数式索引

本文に置かれた表示数式を、講義ノートの章立てに沿って並べています。各項目を押すと、その数式が置かれた本文中の位置へ移動します。

1日目: 量子アニーリングの基礎

1. 最適化問題をIsingハミルトニアンへ写す

• $${\hat{H}}_0=-\sum _{⟨i,j⟩}{J}_{ij}{\hat{\sigma }}_i^z{\hat{\sigma }}_j^z-\sum _{i=1}^N{h}_i{\hat{\sigma }}_i^z$$

• $$E(\mathit{\sigma })=-\sum _{⟨i,j⟩}{J}_{ij}{\sigma }_i{\sigma }_j-\sum _{i=1}^N{h}_i{\sigma }_i$$

• $${\hat{H}}_{\mathrm{H}\mathrm{e}\mathrm{i}\mathrm{s}}=-J\sum _{⟨i,j⟩}({\hat{\sigma }}_i^x{\hat{\sigma }}_j^x+{\hat{\sigma }}_i^y{\hat{\sigma }}_j^y+{\hat{\sigma }}_i^z{\hat{\sigma }}_j^z)$$

• $${\hat{H}}_{\mathrm{I}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{g}}=-J\sum _{⟨i,j⟩}{\hat{\sigma }}_i^z{\hat{\sigma }}_j^z$$

• $${\hat{\sigma }}^x=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right),{\hat{\sigma }}^y=\left(\begin{array}{cc}0& -i\\ i& 0\end{array}\right),{\hat{\sigma }}^z=\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right)$$

• $$[{\hat{\sigma }}^x,{\hat{\sigma }}^y]=2i{\hat{\sigma }}^z,[{\hat{\sigma }}^z,{\hat{\sigma }}^x]=2i{\hat{\sigma }}^y$$

• $${\hat{\sigma }}^z|0⟩=+|0⟩,{\hat{\sigma }}^z|1⟩=-|1⟩$$

2. 横磁場ドライバーから始める

• $${\hat{H}}_1=-\mathrm{\Gamma }\sum _i{\hat{\sigma }}_i^x$$

• $$|s_i⟩=\frac{1}{\sqrt{2}}(|0⟩+|1⟩)$$

• $$|s⟩=\underset{i=1}{\overset{N}{⨂}}|s_i⟩=\underset{i=1}{\overset{N}{⨂}}\frac{1}{\sqrt{2}}(|0⟩+|1⟩)$$

3. 時間依存ハミルトニアンで目的状態へ運ぶ

• $$\hat{H}(t)=f(t){\hat{H}}_0+\{1-f(t)\}{\hat{H}}_1$$

• $$f(0)=0,f(T)=1$$

• $$f(t)=\frac{t}{T}$$

• $$s=\frac{t}{T}(0\le s\le 1)$$

• $$\hat{H}(s)=s{\hat{H}}_0+(1-s){\hat{H}}_1,\frac{d\hat{H}}{ds}={\hat{H}}_0-{\hat{H}}_1$$

• $$i\hslash \frac{d}{dt}|\psi (t)⟩=\hat{H}(t)|\psi (t)⟩$$

• $$i\hslash \frac{d}{ds}|\psi (s)⟩=T\hat{H}(s)|\psi (s)⟩$$

4. 瞬間固有状態で展開する

• $$\hat{H}(s)|j(s)⟩=E_j(s)|j(s)⟩$$

• $$|\psi (s)⟩=\sum _j{\tilde{C}}_j(s)\exp [-\frac{iT}{\hslash }{\int }_0^s{E}_j(u)du]|j(s)⟩$$

• $${\varphi }_j(s)={\int }_0^s{E}_j(u)du$$

• $$\frac{d{\tilde{C}}_k}{ds}=-\sum _j{\tilde{C}}_j(s)\exp [\frac{iT}{\hslash }\{{\varphi }_k(s)-{\varphi }_j(s)\}]⟨k(s)\left|\frac{d}{ds}\right|j(s)⟩$$

• $$⟨k(s)\left|\frac{d}{ds}\right|j(s)⟩$$

• $$\frac{d\hat{H}}{ds}|j⟩+\hat{H}\frac{d}{ds}|j⟩=\frac{d{E}_j}{ds}|j⟩+E_j\frac{d}{ds}|j⟩$$

• $$⟨k\left|\frac{d}{ds}\right|j⟩=\frac{⟨k\left|\frac{d\hat{H}}{ds}\right|j⟩}{E_j-E_k}$$

• $$\frac{\left|⟨k\left|\frac{d\hat{H}}{ds}\right|j⟩\right|}{|E_j-E_k|}$$

5. 励起振幅を評価する

• $${\tilde{C}}_0(0)=1,{\tilde{C}}_k(0)=0(k\ne 0)$$

• $${\tilde{C}}_k(s)\simeq {\int }_0^{s}du\exp [\frac{iT}{\hslash }\mathrm{\Delta }{\varphi }_{k0}(u)]\frac{⟨k(u)\left|\frac{d\hat{H}}{du}\right|0(u)⟩}{{\mathrm{\Delta }}_{k0}(u)}$$

• $${\mathrm{\Delta }}_{k0}(u)=E_k(u)-E_0(u),\mathrm{\Delta }{\varphi }_{k0}(u)={\varphi }_k(u)-{\varphi }_0(u)$$

• $$\frac{d}{du}\exp [\frac{iT}{\hslash }\mathrm{\Delta }{\varphi }_{k0}(u)]=\frac{iT}{\hslash }{\mathrm{\Delta }}_{k0}(u)\exp [\frac{iT}{\hslash }\mathrm{\Delta }{\varphi }_{k0}(u)]$$

• $$|{\tilde{C}}_k(s)|\lesssim \frac{2\hslash }{T}\underset{s}{max}\left|\frac{⟨k(s)\left|\frac{d\hat{H}}{ds}\right|0(s)⟩}{{\mathrm{\Delta }}_{k0}(s{)}^2}\right|$$

• $$T\gtrsim \frac{2\hslash }{ϵ}\underset{s}{max}\left|\frac{⟨k(s)\left|\frac{d\hat{H}}{ds}\right|0(s)⟩}{{\mathrm{\Delta }}_{k0}(s{)}^2}\right|$$

6. 基底状態への収束を保証する横磁場の減衰

• $${\frac{d^m\hat{H}}{d{s}^m}|}_{s=0,1}=0$$

• $$|{\tilde{C}}_k(1){|}^2\sim \frac{1}{T^{2(m+1)}}$$

7. 反断熱項を厳密に書く

• $$|\psi (t)⟩=e^{i{\varphi }_n(t)}|n(t)⟩$$

• $${\hat{H}}^{\ast }(t)=\hat{H}(t)+{\hat{H}}_{\mathrm{C}\mathrm{D}}(t)$$

• $${\hat{H}}^{\ast }(t)|n(t)⟩=-\hslash {\dot{\varphi }}_n(t)|n(t)⟩+i\hslash |\dot{n}(t)⟩$$

