这是一大堆公式来袭！！！！

神经网络量子态

设 Q 1 , Q 2 , … , Q N Q_{1}, Q_{2}, \ldots, Q_{N} Q1​,Q2​,…,QN​是具有尺寸为 d 1 , d 2 , … , d N d_{1}, d_{2}, \ldots, d_{N} d1​,d2​,…,dN​ 的状态空间 H 1 , H 2 , … , H N H_{1}, H_{2}, \ldots, H_{N} H1​,H2​,…,HN​ 的 N N N 个量子系统。考虑Q1，Q2，…，QN的复合系统 Q 的状态空间为 H = H 1 ⊗ H 2 ⊗ … ⊗ H N \mathcal{H}=\mathcal{H}_{1} \otimes \mathcal{H}_{2} \otimes \ldots \otimes \mathcal{H}_{N} H=H1​⊗H2​⊗…⊗HN​。设 S 1 , S 2 , … , S N S_{1}, S_{2}, \ldots, S_{N} S1​,S2​,…,SN​分别是系统 Q 1 , Q 2 , … , Q N Q_{1}, Q_{2}, \ldots, Q_{N} Q1​,Q2​,…,QN​的非简并观测量，那么 S = S 1 ⊗ S 2 ⊗ … ⊗ S N \mathcal{S}=\mathcal{S}_{1} \otimes \mathcal{S}_{2} \otimes \ldots \otimes \mathcal{S}_{N} S=S1​⊗S2​⊗…⊗SN​是复合系统 Q Q Q 的可观测值，于是

S j ∣ ψ k j ⟩ = λ k j ∣ ψ k j ⟩ ( k j = 0 , 1 , … , d j − 1 ) S_{j}\left|\psi_{k_{j}}\right\rangle=\lambda_{k_{j}}\left|\psi_{k_{j}}\right\rangle\left(k_{j}=0,1, \ldots, d_{j}-1\right) Sj​∣∣​ψkj​​⟩=λkj​​∣∣​ψkj​​⟩(kj​=0,1,…,dj​−1)

很容易知道 S = S 1 ⊗ S 2 ⊗ … ⊗ S N S=S_{1} \otimes S_{2} \otimes \ldots \otimes S_{N} S=S1​⊗S2​⊗…⊗SN​对应的特征值 λ k 1 λ k 2 … λ k N \lambda_{k_{1}} \lambda_{k_{2}} \ldots \lambda_{k_{N}} λk1​​λk2​​…λkN​​和本征基 ∣ ψ k 1 ⟩ ⊗ ∣ ψ k 2 ⟩ ⊗ … ⊗ ∣ ψ k N ⟩ ( k j = 0 , 1 , … , d j − 1 ) \left|\psi_{k_{1}}\right\rangle \otimes\left|\psi_{k_{2}}\right\rangle \otimes \ldots \otimes\left|\psi_{k_{N}}\right\rangle\left(k_{j}=0,1, \ldots, d_{j}-1\right) ∣ψk1​​⟩⊗∣ψk2​​⟩⊗…⊗∣ψkN​​⟩(kj​=0,1,…,dj​−1)

令输入空间为;
V ( S ) = { Λ k 1 k 2 … k N ≡ ( λ k 1 , λ k 2 , … , λ k N ) T : k j = 0 , 1 , … , d j − 1 } V(S)=\left\{\Lambda_{k_{1} k_{2} \ldots k_{N}} \equiv\left(\lambda_{k_{1}}, \lambda_{k_{2}}, \ldots, \lambda_{k_{N}}\right)^{\mathrm{T}}: k_{j}=0,1, \ldots, d_{j}-1\right\} V(S)={Λk1​k2​…kN​​≡(λk1​​,λk2​​,…,λkN​​)T:kj​=0,1,…,dj​−1}
其中参数 a = ( a 1 , a 2 , … , a N ) T ∈ C N a=\left(a_{1}, a_{2}, \ldots, a_{N}\right)^{\mathrm{T}} \in \mathbb{C}^{N} a=(a1​,a2​,…,aN​)T∈CN, b = ( b 1 , b 2 , … , b M ) T ∈ C M b=\left(b_{1}, b_{2}, \ldots, b_{M}\right)^{\mathrm{T}} \in \mathbb{C}^{M} b=(b1​,b2​,…,bM​)T∈CM, W = [ W i j ] ∈ C M × N W=\left[W_{i j}\right] \in \mathbb{C}^{M \times N} W=[Wij​]∈CM×N
令 Ω = ( a , b , W ) \Omega=(a, b, W) Ω=(a,b,W)，于是可以获得神经网络量子波函数neural network quantum wave function（NNQWF）：

Ψ S , Ω ( λ k 1 , λ k 2 , … , λ k N ) = ∑ h i = ± 1 exp ⁡ ( ∑ j = 1 N a j λ k j + ∑ i = 1 M b i h i + ∑ i = 1 M ∑ j = 1 N W i j h i λ k j ) \begin{array}{l} \Psi_{S, \Omega}\left(\lambda_{k_{1}}, \lambda_{k_{2}}, \ldots, \lambda_{k_{N}}\right) \\ =\sum_{h_{i}=\pm 1} \exp \left(\sum_{j=1}^{N} a_{j} \lambda_{k_{j}}+\sum_{i=1}^{M} b_{i} h_{i}+\sum_{i=1}^{M} \sum_{j=1}^{N} W_{i j} h_{i} \lambda_{k_{j}}\right) \end{array} ΨS,Ω​(λk1​​,λk2​​,…,λkN​​)=∑hi​=±1​exp(∑j=1N​aj​λkj​​+∑i=1M​bi​hi​+∑i=1M​∑j=1N​Wij​hi​λkj​​)​

