> \left< w \right| \right) ^{\dagger}=\left| w \right> \left< v \right| ∣ v ⟩ † = ⟨ v ∣ , ( ∣ v ⟩ ⟨ w ∣ ) † = ∣ w ⟩ ⟨ v ∣
投影算子P P P C n → C m C^n\rightarrow C^m C n → C m m < n m<n m < n P = ∑ i = 1 m ∣ i > < i ∣ P=\sum_{i=1}^m{\left| i \right>}\left< i \right| P = ∑ i = 1 m ∣ i ⟩ ⟨ i ∣ P † = P , P 2 = P
P^{\dagger}=P, P^2=P
P † = P , P 2 = P P 2 = ( ∑ i = 1 k ∣ i > < i ∣ ) 2 = 乘法分配律 ∑ i , j = 1 k ∣ i > < i ∣ j > < j ∣ = ∑ i , j = 1 k ∣ i > δ i j < j ∣ = ∑ i = 1 k ∣ i > < i ∣ = P P^2=\left( \sum_{i=1}^k{\left| i \right> \left< i \right|} \right) ^2\xlongequal{\text{乘法分配律}}\sum_{i,j=1}^k{\left| i \right>}\left< i \mid j \right> \left< j \right|=\sum_{i,j=1}^k{\left| i \right>}\delta _{ij}\left< j \right|=\sum_{i=1}^k{\left| i \right> \left< i \right|}=P P 2 = ( ∑ i = 1 k ∣ i ⟩ ⟨ i ∣ ) 2 乘法分配律 ∑ i , j = 1 k ∣ i ⟩ ⟨ i ∣ j ⟩ ⟨ j ∣ = ∑ i , j = 1 k ∣ i ⟩ δ i j ⟨ j ∣ = ∑ i = 1 k ∣ i ⟩ ⟨ i ∣ = P
正规算子A A A A † A = A A † A^{\dagger}A=AA^{\dagger} A † A = A A † P † = P P^{\dagger}=P P † = P
酉算子（酉矩阵）（所有的量子逻辑门都是由酉算子表示的）U † U = U U † = I U^{\dagger}U=UU^{\dagger}=I U † U = U U † = I U U U
一个算子 A A A v v v ∣ v > \left| v \right> ∣ v ⟩ A ∣ v > = v ∣ v > A\left| v \right> =v\left| v \right> A ∣ v ⟩ = v ∣ v ⟩ A = ∑ i λ i ∣ i > < i ∣ A=\sum_i{\lambda _i\left| i \right>}\left< i \right| A = ∑ i λ i ∣ i ⟩ ⟨ i ∣ λ i \lambda _i λ i A A A i i i ∣ i > \left| i \right> ∣ i ⟩ ∣ i > \left| i \right> ∣ i ⟩
谱分解就是线性代数中的特征值分解（Eigenvalue Decomposition）。A = ∑ a a ∣ a > < a ∣ A=\sum_a{a\left| a \right>}\left< a \right| A = ∑ a a ∣ a ⟩ ⟨ a ∣
当我们需要对一个算子进行函数运算，比如给定算子 A A A f ( A ) f(A) f ( A ) A A A f ( A ) = ∑ a f ( a ) ∣ a > < a ∣ f\left( A \right) =\sum_a{f\left( a \right) \left| a \right>}\left< a \right| f ( A ) = ∑ a f ( a ) ∣ a ⟩ ⟨ a ∣
求解幂函数A n = ∑ a a n ∣ a > < a ∣
A^n=\sum_a{a^n\left| a \right>}\left< a \right|
A n = a ∑ a n ∣ a ⟩ ⟨ a ∣
开方操作A 1 n = ∑ a a 1 n ∣ a > < a ∣
A^{\frac{1}{n}}=\sum_a{a^{\frac{1}{n}}\left| a \right>}\left< a \right|
A n 1 = a ∑ a n 1 ∣ a ⟩ ⟨ a ∣
向量 ∣ v > \left| v \right> ∣ v ⟩ ∣ w > \left| w \right> ∣ w ⟩ C n C^n C n C m C^m C m C n × m C^{n\times m} C n × m ∣ v > ⊗ ∣ w > = ∣ v > ∣ w > = ∣ v w > \left| v \right> \otimes \left| w \right> =\left| v \right> \left| w \right> =\left| vw \right> ∣ v ⟩ ⊗ ∣ w ⟩ = ∣ v ⟩ ∣ w ⟩ = ∣ v w ⟩ [ v 1 v 2 ] ⊗ [ w 1 w 2 ] = [ v 1 w 1 v 1 w 2 v 2 w 1 v 2 w 2 ]