• $$⟨n(t)|{\hat{H}}_{\mathrm{C}\mathrm{D}}(t)|n(t)⟩=0$$

• $${\dot{\varphi }}_n(t)=-\frac{1}{\hslash }{E}_n(t)+i⟨n(t)|\dot{n}(t)⟩$$

• $${\hat{H}}_{\mathrm{C}\mathrm{D}}(t)=i\hslash \sum _n(|\dot{n}(t)⟩⟨n(t)|-⟨n(t)|\dot{n}(t)⟩|n(t)⟩⟨n(t)|)$$

• $${\hat{H}}_{\mathrm{C}\mathrm{D}}(t)=i\hslash \sum _{m\ne n}|m(t)⟩⟨m(t)|\dot{n}(t)⟩⟨n(t)|$$

• $$⟨m(t)|\dot{n}(t)⟩=\frac{⟨m(t)|\dot{\hat{H}}(t)|n(t)⟩}{E_n(t)-E_m(t)}(m\ne n)$$

• $${\hat{H}}_{\mathrm{C}\mathrm{D}}(t)=i\hslash \sum _{m\ne n}\frac{⟨m(t)|\dot{\hat{H}}(t)|n(t)⟩}{E_n(t)-E_m(t)}|m(t)⟩⟨n(t)|$$

8. 断熱ゲージポテンシャルの変分原理

• $$G(t)=\frac{d\hat{H}}{dt}+i[{\hat{H}}_{\mathrm{C}\mathrm{D}}(t),\hat{H}(t)]$$

• $$i⟨m|\dot{\hat{H}}|n⟩=⟨m|[{\hat{H}}_{\mathrm{C}\mathrm{D}},\hat{H}]|n⟩$$

• $$G_{\lambda }({\hat{A}}_{\lambda })={\partial }_{\lambda }\hat{H}+i[{\hat{A}}_{\lambda },\hat{H}]$$

• $$\underset{{\hat{A}}_{\lambda }}{min}\mathrm{Tr}[G_{\lambda }({\hat{A}}_{\lambda }{)}^2]$$

• $$\delta G_{\lambda }=i[\delta {\hat{A}}_{\lambda },\hat{H}]$$

• $$2\mathrm{Tr}\left[G_{\lambda }\delta G_{\lambda }\right]=2i\mathrm{Tr}[G_{\lambda }[\delta {\hat{A}}_{\lambda },\hat{H}]]=0$$

• $$\mathrm{Tr}[\hat{A}\hat{B}]=\mathrm{Tr}[\hat{B}\hat{A}],\mathrm{Tr}[\hat{A}[\hat{B},\hat{C}]]=\mathrm{Tr}[[\hat{C},\hat{A}]\hat{B}]$$

• $$[\hat{H},G_{\lambda }]=0$$

• $$\hat{H}(\lambda )=\lambda {\hat{H}}_0+(1-\lambda ){\hat{H}}_1$$

• $${\partial }_{\lambda }\hat{H}={\hat{H}}_0-{\hat{H}}_1$$

• $${\hat{A}}_{\lambda }\propto i[\hat{H},{\partial }_{\lambda }\hat{H}]\propto i[{\hat{H}}_0,{\hat{H}}_1]$$

• $$[{\hat{\sigma }}^z,{\hat{\sigma }}^x]=2i{\hat{\sigma }}^y$$

9. QAOAへ続く橋

• $$U(T)=\exp (-\frac{i}{\hslash }{\int }_0^{T}dt\hat{H}(t))$$

• $$U(T)\simeq \prod _{k=1}^n\exp (-\frac{i\mathrm{\Delta }t}{\hslash }{\hat{H}}_k),{\hat{H}}_k=\hat{H}(k\mathrm{\Delta }t)$$

• $$U(T)\simeq \prod _{k=1}^n\exp (-\frac{i\mathrm{\Delta }t}{\hslash }{f}_k{\hat{H}}_0)\exp (-\frac{i\mathrm{\Delta }t}{\hslash }(1-f_k){\hat{H}}_1)$$

2日目: QAOA、量子モンテカルロ、p-spin模型

1. QAOAの状態と評価関数

• $$|s⟩=\underset{i=1}{\overset{N}{⨂}}\frac{1}{\sqrt{2}}(|1{⟩}_i+|-1{⟩}_i)$$

• $$|\mathit{\gamma },\mathit{\beta }⟩=\prod _{k=1}^p\left(e^{-i{\beta }_k{\hat{H}}_1}{e}^{-i{\gamma }_k{\hat{H}}_0}\right)|s⟩$$

• $$F_p(\mathit{\gamma },\mathit{\beta })=⟨\mathit{\gamma },\mathit{\beta }|{\hat{H}}_0|\mathit{\gamma },\mathit{\beta }⟩$$

2. QAからQAOAへ: 時間発展の離散化

• $$U(T)=\exp (-\frac{i}{\hslash }{\int }_0^{T}dt\hat{H}(t))$$

• $$U(T)\simeq \prod _{k=1}^n\exp (-\frac{i\mathrm{\Delta }t}{\hslash }{\hat{H}}_k)$$

• $${\hat{H}}_k=f_k{\hat{H}}_0+(1-f_k){\hat{H}}_1$$

• $$U(T)\simeq \prod _{k=1}^n\exp (-\frac{i\mathrm{\Delta }t}{\hslash }{f}_k{\hat{H}}_0)\exp (-\frac{i\mathrm{\Delta }t}{\hslash }(1-f_k){\hat{H}}_1)$$

• $${\gamma }_k=\frac{\mathrm{\Delta }t}{\hslash }{f}_k,{\beta }_k=\frac{\mathrm{\Delta }t}{\hslash }(1-f_k)$$

3. Lie-Trotter、Suzuki-Trotter、BCH

• $$e^{t(\hat{A}+\hat{B})}\simeq {\left(e^{\frac{t}{n}\hat{A}}{e}^{\frac{t}{n}\hat{B}}\right)}^n(n\to \mathrm{\infty })$$

• $$e^{t(\hat{A}+\hat{B})}\simeq {\left(e^{\frac{t}{2n}\hat{A}}{e}^{\frac{t}{n}\hat{B}}{e}^{\frac{t}{2n}\hat{A}}\right)}^n$$

• $$e^{t(\hat{A}+\hat{B})}=I+t(\hat{A}+\hat{B})+\frac{t^2}{2}({\hat{A}}^2+\hat{A}\hat{B}+\hat{B}\hat{A}+{\hat{B}}^2)+\cdots$$

• $$e^{t\hat{A}}{e}^{t\hat{B}}=I+t(\hat{A}+\hat{B})+t^2\hat{A}\hat{B}+\frac{t^2}{2}({\hat{A}}^2+{\hat{B}}^2)+\cdots$$

• $$e^{t\hat{A}}{e}^{t\hat{B}}-e^{t(\hat{A}+\hat{B})}=\frac{t^2}{2}[\hat{A},\hat{B}]+\cdots$$