对于一些 Λ k 1 k 2 … k N \Lambda_{k_{1}} k_{2} \ldots k_{N} Λk1​​k2​…kN​，假设 Ψ S , Ω ( λ k 1 , λ k 2 , … , λ k N ) ≠ 0 \Psi_{S, \Omega}\left(\lambda_{k_{1}}, \lambda_{k_{2}}, \ldots, \lambda_{k_{N}}\right) \neq 0 ΨS,Ω​(λk1​​,λk2​​,…,λkN​​)​=0，定义：
∣ Ψ S , Ω ⟩ = ∑ Λ k 1 k 2 … k N ∈ V ( S ) Ψ S , Ω ( λ k 1 , λ k 2 , … , λ k N ) ⋅ ∣ ψ k 1 ⟩ ⊗ ∣ ψ k 2 ⟩ ⊗ … ⊗ ∣ ψ k N ⟩ \begin{aligned} \left|\Psi_{S, \Omega}\right\rangle=& \sum_{\Lambda_{k_{1} k_{2} \ldots k_{N}} \in V(S)} \Psi_{S, \Omega}\left(\lambda_{k_{1}}, \lambda_{k_{2}}, \ldots, \lambda_{k_{N}}\right) \cdot\left|\psi_{k_{1}}\right\rangle \otimes\left|\psi_{k_{2}}\right\rangle \otimes \ldots \otimes\left|\psi_{k_{N}}\right\rangle \end{aligned} ∣ΨS,Ω​⟩=​Λk1​k2​…kN​​∈V(S)∑​ΨS,Ω​(λk1​​,λk2​​,…,λkN​​)⋅∣ψk1​​⟩⊗∣ψk2​​⟩⊗…⊗∣ψkN​​⟩​

它是希尔伯特空间H的非零向量（不一定归一化）。我们称其为由参数 Ω = ( a , b , W ) \Omega=(a, b, W) Ω=(a,b,W) 和输入可观察到的 S 1 , S 2 , … , S N S_{1}, S_{2}, \ldots, S_{N} S1​,S2​,…,SN​ 引起的神经网络量子态（NNQS）:

对NNQS进行编码的人工神经网络，具有一组N个可见的人工神经元（蓝色磁盘）和一组M个隐藏的神经元（黄色磁盘）， 对于输入的可观察到的S的每个值 Λ k 1 k 2 … k N \Lambda_{k_{1}} k_{2} \ldots k_{N} Λk1​​k2​…kN​，神经网络计算 Ψ S , Ω ( λ k 1 , λ k 2 , … , λ k N ) \Psi_{S, \Omega}\left(\lambda_{k_{1}}, \lambda_{k_{2}}, \ldots, \lambda_{k_{N}}\right) ΨS,Ω​(λk1​​,λk2​​,…,λkN​​)的值。

使用内积符号，可以写成:
∑ j = 1 N a j λ k j + ∑ i = 1 M b i h i + ∑ i = 1 M ∑ j = 1 N W i j h i λ k j = ⟨ Λ k 1 k 2 … k N , a ⟩ + ⟨ h , b + W Λ k 1 k 2 … k N ⟩ \begin{array}{l} \sum_{j=1}^{N} a_{j} \lambda_{k_{j}}+\sum_{i=1}^{M} b_{i} h_{i}+\sum_{i=1}^{M} \sum_{j=1}^{N} W_{i j} h_{i} \lambda_{k_{j}} \\ =\left\langle\Lambda_{k_{1} k_{2} \ldots k_{N}}, a\right\rangle+\left\langle h, b+W \Lambda_{k_{1} k_{2} \ldots k_{N}}\right\rangle \end{array} ∑j=1N​aj​λkj​​+∑i=1M​bi​hi​+∑i=1M​∑j=1N​Wij​hi​λkj​​=⟨Λk1​k2​…kN​​,a⟩+⟨h,b+WΛk1​k2​…kN​​⟩​
即 Ψ S , Ω ( λ k 1 , λ k 2 , … , λ k N ) = e ⟨ Λ k 1 k 2 , k N , a ⟩ ⋅ ∑ h i = ± 1 e ⟨ h , b + W Λ k 1 k 2 … k N ⟩ \Psi_{S, \Omega}\left(\lambda_{k_{1}}, \lambda_{k_{2}}, \ldots, \lambda_{k_{N}}\right)=\mathrm{e}^{\left\langle\Lambda_{k_{1} k_{2}, k_{N}}, a\right\rangle} \cdot \sum_{h_{i}=\pm 1} \mathrm{e}^{\left\langle h, b+W \Lambda_{k_{1}} k_{2} \ldots k_{N}\right\rangle} ΨS,Ω​(λk1​​,λk2​​,…,λkN​​)=e⟨Λk1​k2​,kN​​,a⟩⋅∑hi​=±1​e⟨h,b+WΛk1​​k2​…kN​⟩