\left[ \begin{array}{c}
v_1\\
v_2\\
\end{array} \right] \otimes \left[ \begin{array}{c}
w_1\\
w_2\\
\end{array} \right] =\left[ \begin{array}{c}
v_1w_1\\
v_1w_2\\
v_2w_1\\
v_2w_2\\
\end{array} \right]
[ v 1 v 2 ] ⊗ [ w 1 w 2 ] = ⎣ ⎢ ⎢ ⎡ v 1 w 1 v 1 w 2 v 2 w 1 v 2 w 2 ⎦ ⎥ ⎥ ⎤
矩阵 A A A B B B A ⊗ B A\otimes B A ⊗ B A ⊗ B ( ∣ v > ⊗ ∣ w > ) = A ∣ v > ⊗ B ∣ w >
A\otimes B\left( \left| v \right> \otimes \left| w \right> \right) =A\left| v \right> \otimes B\left| w \right>
A ⊗ B ( ∣ v ⟩ ⊗ ∣ w ⟩ ) = A ∣ v ⟩ ⊗ B ∣ w ⟩ A A A ∣ v > \left| v \right> ∣ v ⟩ B B B ∣ w > \left| w \right> ∣ w ⟩ A A A B B B A = [ a 11 a 12 a 21 a 22 ] B = [ b 11 b 12 b 21 b 22 ]
A=\left[ \begin{matrix}
a_{11}& a_{12}\\
a_{21}& a_{22}\\
\end{matrix} \right] \,\, B=\left[ \begin{matrix}
b_{11}& b_{12}\\
b_{21}& b_{22}\\
\end{matrix} \right]
A = [ a 1 1 a 2 1 a 1 2 a 2 2 ] B = [ b 1 1 b 2 1 b 1 2 b 2 2 ] A A A B B B A ⊗ B = [ a 11 B a 12 B a 21 B a 22 B ] = [ a 11 b 11 a 11 b 12 a 12 b 11 a 12 b 12 a 11 b 21 a 11 b 22 a 12 b 21 a 12 b 22 a 21 b 11 a 21 b 12 a 22 b 11 a 22 b 12 a 21 b 21 a 21 b 22 a 22 b 21 a 22 b 22 ]
A\otimes B=\left[ \begin{matrix}
a_{11}B& a_{12}B\\
a_{21}B& a_{22}B\\
\end{matrix} \right] =\left[ \begin{matrix}
a_{11}b_{11}& a_{11}b_{12}& a_{12}b_{11}& a_{12}b_{12}\\
a_{11}b_{21}& a_{11}b_{22}& a_{12}b_{21}& a_{12}b_{22}\\
a_{21}b_{11}& a_{21}b_{12}& a_{22}b_{11}& a_{22}b_{12}\\
a_{21}b_{21}& a_{21}b_{22}& a_{22}b_{21}& a_{22}b_{22}\\
\end{matrix} \right]
A ⊗ B = [ a 1 1 B a 2 1 B a 1 2 B a 2 2 B ] = ⎣ ⎢ ⎢ ⎡ a 1 1 b 1 1 a 1 1 b 2 1 a 2 1 b 1 1 a 2 1 b 2 1 a 1 1 b 1 2 a 1 1 b 2 2 a 2 1 b 1 2 a 2 1 b 2 2 a 1 2 b 1 1 a 1 2 b 2 1 a 2 2 b 1 1 a 2 2 b 2 1 a 1 2 b 1 2 a 1 2 b 2 2 a 2 2 b 1 2 a 2 2 b 2 2 ⎦ ⎥ ⎥ ⎤
( A ⊗ B ) † = A † ⊗ B †
\left( A\otimes B \right) ^{\dagger}=A^{\dagger}\otimes B^{\dagger}
( A ⊗ B ) † = A † ⊗ B †
一个矩阵 A A A t r ( A ) = ∑ i A i i tr\left( A \right) =\sum_i{A_{ii}} t r ( A ) = ∑ i A i i A A A B B B C C C
循环性：t r ( A B C ) = t r ( C A B ) = t r ( B C A ) tr\left( ABC \right) =tr\left( CAB \right) =tr\left( BCA \right) t r ( A B C ) = t r ( C A B ) = t r ( B C A )
外积公式：t r ( A ∣ ψ > < ψ ∣ ) = < ψ ∣ A ∣ ψ > tr\left( A\left| \psi \right> \left< \psi \right| \right) =\left< \psi \right|\begin{matrix}
A\\
\end{matrix}\left| \psi \right> t r ( A ∣ ψ ⟩ ⟨ ψ ∣ ) = ⟨ ψ ∣ A ∣ ψ ⟩