• $$e^{t\hat{A}}{e}^{t\hat{B}}=\exp [t(\hat{A}+\hat{B})+\frac{t^2}{2}[\hat{A},\hat{B}]+O(t^3)]$$

4. Pontryagin原理とbang-bang制御

• $$\frac{dx}{dt}=f(x,g)$$

• $$F(x(T))$$

• $$J=F(x(T))+{\int }_0^T\lambda (t)[f(x,g)-\frac{dx}{dt}]dt$$

• $$P(x,\lambda ,g)=\lambda f(x,g)$$

• $$\frac{dx}{dt}=\frac{\partial P}{\partial \lambda },\frac{d\lambda }{dt}=-\frac{\partial P}{\partial x}$$

• $$\lambda (T)=-\mathrm{\nabla }F(x(T))$$

• $$g^{\ast }(t)=\arg \underset{g}{min}P(x^{\ast }(t),{\lambda }^{\ast }(t),g)$$

• $$F(T)=⟨\psi (T)|{\hat{H}}_0|\psi (T)⟩$$

• $$i\hslash \frac{d}{dt}|\psi (t)⟩=\hat{H}(t)|\psi (t)⟩$$

• $$|\chi (T)⟩\propto {\hat{H}}_0|\psi (T)⟩$$

• $$\hat{H}(t)=f(t){\hat{H}}_0+\{1-f(t)\}{\hat{H}}_1$$

• $$P(\psi ,\chi ,f)=⟨\chi |\hat{H}(t)|\psi ⟩+⟨\psi |\hat{H}(t)|\chi ⟩=2\mathrm{R}\mathrm{e}⟨\chi |\hat{H}(t)|\psi ⟩$$

• $$\frac{\partial P}{\partial f}=2\mathrm{R}\mathrm{e}⟨\chi (t)|{\hat{H}}_0-{\hat{H}}_1|\psi (t)⟩\equiv \varphi (t)$$

• $$\varphi (t)>0\Rightarrow f(t)=0,\varphi (t)<0\Rightarrow f(t)=1$$

5. Quantum Schrodinger Bridgeの形

• $$J={\int }_0^T\mathrm{T}\mathrm{r}[\hat{H}(t{)}^2]dt+{\int }_0^{T}dt\{⟨\chi |i\hat{H}+\frac{d}{dt}|\psi ⟩+⟨\psi |-i\hat{H}+\frac{d}{dt}|\chi ⟩\}$$

• $$\delta J=2\mathrm{T}\mathrm{r}[\hat{H}\delta \hat{H}]-2\mathrm{I}\mathrm{m}⟨\chi |\delta \hat{H}|\psi ⟩=0$$

• $$\hat{H}(t)=i(|\psi (t)⟩⟨\chi (t)|-|\chi (t)⟩⟨\psi (t)|)$$

• $$\frac{d\hat{H}}{dt}=0$$

6. 有限温度の量子アニーリング

• $$P(\mathit{\sigma })=\frac{1}{Z}\exp [-\beta E(\mathit{\sigma })],Z=\sum _{\mathit{\sigma }}\exp [-\beta E(\mathit{\sigma })]$$

• $$\hat{\rho }=\frac{1}{Z}\exp [-\beta \hat{H}],Z=\mathrm{T}\mathrm{r}\exp [-\beta \hat{H}]$$

• $$\hat{H}={\hat{H}}_0({\hat{\sigma }}^z)+{\hat{H}}_1({\hat{\sigma }}^x),{\hat{H}}_1=-\mathrm{\Gamma }\sum _i{\hat{\sigma }}_i^x$$

• $$Z=\mathrm{T}\mathrm{r}\exp [-\beta ({\hat{H}}_0+{\hat{H}}_1)]\simeq \mathrm{T}\mathrm{r}[\prod _{k=1}^M\exp (-\frac{\beta }{M}{\hat{H}}_0)\exp (-\frac{\beta }{M}{\hat{H}}_1)]$$

• $$I=\sum _{{\mathit{\sigma }}_k}|{\mathit{\sigma }}_k⟩⟨{\mathit{\sigma }}_k|$$

• $$\exp \left(\frac{\beta \mathrm{\Gamma }}{M}{\hat{\sigma }}^x\right)=\cosh \left(\frac{\beta \mathrm{\Gamma }}{M}\right)I+\sinh \left(\frac{\beta \mathrm{\Gamma }}{M}\right){\hat{\sigma }}^x$$

• $$⟨{\sigma }_{i,k}|e^{\frac{\beta \mathrm{\Gamma }}{M}{\hat{\sigma }}_i^x}|{\sigma }_{i,k+1}⟩=C{e}^{K{\sigma }_{i,k}{\sigma }_{i,k+1}}$$

• $$e^{-2K}=\tanh \left(\frac{\beta \mathrm{\Gamma }}{M}\right),K=-\frac{1}{2}\log \tanh \left(\frac{\beta \mathrm{\Gamma }}{M}\right)$$

• $$Z\propto \sum _{\{{\sigma }_{i,k}\}}\exp [-\frac{\beta }{M}\sum _{k=1}^M{H}_0({\mathit{\sigma }}_k)+K\sum _{i=1}^N\sum _{k=1}^M{\sigma }_{i,k}{\sigma }_{i,k+1}]$$

7. p-spin模型の平均場解析

• $${\hat{H}}_0=-\frac{J}{N^{p-1}}\sum _{i_1,\dots ,i_p}{\hat{\sigma }}_{i_1}^z\cdots {\hat{\sigma }}_{i_p}^z=-NJ{\left(\frac{1}{N}\sum _i{\hat{\sigma }}_i^z\right)}^p$$

• $$m=\frac{1}{N}\sum _i{\sigma }_i$$

• $$m_k=\frac{1}{N}\sum _i{\sigma }_{i,k}$$

• $$1=\int d{m}_k\delta (N{m}_k-\sum _i{\sigma }_{i,k})$$

• $$\delta (N{m}_k-\sum _i{\sigma }_{i,k})\propto \int d{\tilde{m}}_k\exp [-{\tilde{m}}_k(N{m}_k-\sum _i{\sigma }_{i,k})]$$

• $$Z\propto \int \prod _{k}d{m}_{k}d{\tilde{m}}_k\exp [N\frac{\beta J}{M}\sum _k{m}_k^p-N\sum _k{m}_k{\tilde{m}}_k]{\left(\sum _{\{{\sigma }_k\}}{e}^{\sum _k{\tilde{m}}_k{\sigma }_k+K\sum _k{\sigma }_k{\sigma }_{k+1}}\right)}^N$$

• $$m_k\to m,{\tilde{m}}_k\to \frac{\beta }{M}\tilde{m}$$

• $$\hat{A}=\frac{\beta }{M}\left(\begin{array}{cc}\tilde{m}& \mathrm{\Gamma }\\ \mathrm{\Gamma }& -\tilde{m}\end{array}\right)$$