NNQWF的另外一种形式
已知 e ⟨ Λ k 1 k 2 … k N , a ⟩ = e ∑ j = 1 N a j λ k j = ∏ j = 1 N e a j λ k j \mathrm{e}^{\left\langle\Lambda_{k_{1} k_{2} \ldots k_{N}}, a\right\rangle}=\mathrm{e}^{\sum_{j=1}^{N} a_{j} \lambda_{k_{j}}}=\prod_{j=1}^{N} \mathrm{e}^{a_{j} \lambda_{k_{j}}} e⟨Λk1​k2​…kN​​,a⟩=e∑j=1N​aj​λkj​​=∏j=1N​eaj​λkj​​
令 x i = b i + ∑ j = 1 N W i j λ k j x_{i}=b_{i}+\sum_{j=1}^{N} W_{i j} \lambda_{k_{j}} xi​=bi​+∑j=1N​Wij​λkj​​

∑ h i = ± 1 e ⟨ h , b + W Λ k 1 k 2 … k N ⟩ = ∑ h i = ± 1 e ∑ i = 1 M h i [ b i + ∑ j = 1 N W i j λ k j ] = ∑ h i = ± 1 ∏ i = 1 M e x i h i = ∑ h 1 = ± 1 … ∑ h M = ± 1 e x 1 h 1 e x 2 h 2 ⋯ e x M h M = ( e x 1 + e − x 1 ) ( e x 2 + e − x 2 ) ⋯ ( e x M + e − x M ) = ∏ i = 1 M 2 cosh ⁡ ( x i ) \begin{array}{l} \sum_{h_{i}=\pm 1} \mathrm{e}^{\left\langle h, b+W \Lambda_{k_{1} k_{2} \ldots k_{N}}\right\rangle} \\ =\sum_{h_{i}=\pm 1} \mathrm{e}^{\sum_{i=1}^{M} h_{i}\left[b_{i}+\sum_{j=1}^{N} W_{i j} \lambda_{k_{j}}\right]} \\ =\sum_{h_{i}=\pm 1} \prod_{i=1}^{M} \mathrm{e}^{x_{i} h_{i}} \\ =\sum_{h_{1}=\pm 1} \ldots \sum_{h_{M}=\pm 1} \mathrm{e}^{x_{1} h_{1}} \mathrm{e}^{x_{2} h_{2}} \cdots \mathrm{e}^{x_{M} h_{M}} \\ =\left(\mathrm{e}^{x_{1}}+\mathrm{e}^{-x_{1}}\right)\left(\mathrm{e}^{x_{2}}+\mathrm{e}^{-x_{2}}\right) \cdots\left(\mathrm{e}^{x_{M}}+\mathrm{e}^{-x_{M}}\right) \\ =\prod_{i=1}^{M} 2 \cosh \left(x_{i}\right) \end{array} ∑hi​=±1​e⟨h,b+WΛk1​k2​…kN​​⟩=∑hi​=±1​e∑i=1M​hi​[bi​+∑j=1N​Wij​λkj​​]=∑hi​=±1​∏i=1M​exi​hi​=∑h1​=±1​…∑hM​=±1​ex1​h1​ex2​h2​⋯exM​hM​=(ex1​+e−x1​)(ex2​+e−x2​)⋯(exM​+e−xM​)=∏i=1M​2cosh(xi​)​

注释： cosh ⁡ x = e x + e − x 2 = e 2 x + 1 2 e x = 1 + e − 2 x 2 e − x \cosh x=\frac{e^{x}+e^{-x}}{2}=\frac{e^{2 x}+1}{2 e^{x}}=\frac{1+e^{-2 x}}{2 e^{-x}} coshx=2ex+e−x​=2exe2x+1​=2e−x1+e−2x​

于是：
Ψ S , Ω ( λ k 1 , λ k 2 , … , λ k N ) = ∏ j = 1 N e a j λ k j ⋅ ∏ i = 1 M 2 cosh ⁡ ( b i + ∑ j = 1 N W i j λ k j ) . \begin{array}{l} \Psi_{S, \Omega}\left(\lambda_{k_{1}}, \lambda_{k_{2}}, \ldots, \lambda_{k_{N}}\right) \\ =\prod_{j=1}^{N} \mathrm{e}^{a_{j} \lambda_{k_{j}}} \cdot \prod_{i=1}^{M} 2 \cosh \left(b_{i}+\sum_{j=1}^{N} W_{i j} \lambda_{k_{j}}\right) . \end{array} ΨS,Ω​(λk1​​,λk2​​,…,λkN​​)=∏j=1N​eaj​λkj​​⋅∏i=1M​2cosh(bi​+∑j=1N​Wij​λkj​​).​

我们将此网络称为量子人工神经网络，因为它的输入是量子可观察量的特征值，结果是NNQWF的值，而它的网络结构类似于通常的人工神经网络:

NNQS的张量积

当 M > N M>N M>N ， a ∈ C N , b ∈ C N , W ∈ C N × N a \in \mathbb{C}^{N}, b \in \mathbb{C}^{N}, W \in \mathbb{C}^{N \times N} a∈CN,b∈CN,W∈CN×N
令
b ′ ′ = ( π i 3 π i 3 ⋮ π i 3 ) ∈ C M − N b^{\prime \prime}=\left(\begin{array}{c} \frac{\pi \mathrm{i}}{3} \\ \frac{\pi \mathrm{i}}{3} \\ \vdots \\ \frac{\pi \mathrm{i}}{3} \end{array}\right) \in \mathbb{C}^{M-N} b′′=⎝⎜⎜⎜⎛​3πi​3πi​⋮3πi​​⎠⎟⎟⎟⎞​∈CM−N

b ′ = ( b b ′ ′ ) ∈ C M b^{\prime}=\left(\begin{array}{c} b \\ b^{\prime \prime} \end{array}\right) \in \mathbb{C}^{M} b′=(bb′′​)∈CM

W ′ = ( W 0 ) = [ W i j ′ ] ∈ C M × N W^{\prime}=\left(\begin{array}{c} W \\ 0 \end{array}\right)=\left[W_{i j}^{\prime}\right] \in \mathbb{C}^{M \times N} W′=(W0​)=[Wij′​]∈CM×N

定义一个新的参数： Ω ′ = ( a , b ′ , W ′ ) \Omega^{\prime}=\left(a, b^{\prime}, W^{\prime}\right) Ω′=(a,b′,W′)

b i ′ = b i ( i = 1 , 2 , … , N ) , b i ′ = π i 3 ( i = N + 1 , N + 2 , … , M ) W i j ′ = W i j ( 1 ≤ i ≤ N , 1 ≤ j ≤ N ) W i j ′ = 0 ( i = N + 1 , N + 2 , … , M , j = 1 , 2 , … , N ) \begin{array}{l} b_{i}^{\prime}=b_{i}(i=1,2, \ldots, N),\\ b_{i}^{\prime}=\frac{\pi \mathrm{i}}{3}(i=N+1, N+2, \ldots, M) \\ W_{i j}^{\prime}=W_{i j}(1 \leq i \leq N, 1 \leq j \leq N) \\ W_{i j}^{\prime}=0(i=N+1, N+2, \ldots, M, j=1,2, \ldots, N) \end{array} bi′​=bi​(i=1,2,…,N),bi′​=3πi​(i=N+1,N+2,…,M)Wij′​=Wij​(1≤i≤N,1≤j≤N)Wij′​=0(i=N+1,N+2,…,M,j=1,2,…,N)​

当 i i i大于 N N N时
2 cosh ⁡ ( b i ′ + ∑ j = 1 N W i j ′ λ k j ) = 2 cosh ⁡ π i 3 = 2 cos ⁡ π 3 = 1 2 \cosh \left(b_{i}^{\prime}+\sum_{j=1}^{N} W_{i j}^{\prime} \lambda_{k_{j}}\right)=2 \cosh \frac{\pi \mathrm{i}}{3}=2 \cos \frac{\pi}{3}=1 2cosh(bi′​+j=1∑N​Wij′​λkj​​)=2cosh3πi​=2cos3π​=1

所以
∏ i = 1 M 2 cosh ⁡ ( b i ′ + ∑ j = 1 N W i j ′ λ k j ) = ∏ i = 1 N 2 cosh ⁡ ( b i + ∑ j = 1 N W i j λ k j ) \prod_{i=1}^{M} 2 \cosh \left(b_{i}^{\prime}+\sum_{j=1}^{N} W_{i j}^{\prime} \lambda_{k_{j}}\right)=\prod_{i=1}^{N} 2 \cosh \left(b_{i}+\sum_{j=1}^{N} W_{i j} \lambda_{k_{j}}\right) i=1∏M​2cosh(bi′​+j=1∑N​Wij′​λkj​​)=i=1∏N​2cosh(bi​+j=1∑N​Wij​λkj​​)

Ψ S , Ω ′ ( λ k 1 , λ k 2 , … , λ k N ) = Ψ S , Ω ( λ k 1 , λ k 2 , … , λ k N ) , ∀ Λ k 1 k 2 … k N \Psi_{S, \Omega^{\prime}}\left(\lambda_{k_{1}}, \lambda_{k_{2}}, \ldots, \lambda_{k_{N}}\right)=\Psi_{S, \Omega}\left(\lambda_{k_{1}}, \lambda_{k_{2}}, \ldots, \lambda_{k_{N}}\right), \quad \forall \Lambda_{k_{1} k_{2} \ldots k_{N}} ΨS,Ω′​(λk1​​,λk2​​,…,λkN​​)=ΨS,Ω​(λk1​​,λk2​​,…,λkN​​),∀Λk1​k2​…kN​​