• $${\lambda }_{\pm }=\pm \frac{\beta }{M}\sqrt{{\tilde{m}}^2+{\mathrm{\Gamma }}^2}$$

• $$\mathrm{T}\mathrm{r}{e}^{M\hat{A}}=2\cosh \left[\beta \sqrt{{\tilde{m}}^2+{\mathrm{\Gamma }}^2}\right]$$

• $$Z=\int dmd\tilde{m}\exp \left\{N[\beta J{m}^p-\beta m\tilde{m}+\log 2\cosh \left(\beta \sqrt{{\tilde{m}}^2+{\mathrm{\Gamma }}^2}\right)]\right\}$$

• $$I=\int dx{e}^{Nf(x)}\simeq e^{Nf(x^{\ast })},{\frac{\partial f}{\partial x}|}_{x=x^{\ast }}=0$$

• $$\tilde{m}=pJ{m}^{p-1}$$

• $$m=\frac{\tilde{m}}{\sqrt{{\tilde{m}}^2+{\mathrm{\Gamma }}^2}}\tanh \left[\beta \sqrt{{\tilde{m}}^2+{\mathrm{\Gamma }}^2}\right]$$

• $$\beta \to \mathrm{\infty }$$

• $$\mathrm{\Gamma }\to 0$$

7.1 non-stoquastic系としての反強磁性XX模型

• $$\hat{H}(s,\lambda )=s\{\lambda {\hat{H}}_0+(1-\lambda ){\hat{V}}_{\mathrm{X}\mathrm{X}}\}+(1-s){\hat{V}}_{\mathrm{T}\mathrm{F}}$$

• $$\begin{array}{rl}f& =(p-1)s\lambda (m^z{)}^p-s(1-\lambda )(m^x{)}^2\\ & -\frac{1}{\beta }\log 2\cosh \beta \sqrt{\{ps\lambda (m^z{)}^{p-1}{\}}^2+\{1-s-2s(1-\lambda )m^x{\}}^2}\end{array}$$

• $$m^z=\frac{h_z}{\sqrt{h_z^2+h_x^2}}\tanh \beta \sqrt{h_z^2+h_x^2},m^x=\frac{h_x}{\sqrt{h_z^2+h_x^2}}\tanh \beta \sqrt{h_z^2+h_x^2}$$

8. QAOAをp-spin模型で見る

• $$P_{\mathit{\alpha },\mathit{\gamma }}(\mathit{\sigma })={|⟨\mathit{\sigma }|\mathit{\alpha },\mathit{\gamma }⟩|}^2$$

• $$Z=\sum _{\mathit{\sigma }}\int d\mathit{\alpha }d\mathit{\gamma }\exp [-\beta E(\mathit{\sigma })]P_{\mathit{\alpha },\mathit{\gamma }}(\mathit{\sigma })$$

• $$|\alpha ,\gamma ⟩=e^{-i\alpha {\hat{H}}_1}{e}^{-i\gamma {\hat{H}}_0}|s⟩$$

• $$|\alpha ,\gamma ⟩={\left(\frac{1}{\sqrt{2}}\right)}^N\sum _{\mathit{\sigma }}{e}^{-i\alpha {\hat{H}}_1}{e}^{-i\gamma H_0(\mathit{\sigma })}|\mathit{\sigma }⟩$$

• $$e^{-i\gamma H_0(\mathit{\sigma })}=\int dmd\tilde{m}\exp [-iN\gamma J{m}^p-Nm\tilde{m}+\tilde{m}\sum _i{\sigma }_i]$$

• $$e^{-i\alpha {\hat{H}}_1}=\prod _i(\cos \alpha I+i\sin \alpha {\hat{\sigma }}_i^x)$$

• $$e^{\tilde{m}{\hat{\sigma }}^z}=\cosh \tilde{m}I+\sinh \tilde{m}{\hat{\sigma }}^z$$

• $$\hat{A}(\alpha )=\cos \alpha I+i\sin \alpha {\hat{\sigma }}^x,\hat{B}(\gamma ,\tilde{m},m)=e^{-i\gamma J{m}^p-m\tilde{m}}{e}^{\tilde{m}{\hat{\sigma }}^z}$$

• $$Z=\sum _{{\sigma }_0,{\sigma }_1,{\sigma }_2}\int d\gamma d\alpha \prod _{a=0}^{2}d{m}_{a}d{\tilde{m}}_a{e}^{N\mathrm{\Phi }(m_a,{\tilde{m}}_a;\alpha ,\gamma )}{[\mathrm{T}\mathrm{r}{\hat{B}}^{†}(\gamma ,{\tilde{m}}_2,m_2){\hat{A}}^{†}(\alpha )e^{{\tilde{m}}_0{\hat{\sigma }}^z}\hat{A}(\alpha )\hat{B}(\gamma ,{\tilde{m}}_1,m_1)]}^N$$

• $$N\beta J{m}_0^p-iN\gamma J{m}_1^p+iN\gamma J{m}_2^p-N\sum _{a=0}^2{m}_a{\tilde{m}}_a$$

3日目: 拡散モデルと量子アニーリング

1. 実時間から虚時間へ

• $$\hat{H}(x,t)=-D\frac{{\partial }^2}{\partial x^2}+V(x,t)$$

• $$i\hslash \frac{\partial }{\partial t}\psi (x,t)=\hat{H}(x,t)\psi (x,t)$$

• $$t=-i\tau$$

• $$-\frac{\partial }{\partial \tau }\psi (x,\tau )=\hat{H}(x,\tau )\psi (x,\tau )$$

• $$\psi (x,\tau )=\exp [-{\int }_0^{\tau }d{\tau }^{\prime }\hat{H}(x,{\tau }^{\prime })]\psi (x,0)$$

• $$-\frac{\partial }{\partial \tau }\psi =\hat{H}\psi \mathrm{f}\mathrm{o}\mathrm{r}\mathrm{w}\mathrm{a}\mathrm{r}\mathrm{d}$$

• $$-\frac{\partial }{\partial \tau }\tilde{\psi }=-\hat{H}\tilde{\psi }\mathrm{b}\mathrm{a}\mathrm{c}\mathrm{k}\mathrm{w}\mathrm{a}\mathrm{r}\mathrm{d},\mathrm{a}\mathrm{d}\mathrm{j}\mathrm{o}\mathrm{i}\mathrm{n}\mathrm{t}$$

• $$\frac{\partial }{\partial \tau }\tilde{\psi }=\hat{H}\tilde{\psi }$$

• $$P(x,\tau )=\psi (x,\tau )\tilde{\psi }(x,\tau )$$

2. 確率密度から確率流を作る

• $$\frac{\partial P}{\partial \tau }=\frac{\partial \psi }{\partial \tau }\tilde{\psi }+\psi \frac{\partial \tilde{\psi }}{\partial \tau }$$