同理
S ′ = S 1 ′ ⊗ … ⊗ S N ′ ′ , S ′ ′ = S 1 ′ ′ ⊗ … ⊗ S N ′ ′ ′ ′ Ω ′ = ( a ′ , b ′ , W ′ ) , Ω ′ ′ = ( a ′ ′ , b ′ ′ , W ′ ′ ) \begin{array}{l} S^{\prime}=S_{1}^{\prime} \otimes \ldots \otimes S_{N^{\prime}}^{\prime}, S^{\prime \prime}=S_{1}^{\prime \prime} \otimes \ldots \otimes S_{N^{\prime \prime}}^{\prime \prime} \\ \Omega^{\prime}=\left(a^{\prime}, b^{\prime}, W^{\prime}\right), \Omega^{\prime \prime}=\left(a^{\prime \prime}, b^{\prime \prime}, W^{\prime \prime}\right) \end{array} S′=S1′​⊗…⊗SN′′​,S′′=S1′′​⊗…⊗SN′′′′​Ω′=(a′,b′,W′),Ω′′=(a′′,b′′,W′′)​

对于
S = S ′ ⊗ S ′ ′ , N = N ′ + N ′ ′ , M = M ′ + M ′ ′ a = ( a ′ a ′ ′ ) , b = ( b ′ b ′ ′ ) , W = [ W i j ] = ( W M ′ × N ′ ′ 0 0 W M ′ ′ × N ′ ′ ′ ′ ) , Ω = ( a , b , W ) \begin{array}{l} S=S^{\prime} \otimes S^{\prime \prime}, N=N^{\prime}+N^{\prime \prime}, M=M^{\prime}+M^{\prime \prime} \\ a=\left(\begin{array}{c} a^{\prime} \\ a^{\prime \prime} \end{array}\right), b=\left(\begin{array}{c} b^{\prime} \\ b^{\prime \prime} \end{array}\right), \\ W=\left[W_{i j}\right]=\left(\begin{array}{cc} W_{M^{\prime} \times N^{\prime}}^{\prime} & 0 \\ 0 & W_{M^{\prime \prime} \times N^{\prime \prime}}^{\prime \prime} \end{array}\right), \Omega=(a, b, W) \end{array} S=S′⊗S′′,N=N′+N′′,M=M′+M′′a=(a′a′′​),b=(b′b′′​),W=[Wij​]=(WM′×N′′​0​0WM′′×N′′′′​​),Ω=(a,b,W)​

∣ Ψ S ′ , Ω ′ ′ ⟩ = ∑ ( λ k 1 , λ k 2 , … , λ k N ′ ) T ∈ V ( S ′ ) Ψ S ′ , Ω ′ ′ ( λ k 1 , λ k 2 , … , λ k N ′ ) ⋅ ∣ ψ k 1 ⟩ ⊗ ∣ ψ k 2 ⟩ ⊗ … ⊗ ∣ ψ k N ′ ⟩ \begin{aligned} \left|\Psi_{S^{\prime}, \Omega^{\prime}}^{\prime}\right\rangle=& \sum_{\left(\lambda_{k_{1}}, \lambda_{k_{2}}, \ldots, \lambda_{k_{N^{\prime}}}\right)^{\mathrm{T}} \in V\left(S^{\prime}\right)} \Psi_{S^{\prime}, \Omega^{\prime}}^{\prime}\left(\lambda_{k_{1}}, \lambda_{k_{2}}, \ldots, \lambda_{k_{N^{\prime}}}\right) \cdot\left|\psi_{k_{1}}\right\rangle \otimes\left|\psi_{k_{2}}\right\rangle \otimes \ldots \otimes\left|\psi_{k_{N^{\prime}}}\right\rangle \end{aligned} ∣∣​ΨS′,Ω′′​⟩=​(λk1​​,λk2​​,…,λkN′​​)T∈V(S′)∑​ΨS′,Ω′′​(λk1​​,λk2​​,…,λkN′​​)⋅∣ψk1​​⟩⊗∣ψk2​​⟩⊗…⊗∣∣​ψkN′​​⟩​

∣ Ψ S ′ ′ , Ω ′ ′ ′ ′ ⟩ = ∑ ( μ m 1 , μ m 2 , … , μ m N ′ ′ ) T ∈ V ( S ′ ′ ) Ψ S ′ ′ , Ω ′ ′ ′ ′ ( μ m 1 , μ m 2 , … , μ m N ′ ′ ) ⋅ ∣ ϕ m 1 ⟩ ⊗ ∣ ϕ m 2 ⟩ ⊗ … ⊗ ∣ ϕ m N ′ ′ ⟩ \begin{aligned} \left|\Psi_{S^{\prime \prime}, \Omega^{\prime \prime}}^{\prime \prime}\right\rangle=& \sum_{\left(\mu_{m_{1}}, \mu_{m_{2}}, \ldots, \mu_{m_{N^{\prime \prime}}}\right)^{\mathrm{T}} \in V\left(S^{\prime \prime}\right)} \Psi_{S^{\prime \prime}, \Omega^{\prime \prime}}^{\prime \prime}\left(\mu_{m_{1}}, \mu_{m_{2}}, \ldots, \mu_{m_{N^{\prime \prime}}}\right)\cdot\left|\phi_{m_{1}}\right\rangle \otimes\left|\phi_{m_{2}}\right\rangle \otimes \ldots \otimes\left|\phi_{m_{N^{\prime \prime}}}\right\rangle \end{aligned} ∣∣​ΨS′′,Ω′′′′​⟩=​(μm1​​,μm2​​,…,μmN′′​​)T∈V(S′′)∑​ΨS′′,Ω′′′′​(μm1​​,μm2​​,…,μmN′′​​)⋅∣ϕm1​​⟩⊗∣ϕm2​​⟩⊗…⊗∣∣​ϕmN′′​​⟩​

Ψ S ′ , Ω ′ ′ ( λ k 1 , λ k 2 , … , λ k N ′ ) ⋅ Ψ S ′ ′ , Ω ′ ′ ′ ′ ( μ m 1 , μ m 2 , … , μ m N ′ ′ ) \Psi_{S^{\prime}, \Omega^{\prime}}^{\prime}\left(\lambda_{k_{1}}, \lambda_{k_{2}}, \ldots, \lambda_{k_{N^{\prime}}}\right) \cdot \Psi_{S^{\prime \prime}, \Omega^{\prime \prime}}^{\prime \prime}\left(\mu_{m_{1}}, \mu_{m_{2}}, \ldots, \mu_{m_{N^{\prime \prime}}}\right) ΨS′,Ω′′​(λk1​​,λk2​​,…,λkN′​​)⋅ΨS′′,Ω′′′′​(μm1​​,μm2​​,…,μmN′′​​)

= ( ∏ j = 1 N ′ e a j ′ λ k j ) ⋅ ( ∏ j = 1 N ′ ′ e a j ′ ′ μ m j ) ⋅ ( ∏ i = 1 M ′ 2 cosh ⁡ ( b i ′ + ∑ j = 1 N ′ W i j ′ λ k j ) ) ⋅ ( ∏ i = 1 M ′ ′ 2 cosh ⁡ ( b i ′ ′ + ∑ j = 1 N ′ ′ W i j ′ ′ μ m j ) ) = ( ∏ j = 1 N e a j ξ j j ) ⋅ ( ∏ i = 1 M 2 cosh ⁡ ( b i + ∑ j = 1 N W i j ξ j ) ) = Ψ S , Ω ( ξ l 1 , ξ l 2 , … , ξ l N ) \begin{aligned} =&\left(\prod_{j=1}^{N^{\prime}} \mathrm{e}^{a_{j}^{\prime} \lambda_{k_{j}}}\right) \cdot\left(\prod_{j=1}^{N^{\prime \prime}} \mathrm{e}^{a_{j}^{\prime \prime} \mu_{m_{j}}}\right) \cdot\left(\prod_{i=1}^{M^{\prime}} 2 \cosh \left(b_{i}^{\prime}+\sum_{j=1}^{N^{\prime}} W_{i j}^{\prime} \lambda_{k_{j}}\right)\right) \\ & \cdot\left(\prod_{i=1}^{M^{\prime \prime}} 2 \cosh \left(b_{i}^{\prime \prime}+\sum_{j=1}^{N^{\prime \prime}} W_{i j}^{\prime \prime} \mu_{m_{j}}\right)\right) \\ =&\left(\prod_{j=1}^{N} \mathrm{e}^{a_{j} \xi_{j_{j}}}\right) \cdot\left(\prod_{i=1}^{M} 2 \cosh \left(b_{i}+\sum_{j=1}^{N} W_{i j} \xi_{j}\right)\right) \\ =& \Psi_{S, \Omega}\left(\xi_{l_{1}}, \xi_{l_{2}}, \ldots, \xi_{l_{N}}\right) \end{aligned} ===​⎝⎛​j=1∏N′​eaj′​λkj​​⎠⎞​⋅⎝⎛​j=1∏N′′​eaj′′​μmj​​⎠⎞​⋅⎝⎛​i=1∏M′​2cosh⎝⎛​bi′​+j=1∑N′​Wij′​λkj​​⎠⎞​⎠⎞​⋅⎝⎛​i=1∏M′′​2cosh⎝⎛​bi′′​+j=1∑N′′​Wij′′​μmj​​⎠⎞​⎠⎞​(j=1∏N​eaj​ξjj​​)⋅(i=1∏M​2cosh(bi​+j=1∑N​Wij​ξj​))ΨS,Ω​(ξl1​​,ξl2​​,…,ξlN​​)​
其中 ( ξ l 1 , ξ l 2 , … , ξ l N ) = ( λ k 1 , λ k 2 , … , λ k N ′ , μ m 1 , μ m 2 , … , μ m N ′ ′ ) \left(\xi_{l_{1}}, \xi_{l_{2}}, \ldots, \xi_{l_{N}}\right)=\left(\lambda_{k_{1}}, \lambda_{k_{2}}, \ldots, \lambda_{k_{N^{\prime}}}, \mu_{m_{1}}, \mu_{m_{2}}, \ldots, \mu_{m_{N^{\prime \prime}}}\right) (ξl1​​,ξl2​​,…,ξlN​​)=(λk1​​,λk2​​,…,λkN′​​,μm1​​,μm2​​,…,μmN′′​​)