• $$\frac{\partial P}{\partial \tau }=D(\frac{{\partial }^2\psi }{\partial x^2}\tilde{\psi }-\psi \frac{{\partial }^2\tilde{\psi }}{\partial x^2})$$

• $$\frac{\partial P}{\partial \tau }=-\frac{\partial }{\partial x}J(x,\tau )$$

• $$J(x,\tau )=-D\frac{\partial P}{\partial x}+f(x,\tau )P(x,\tau )$$

• $$\frac{\partial P}{\partial \tau }=-\frac{\partial }{\partial x}[f(x,\tau )P(x,\tau )]+D\frac{{\partial }^{2}P}{\partial x^2}$$

• $$dx=f(x,\tau )d\tau +\sqrt{2D}d{W}_{\tau }$$

• $$f(x)=-\frac{\partial V(x)}{\partial x}$$

• $$J(x)=0$$

• $$P_{\mathrm{e}\mathrm{q}}(x)\propto \exp [-\frac{1}{D}V(x)]$$

• $$\frac{\partial P}{\partial t}=0$$

• $$J=0$$

3. forward FPとbackward FP

• $$\psi =\frac{P}{\tilde{\psi }}$$

• $$\frac{\partial \psi }{\partial x}=\frac{1}{\tilde{\psi }}\frac{\partial P}{\partial x}-\frac{P}{{\tilde{\psi }}^2}\frac{\partial \tilde{\psi }}{\partial x}$$

• $$\frac{\partial P}{\partial \tau }=D\frac{{\partial }^{2}P}{\partial x^2}-2D\frac{\partial }{\partial x}[P(x,\tau )\frac{\partial }{\partial x}\log \tilde{\psi }(x,\tau )]$$

• $$f_{+}(x,\tau )=2D\frac{\partial }{\partial x}\log \tilde{\psi }(x,\tau )$$

• $$\tilde{\psi }=\frac{P}{\psi }$$

• $$\frac{\partial P}{\partial \tau }=-D\frac{{\partial }^{2}P}{\partial x^2}-2D\frac{\partial }{\partial x}[P(x,\tau )\frac{\partial }{\partial x}\{-\log \psi (x,\tau )\}]$$

• $$f_{-}(x,\tau )=2D\frac{\partial }{\partial x}\log \psi (x,\tau )$$

• $$dx=f_{-}(x,\tau )d\tau +\sqrt{2D}dW(d\tau <0)$$

• $$dy=f_{-}(y,T-s)ds+\sqrt{2D}dW$$

4. SUSY量子力学として見る

• $$(-\frac{{\partial }^2}{\partial x^2}+V_{\mathrm{q}}(x))\psi (x)=E\psi (x)$$

• $$\hat{A}=\frac{\partial }{\partial x}+\frac{1}{2}{V}^{\prime }(x),{\hat{A}}^{†}=-\frac{\partial }{\partial x}+\frac{1}{2}{V}^{\prime }(x)$$

• $$\hat{A}{\psi }_0(x)=0$$

• $$\frac{\partial {\psi }_0}{\partial x}=-\frac{1}{2}{V}^{\prime }(x){\psi }_0(x)$$

• $${\psi }_0(x)\propto \exp [-\frac{1}{2}V(x)]$$

• $${\hat{A}}^{†}\hat{A}=(-\frac{\partial }{\partial x}+\frac{1}{2}{V}^{\prime }(x))(\frac{\partial }{\partial x}+\frac{1}{2}{V}^{\prime }(x))$$

• $${\hat{A}}^{†}\hat{A}=-\frac{{\partial }^2}{\partial x^2}+\frac{1}{4}\{V^{\prime }(x){\}}^2-\frac{1}{2}{V}^{″}(x)\equiv {\hat{H}}_{\mathrm{S}\mathrm{U}\mathrm{S}\mathrm{Y}}$$

• $${\hat{H}}_{\mathrm{S}\mathrm{U}\mathrm{S}\mathrm{Y}}={\hat{A}}^{†}\hat{A}$$

• $${\hat{H}}_{\mathrm{S}\mathrm{U}\mathrm{S}\mathrm{Y}}{\psi }_0(x)=0,{\psi }_0(x)\propto \exp [-\frac{1}{2}V(x)]$$

• $$V(x)=\frac{1}{2}{x}^2$$

• $$V^{\prime }(x)=x$$

5. 分布を基底状態として持つハミルトニアン

• $${\psi }_t(x)=\sqrt{P_t(x)}$$

• $$P_t(x)={\psi }_t(x){\tilde{\psi }}_t(x)$$

• $$V_t(x)=-\log P_t(x)$$

• $${\psi }_t(x)=\exp [-\frac{1}{2}{V}_t(x)]=e^{\frac{1}{2}\log P_t(x)}=\sqrt{P_t(x)}$$

• $${\hat{H}}_{\mathrm{S}\mathrm{U}\mathrm{S}\mathrm{Y}}(t)=-\frac{{\partial }^2}{\partial x^2}+\frac{1}{4}\{V_t^{\prime }(x){\}}^2-\frac{1}{2}{V}_t^{″}(x)$$

• $${\hat{H}}_{\mathrm{S}\mathrm{U}\mathrm{S}\mathrm{Y}}(t)\sqrt{P_t(x)}=0$$

6. 拡散モデルのforward過程

• $$P_0(x):\mathrm{デ}\mathrm{ー}\mathrm{タ}\mathrm{分}\mathrm{布}⟶P_T(x):\mathrm{作}\mathrm{り}\mathrm{や}\mathrm{す}\mathrm{い}\mathrm{分}\mathrm{布}$$

• $$dx=f(x,t)dt+\sqrt{2D_t}d{W}_t$$

• $$\frac{\partial P_t}{\partial t}=-\frac{\partial }{\partial x}[f(x,t)P_t(x)]+D_t\frac{{\partial }^2{P}_t}{\partial x^2}$$

7. scoreを学ぶ

• $$S(x,t)=\frac{\partial }{\partial x}\log P(x,t)$$

• $$\log P(x,t)=\log \psi (x,t)+\log \tilde{\psi }(x,t)$$

• $$S(x,t)=\frac{\partial }{\partial x}\log \psi (x,t)+\frac{\partial }{\partial x}\log \tilde{\psi }(x,t)$$

• $$f_{+}(x,t)=2D\frac{\partial }{\partial x}\log \tilde{\psi }(x,t)$$

• $$f_{+}(x,t)=f(x,t)$$

• $$f_{-}(x,t)=2D\frac{\partial }{\partial x}\log \psi (x,t)$$

• $$f_{+}(x,t)+f_{-}(x,t)=2D\frac{\partial }{\partial x}\log P(x,t)=2DS(x,t)$$

• $$f_{-}(x,t)=-f_{+}(x,t)+2D\frac{\partial }{\partial x}\log P(x,t)$$

• $$dx=[f(x,t)-2D\frac{\partial }{\partial x}\log P(x,t)]dt+\sqrt{2D}d{W}_t$$

• $$-2D\frac{\partial }{\partial x}\log P(x,t)$$

8. 反断熱項としてのscore補正

• $${\mathrm{\nabla }}_x\log P_t(x)$$

• $$d{X}_t=f(X_t,t)dt+g(t)d{W}_t$$

• $$d{X}_t=[f(X_t,t)-g(t{)}^2{\mathrm{\nabla }}_x\log p_t(X_t)]dt+g(t)d{\overline{W}}_t$$