∣ Ψ S ′ , Ω ′ ′ ⟩ ⊗ ∣ Ψ S ′ ′ , Ω ′ ′ ′ ′ ⟩ = ∑ ( λ 1 , λ k 2 , … , λ N ′ ) T ∈ V ( S ′ ) ( μ m 1 , μ m 2 , … , μ m N ′ ′ ) T ∈ V ( S ′ ′ ) Ψ S ′ , Ω ′ ′ ( λ k 1 , λ k 2 , … , λ k N ′ ) ⋅ Ψ S ′ ′ , Ω ′ ′ ′ ′ ( μ m 1 , μ m 2 , … , μ m N ′ ′ ) ⋅ ∣ ψ k 1 ⟩ ⊗ ∣ ψ k 2 ⟩ ⊗ … ⊗ ∣ ψ k N ′ ⟩ ⊗ ∣ ϕ m 1 ⟩ ⊗ ∣ ϕ m 2 ⟩ ⊗ … ⊗ ∣ ϕ m N ′ ′ ⟩ = ∑ ( ξ l , ξ l 2 , … , ξ l N ) T ∈ V ( S ) Ψ S , Ω ( ξ l 1 , ξ l 2 , … , ξ l N ) ∣ φ l 1 ⟩ ⊗ … ⊗ ∣ φ l N ⟩ = ∣ Φ S , Ω ⟩ \begin{array}{l} \left|\Psi_{S^{\prime}, \Omega^{\prime}}^{\prime}\right\rangle \otimes\left|\Psi_{S^{\prime \prime}, \Omega^{\prime \prime}}^{\prime \prime}\right\rangle \\ =\sum_{\left(\lambda_{1}, \lambda_{k_{2}}, \ldots, \lambda_{N^{\prime}}\right)^{\mathrm{T}} \in V\left(S^{\prime}\right)\left(\mu_{m_{1}}, \mu_{m_{2}}, \ldots, \mu_{m_{N^{\prime \prime}}}\right)^{\mathrm{T}} \in V\left(S^{\prime \prime}\right)} \\ \Psi_{S^{\prime}, \Omega^{\prime}}^{\prime}\left(\lambda_{k_{1}}, \lambda_{k_{2}}, \ldots, \lambda_{k_{N^{\prime}}}\right) \cdot \Psi_{S^{\prime \prime}, \Omega^{\prime \prime}}^{\prime \prime}\left(\mu_{m_{1}}, \mu_{m_{2}}, \ldots, \mu_{m_{N^{\prime \prime}}}\right) \\ \quad \cdot\left|\psi_{k_{1}}\right\rangle \otimes\left|\psi_{k_{2}}\right\rangle \otimes \ldots \otimes\left|\psi_{k_{N^{\prime}}}\right\rangle \otimes\left|\phi_{m_{1}}\right\rangle \otimes\left|\phi_{m_{2}}\right\rangle \otimes \ldots \otimes\left|\phi_{m_{N^{\prime \prime}}}\right\rangle \\ =\quad \sum_{\left(\xi_{l}, \xi_{l_{2}}, \ldots, \xi_{l_{N}}\right)^{\mathrm{T}} \in V(S)} \Psi_{S, \Omega}\left(\xi_{l_{1}}, \xi_{l_{2}}, \ldots, \xi_{l_{N}}\right)\left|\varphi_{l_{1}}\right\rangle \otimes \ldots \otimes\left|\varphi_{l_{N}}\right\rangle \\ =\left|\Phi_{S, \Omega}\right\rangle \end{array} ∣∣​ΨS′,Ω′′​⟩⊗∣∣​ΨS′′,Ω′′′′​⟩=∑(λ1​,λk2​​,…,λN′​)T∈V(S′)(μm1​​,μm2​​,…,μmN′′​​)T∈V(S′′)​ΨS′,Ω′′​(λk1​​,λk2​​,…,λkN′​​)⋅ΨS′′,Ω′′′′​(μm1​​,μm2​​,…,μmN′′​​)⋅∣ψk1​​⟩⊗∣ψk2​​⟩⊗…⊗∣∣​ψkN′​​⟩⊗∣ϕm1​​⟩⊗∣ϕm2​​⟩⊗…⊗∣∣​ϕmN′′​​⟩=∑(ξl​,ξl2​​,…,ξlN​​)T∈V(S)​ΨS,Ω​(ξl1​​,ξl2​​,…,ξlN​​)∣φl1​​⟩⊗…⊗∣φlN​​⟩=∣ΦS,Ω​⟩​

这表明两个NNQS的张量积也是NNQS

局部统一运算（LUO）对NNQS的影响

有前面的内容我们知道

∣ Ψ S , Ω ⟩ = ∑ Λ k 1 , u N ∈ V ( S ) Ψ S , Ω ( λ k 1 , … , λ k N ) ∣ ψ k 1 ⟩ ⊗ … ⊗ ∣ ψ k N ⟩ \left|\Psi_{S, \Omega}\right\rangle=\sum_{\Lambda_{k_{1}, u_{N}} \in V(S)} \Psi_{S, \Omega}\left(\lambda_{k_{1}}, \ldots, \lambda_{k_{N}}\right)\left|\psi_{k_{1}}\right\rangle \otimes \ldots \otimes\left|\psi_{k_{N}}\right\rangle ∣ΨS,Ω​⟩=Λk1​,uN​​∈V(S)∑​ΨS,Ω​(λk1​​,…,λkN​​)∣ψk1​​⟩⊗…⊗∣ψkN​​⟩

令 U = U 1 ⊗ … ⊗ U N U=U_{1} \otimes \ldots \otimes U_{N} U=U1​⊗…⊗UN​,因为 U i S i U i † ∣ ϕ k i ⟩ = U i S i ∣ ψ k i ⟩ = U i ( λ k i ∣ ψ k i ⟩ ) = λ k i ∣ ϕ k i ⟩ U_{i} S_{i} U_{i}^{\dagger}\left|\phi_{k_{i}}\right\rangle=U_{i} S_{i}\left|\psi_{k_{i}}\right\rangle=U_{i}\left(\lambda_{k_{i}}\left|\psi_{k_{i}}\right\rangle\right)=\lambda_{k_{i}}\left|\phi_{k_{i}}\right\rangle Ui​Si​Ui†​∣ϕki​​⟩=Ui​Si​∣ψki​​⟩=Ui​(λki​​∣ψki​​⟩)=λki​​∣ϕki​​⟩

于是
U ∣ Ψ S , Ω ⟩ = ∑ Λ k 1 , k N ∈ V ( S ) Ψ S , Ω ( λ k 1 , … , λ k N ) ∣ U 1 ψ k 1 ⟩ ⊗ … ⊗ ∣ U N ψ k N ⟩ = ∑ Λ k 1 … k N ∈ V ( U S U † ) Ψ U S U † , Ω ( λ k 1 , … , λ k N ) ∣ ϕ k 1 ⟩ ⊗ … ⊗ ∣ ϕ k N ⟩ = ∣ Ψ U S U † , Ω ⟩ \begin{array}{l} U\left|\Psi_{S, \Omega}\right\rangle \\ =\sum_{\Lambda_{k_{1}, k_{N}} \in V(S)} \Psi_{S, \Omega}\left(\lambda_{k_{1}}, \ldots, \lambda_{k_{N}}\right)\left|U_{1} \psi_{k_{1}}\right\rangle \otimes \ldots \otimes\left|U_{N} \psi_{k_{N}}\right\rangle \\ =\sum_{\Lambda_{k_{1} \ldots k_{N}} \in V\left(U S U^{\dagger}\right)} \Psi_{U S U^{\dagger}, \Omega}\left(\lambda_{k_{1}}, \ldots, \lambda_{k_{N}}\right)\left|\phi_{k_{1}}\right\rangle \otimes \ldots \otimes\left|\phi_{k_{N}}\right\rangle \\ =\left|\Psi_{U S U^{\dagger}, \Omega}\right\rangle \end{array} U∣ΨS,Ω​⟩=∑Λk1​,kN​​∈V(S)​ΨS,Ω​(λk1​​,…,λkN​​)∣U1​ψk1​​⟩⊗…⊗∣UN​ψkN​​⟩=∑Λk1​…kN​​∈V(USU†)​ΨUSU†,Ω​(λk1​​,…,λkN​​)∣ϕk1​​⟩⊗…⊗∣ϕkN​​⟩=∣∣​ΨUSU†,Ω​⟩​

这表明结果状态 U ∣ Ψ S , Ω ⟩ U\left|\Psi_{S, \Omega}\right\rangle U∣ΨS,Ω​⟩也是具有输入可观察到的 U S U † U S U^{\dagger} USU†和参数 Ω Ω Ω的NNQS，并且与 U ∣ Ψ S , Ω ⟩ U\left|\Psi_{S, \Omega}\right\rangle U∣ΨS,Ω​⟩具有相同的NNQWF。

当 S = σ 1 z ⊗ σ 2 z ⊗ … ⊗ σ N z S=\sigma_{1}^{z} \otimes \sigma_{2}^{z} \otimes \ldots \otimes \sigma_{N}^{z} S=σ1z​⊗σ2z​⊗…⊗σNz​

∣ Ψ S , Ω ⟩ = ∑ Λ k 1 k 2 , k N ∈ { − 1 , 1 } N ( ∏ i = 1 M 2 Λ i ) ∣ ψ k 1 ⟩ ⊗ ∣ ψ k 2 ⟩ ⊗ … ⊗ ∣ ψ k N ⟩ \left|\Psi_{S, \Omega}\right\rangle=\sum_{\Lambda_{k_{1}} k_{2}, k_{N} \in\{-1,1\}^{N}}\left(\prod_{i=1}^{M} 2 \Lambda_{i}\right)\left|\psi_{k_{1}}\right\rangle \otimes\left|\psi_{k_{2}}\right\rangle \otimes \ldots \otimes\left|\psi_{k_{N}}\right\rangle ∣ΨS,Ω​⟩=Λk1​​k2​,kN​∈{−1,1}N∑​(i=1∏M​2Λi​)∣ψk1​​⟩⊗∣ψk2​​⟩⊗…⊗∣ψkN​​⟩

其中 Λ i = cos ⁡ ( b + ω 1 λ k i − 1 + ω 0 λ k i + ω − 1 λ k i + 1 ) \Lambda_{i}=\cos \left(b+\omega_{1} \lambda_{k_{i-1}}+\omega_{0} \lambda_{k_{i}}+\omega_{-1} \lambda_{k_{i+1}}\right) Λi​=cos(b+ω1​λki−1​​+ω0​λki​​+ω−1​λki+1​​)