• $$S_t(x)={\mathrm{\nabla }}_x\log p_t(x),{\varphi }_t(x)=\sqrt{p_t(x)}$$

• $${\hat{H}}_{\mathrm{s}\mathrm{c}}(t){\varphi }_t=0,{\hat{H}}_{\mathrm{s}\mathrm{c}}(t)=-D{\mathrm{\nabla }}^2+D\{\frac{1}{2}\mathrm{\nabla }\cdot S_t(x)+\frac{1}{4}|S_t(x){|}^2\}$$

• $$p_t(x)={\tilde{\psi }}_t(x){\psi }_t(x)$$

• $${\partial }_t{\psi }_t=-({\hat{H}}_{\tilde{\psi },t}+G_t^{\mathrm{E}\mathrm{C}\mathrm{D}}){\psi }_t,G_t^{\mathrm{E}\mathrm{C}\mathrm{D}}={\partial }_t\log {\tilde{\psi }}_t$$

• $$Q_t{G}_t^{\mathrm{E}\mathrm{C}\mathrm{D}}|{\tilde{\varphi }}_t⟩=Q_t|{\partial }_t{\tilde{\varphi }}_t⟩,Q_t=1-|{\tilde{\varphi }}_t⟩⟨{\tilde{\varphi }}_t|$$

• $${\hat{H}}_{\mathrm{C}\mathrm{D}}^{(\tilde{\psi })}=i(Q_t{G}_t^{\mathrm{E}\mathrm{C}\mathrm{D}}{\mathrm{\Pi }}_t-{\mathrm{\Pi }}_t{G}_t^{\mathrm{E}\mathrm{C}\mathrm{D}}{Q}_t)=i[{\dot{\mathrm{\Pi }}}_t,{\mathrm{\Pi }}_t],{\mathrm{\Pi }}_t=|{\tilde{\varphi }}_t⟩⟨{\tilde{\varphi }}_t|$$

• $${\lambda }_r(s)=1-(1-s{)}^{r+1},{\frac{d^j{\lambda }_r}{d{s}^j}|}_{s=1}=0(j=1,\dots ,r)$$

• $$S_{\lambda }^{\theta }=S_{\lambda }+{ϵ}_{\lambda }$$

• $$\delta {\hat{H}}_{\lambda }=D(\frac{1}{2}\mathrm{\nabla }\cdot {ϵ}_{\lambda }+\frac{1}{2}{S}_{\lambda }\cdot {ϵ}_{\lambda }+\frac{1}{4}|{ϵ}_{\lambda }{|}^2)$$

• $$\parallel Q_1{\varphi }_1{\parallel }^2\lesssim \frac{C_r}{T^{2r+2}}+C_{ϵ}{\eta }_{ϵ}^2$$

• $$\frac{ds}{d\lambda }\propto M(\lambda )\mathrm{\Delta }(\lambda {)}^{-2}(1-\lambda {)}^{-r/(r+1)}$$

補遺: アハラノフの2ベクトル形式からscoreを見る

1. 2つの境界条件で現在を見る

• $$|\psi (t)⟩=U(t,t_i)|{\psi }_i⟩$$

• $$⟨\varphi (t)|=⟨{\varphi }_f|U(t_f,t)$$

• $$⟨\varphi (t)\Vert \psi (t)⟩$$

• $$P(x,t)=\psi (x,t)\tilde{\psi }(x,t)$$

• $$\psi (x,t)=⟨x|\psi (t)⟩,\tilde{\psi }(x,t)=⟨\varphi (t)|x⟩$$

• $$q(x,t)=⟨\varphi (t)|x⟩⟨x|\psi (t)⟩=\tilde{\psi }(x,t)\psi (x,t)$$

• $$({\mathrm{\Pi }}_x{)}_{\mathrm{w}}=\frac{⟨\varphi (t)|x⟩⟨x|\psi (t)⟩}{⟨\varphi (t)|\psi (t)⟩}=\frac{\tilde{\psi }(x,t)\psi (x,t)}{⟨\varphi (t)|\psi (t)⟩}$$

2. 弱値として現在の量を見る

• $$A_{\mathrm{w}}(t)=\frac{⟨\varphi (t)|\hat{A}|\psi (t)⟩}{⟨\varphi (t)|\psi (t)⟩}$$

• $$p_{\mathrm{R}}^{\mathrm{w}}(x,t)=\frac{⟨x|\hat{p}|\psi (t)⟩}{⟨x|\psi (t)⟩}=-i\hslash \frac{\partial }{\partial x}\log \psi (x,t)$$

• $$p_{\mathrm{L}}^{\mathrm{w}}(x,t)=\frac{⟨\varphi (t)|\hat{p}|x⟩}{⟨\varphi (t)|x⟩}=i\hslash \frac{\partial }{\partial x}\log \tilde{\psi }(x,t)$$

3. scoreは2つの局所運動量のずれである

• $$S(x,t)=\frac{\partial }{\partial x}\log P(x,t)$$

• $$S(x,t)=\frac{\partial }{\partial x}\log \psi (x,t)+\frac{\partial }{\partial x}\log \tilde{\psi }(x,t)$$

• $$\boxed{{\displaystyle S(x,t)=\frac{i}{\hslash }[p_{\mathrm{R}}^{\mathrm{w}}(x,t)-p_{\mathrm{L}}^{\mathrm{w}}(x,t)]}}$$

• $$\boxed{{\displaystyle {\mathrm{\nabla }}_{\mathit{x}}\log P(\mathit{x},t)=\frac{i}{\hslash }[{\mathit{p}}_{\mathrm{R}}^{\mathrm{w}}(\mathit{x},t)-{\mathit{p}}_{\mathrm{L}}^{\mathrm{w}}(\mathit{x},t)]}}$$

• $${\pi }_{\mathrm{R}}(x,t)=-\hslash \frac{\partial }{\partial x}\log \psi (x,t),{\pi }_{\mathrm{L}}(x,t)=\hslash \frac{\partial }{\partial x}\log \tilde{\psi }(x,t)$$

• $$S(x,t)=\frac{1}{\hslash }[{\pi }_{\mathrm{L}}(x,t)-{\pi }_{\mathrm{R}}(x,t)]$$

4. scoreはゲージに依らない

• $$\psi (x,t)\to e^{\chi (x,t)}\psi (x,t),\tilde{\psi }(x,t)\to e^{-\chi (x,t)}\tilde{\psi }(x,t)$$

• $$P(x,t)=\psi (x,t)\tilde{\psi }(x,t)$$

• $$\frac{\partial }{\partial x}\log \psi +\frac{\partial }{\partial x}\log \tilde{\psi }=\frac{\partial }{\partial x}\log P$$

• $$p_{\mathrm{R}}^{\mathrm{w}}-p_{\mathrm{L}}^{\mathrm{w}}$$

5. 反断熱項としての意味

• $$f_{+}(x,t)=2D\frac{\partial }{\partial x}\log \tilde{\psi }(x,t),f_{-}(x,t)=2D\frac{\partial }{\partial x}\log \psi (x,t)$$

• $$f_{+}(x,t)+f_{-}(x,t)=2DS(x,t)$$

• $$2DS(x,t)$$

• $$p_{\mathrm{R}}^{\mathrm{w}}(x,t)-p_{\mathrm{L}}^{\mathrm{w}}(x,t)$$

6. まとめ

• $$\mathrm{\nabla }\log P=\frac{i}{\hslash }({\mathit{p}}_{\mathrm{R}}^{\mathrm{w}}-{\mathit{p}}_{\mathrm{L}}^{\mathrm{w}})$$

補遺: ハバード・ストラトノビッチ変換と大関法

1. 二乗を線形にほどく発想

• $$f(\mathit{q})=f_0(\mathit{q})+\frac{1}{2}\sum _k{\lambda }_k{\{F_k(\mathit{q})-C_k\}}^2$$

• $${(\sum _i{q}_i-K)}^2=\sum _i{q}_i^2+2\sum _{i<j}{q}_i{q}_j-2K\sum _i{q}_i+K^2$$

• $$\exp (-\frac{a}{2}{x}^2)=\int Dz\exp \left(i\sqrt{a}zx\right),Dz=\frac{dz}{\sqrt{2\pi }}\exp (-\frac{z^2}{2})$$

2. 制約ペナルティへの適用

• $$Z=\sum _{\mathit{q}}\exp \{-\beta f(\mathit{q})\}$$

• $$Z=\sum _{\mathit{q}}\prod _k\int D{z}_k\exp [i\sum _k\sqrt{\beta {\lambda }_k}{z}_k\{F_k(\mathit{q})-C_k\}-\beta f_0(\mathit{q})]$$

• $$z_k\to -i\sqrt{\frac{\beta }{{\lambda }_k}}{\nu }_k$$

• $$Z\propto \sum _{\mathit{q}}\prod _k\int d{\nu }_k\exp [\sum _k\frac{\beta }{2{\lambda }_k}{\nu }_k^2+\beta \sum _k{\nu }_k\{F_k(\mathit{q})-C_k\}-\beta f_0(\mathit{q})]$$

• $$H(\mathit{q},\mathit{\nu })=f_0(\mathit{q})-\sum _k{\nu }_k\{F_k(\mathit{q})-C_k\}-\sum _k\frac{{\nu }_k^2}{2{\lambda }_k}$$

3. 固定した補助場でアニーラを使う

• $$H_{\mathrm{e}\mathrm{f}\mathrm{f}}(\mathit{q};\mathit{\nu })=f_0(\mathit{q})-\sum _k{\nu }_k{F}_k(\mathit{q})$$

• $$F_k(\mathit{q})=\sum _i{a}_{ki}{q}_i$$

4. 期待値が制約に合うように更新する

• $$Z(\mathit{\nu })=\sum _{\mathit{q}}\exp [-\beta f_0(\mathit{q})+\beta \sum _k{\nu }_k{F}_k(\mathit{q})]$$

• $$Q_{\mathit{\nu }}(\mathit{q})=\frac{1}{Z(\mathit{\nu })}\exp [-\beta f_0(\mathit{q})+\beta \sum _k{\nu }_k{F}_k(\mathit{q})]$$

• $$⟨F_k(\mathit{q}){⟩}_{Q_{\mathit{\nu }}}=C_k$$

• $$G(\mathit{C})=\underset{\mathit{\nu }}{max}\{\sum _k{\nu }_k{C}_k-\log Z(\mathit{\nu })\}$$

• $${\nu }_k^{(t+1)}={\nu }_k^{(t)}+\eta [C_k-⟨F_k(\mathit{q}){⟩}_{Q_t}]$$

5. 一番小さい例: K個だけ選ぶ

• $$f(\mathit{q})=\sum _{i=1}^N{h}_i{q}_i+\frac{\lambda }{2}{(\sum _{i=1}^N{q}_i-K)}^2$$

• $$H_{\mathrm{e}\mathrm{f}\mathrm{f}}(\mathit{q};\nu )=\sum _{i=1}^N(h_i-\nu )q_i$$

• $$f(\mathit{\sigma })=\frac{1}{2}{\left(\sum _{i=1}^N{n}_i{\sigma }_i\right)}^2,{\sigma }_i=\pm 1$$

• $$H_{\mathrm{e}\mathrm{f}\mathrm{f}}(\mathit{\sigma };\nu )=-\nu \sum _{i=1}^N{n}_i{\sigma }_i$$

6. 交通流最適化での読み方

• $$\sum _e{\left(\sum _{\mu ,i}{S}_{\mu ,i,e}{q}_{\mu ,i}\right)}^2$$

• $$\sum _e{\left(\sum _{\mu ,i}{S}_{\mu ,i,e}{q}_{\mu ,i}\right)}^2⟶-\sum _e{\nu }_e\left(\sum _{\mu ,i}{S}_{\mu ,i,e}{q}_{\mu ,i}\right)$$

• $$h_{\mu ,i}=\sum _e{\nu }_e{S}_{\mu ,i,e}$$

補遺: シミュレーテッドアニーリング

1. 最適化問題をBoltzmann分布に埋め込む

• $${\pi }_T(x)=\frac{1}{Z_T}\exp (-\frac{E(x)}{k_{\mathrm{B}}T}),Z_T=\sum _{x\in \mathrm{\Omega }}\exp (-\frac{E(x)}{k_{\mathrm{B}}T})$$

• $$\underset{T\to 0}{lim}{\pi }_T(x)=\{\begin{array}{ll}1/|\mathcal{G}|,& x\in \mathcal{G},\\ 0,& x\notin \mathcal{G}.\end{array}$$

2. MCMCと釣り合い条件

• $${\mu }_{t+1}(y)=\sum _{x\in \mathrm{\Omega }}{\mu }_t(x)P_T(x,y)$$

• $${\pi }_T(y)=\sum _{x\in \mathrm{\Omega }}{\pi }_T(x)P_T(x,y)$$

• $$\sum _{y\in \mathrm{\Omega }}[{\pi }_T(x)P_T(x,y)-{\pi }_T(y)P_T(y,x)]=0$$

3. 詳細釣り合い条件

• $${\pi }_T(x)P_T(x,y)={\pi }_T(y)P_T(y,x)(\mathrm{\forall }x,y)$$

• $$J_T(x,y)={\pi }_T(x)P_T(x,y)-{\pi }_T(y)P_T(y,x)$$

• $$\sum _y{J}_T(x,y)=0$$

4. Metropolis法

• $$A_T(x,y)=min\{1,\exp [-\beta (E(y)-E(x))]\}$$

• $$P_T(x,y)=q(x,y)A_T(x,y)(x\ne y)$$

• $$A_T(x,y)=min\{1,\frac{{\pi }_T(y)q(y,x)}{{\pi }_T(x)q(x,y)}\}$$

5. 熱浴法

• $$\underset{T}{Pr}({\sigma }_i=s\mid {\mathit{\sigma }}_{\setminus i})=\frac{\exp [-\beta E({\sigma }_i=s,{\mathit{\sigma }}_{\setminus i})]}{\sum _{s^{\prime }=\pm 1}\exp [-\beta E({\sigma }_i=s^{\prime },{\mathit{\sigma }}_{\setminus i})]}(s=\pm 1)$$

• $$\underset{T}{Pr}({\sigma }_i=+1\mid {\mathit{\sigma }}_{\setminus i})=\frac{1}{1+\exp (-2\beta h_i^{\mathrm{e}\mathrm{f}\mathrm{f}})}$$

6. SAのアルゴリズム

• $${\mu }_{t+1}={\mu }_t{P}_{T_t},T_0>T_1>T_2>\cdots$$

• $$T_{t+1}=r{T}_t,0<r<1$$

7. Geman--Gemanの定理

• $$d(x)=\underset{y:E(y)<E(x)}{min}[\mathrm{\Phi }(x,y)-E(x)]$$

• $$d_{\ast }=\underset{x\notin \mathcal{G}}{max}d(x)$$

• $$T_t=\frac{c}{\log (t+t_0)}$$

• $$c>d_{\ast }⟹Pr(x_t\in \mathcal{G})\to 1(t\to \mathrm{\infty })$$

• $$\exp (-\frac{d}{k_{\mathrm{B}}{T}_t})=t^{-d/(k_{\mathrm{B}}c)}$$

8. 古典分布を量子基底状態に写す

• $$|\psi (T)⟩=\sum _{x\in \mathrm{\Omega }}\sqrt{{\pi }_T(x)}|x⟩=\frac{1}{\sqrt{Z_T}}\sum _{x\in \mathrm{\Omega }}\exp (-\frac{\beta E(x)}{2})|x⟩$$

• $${\hat{H}}_q(T)=I-S_T$$

• $${\mathrm{\Delta }}_q(T)={\mathrm{\Delta }}_{\mathrm{M}\mathrm{C}}(T)$$

9. 断熱定理からGeman--Geman条件を見る

• $$\left|\frac{⟨m(T)|{\partial }_T{\hat{H}}_q(T)|\psi (T)⟩}{{\mathrm{\Delta }}_m(T{)}^2}\frac{dT}{dt}\right|\ll 1$$

• $$T(t)\simeq \frac{pN}{k_{\mathrm{B}}\log (\alpha t+1)}$$

• $$\text{SAの準平衡追随}⟺\text{写像された量子系の断熱追随}$$

10. QAとの距離感

• $${\hat{H}}_{\mathrm{Q}\mathrm{A}}(\mathrm{\Gamma })=\hat{H}-\mathrm{\Gamma }\sum _i{\hat{\sigma }}_i^x$$

• $$\text{SA: }{\hat{H}}_q(T)\text{ に沿って }T\downarrow 0,\text{QA: }{\hat{H}}_{\mathrm{Q}\mathrm{A}}(\mathrm{\Gamma })\text{ に沿って }\mathrm{\Gamma }\downarrow 0$$

• $$\gamma (t)\simeq {[(2N-1)\overline{\alpha }t]}^{-1/(2N-1)}$$

11. 拡張QAを式で見る

• $$S_{\beta }={\mathrm{\Pi }}_{\beta }^{1/2}{P}_{\beta }{\mathrm{\Pi }}_{\beta }^{-1/2},(S_{\beta }{)}_{xy}=\sqrt{\frac{{\pi }_{\beta }(x)}{{\pi }_{\beta }(y)}}{P}_{\beta }(x,y)$$

• $$(S_{\beta }{)}_{yx}=\sqrt{\frac{{\pi }_{\beta }(y)}{{\pi }_{\beta }(x)}}{P}_{\beta }(y,x)=\sqrt{\frac{{\pi }_{\beta }(x)}{{\pi }_{\beta }(y)}}{P}_{\beta }(x,y)=(S_{\beta }{)}_{xy}$$

• $$|{\psi }_{\beta }⟩=\frac{1}{\sqrt{Z_{\beta }}}{e}^{-\beta \hat{H}/2}\sum _{\mathit{\sigma }}|\mathit{\sigma }⟩$$

• $${\hat{\sigma }}_j^x{e}^{-\beta \hat{H}/2}{\hat{\sigma }}_j^x=e^{-\beta ({\hat{H}}_{\overline{j}}-{\hat{H}}_j)/2}=e^{\beta {\hat{H}}_j}{e}^{-\beta \hat{H}/2}$$

• $${\hat{H}}_q(\beta )=-\chi \sum _j{\hat{H}}_q^j(\beta )=\chi \sum _j(e^{\beta {\hat{H}}_j}-{\hat{\sigma }}_j^x)$$

• $${\tilde{H}}_q(\gamma )=\chi \sum _j{e}^{\beta {\hat{H}}_j}-\gamma \sum _j{\hat{\sigma }}_j^x$$

• $${\tilde{H}}_q(\chi )=\chi \sum _j(e^{\beta {\hat{H}}_j}-{\hat{\sigma }}_j^x)={\hat{H}}_q(\beta )$$

• $$\mathrm{\Delta }(\gamma )\ge 2\sqrt{2\pi N}{e}^{-N}(1+c{)}^{-N}{\gamma }^N$$

• $$\overline{E}[\sigma ]=\frac{\tilde{\beta }}{L}\sum _{k=1}^L\sum _{ij}{\tilde{J}}_{ij}(\beta ){\sigma }_{ik}{\sigma }_{jk}+\xi (\beta ,t)\sum _{k=1}^L\sum _i{\sigma }_{ik}{\sigma }_{i,k+1}$$

12. 平衡分布への追従とMarkov連鎖の緩和

• $$m_{ij}{\overline{\rho }}_j=w_{ij}{e}^{\beta (E_j-E_i)/2}\frac{e^{-\beta E_j}}{Z}=\frac{w_{ij}}{Z}{e}^{-\beta (E_i+E_j)/2}=m_{ji}{\overline{\rho }}_i$$

• $${\psi }_i^{(\alpha )}=\frac{2}{1-\alpha }{\rho }_i^{(1-\alpha )/2}$$

• $${\overline{h}}_{ij}^{(-\alpha )}=\{\begin{array}{ll}{w}_{ij}{e}^{\alpha \beta (E_i-E_j)/2},& (ij)\in L,\\ -{\displaystyle \sum _{k:(ik)\in L}{w}_{ki}{e}^{\beta (E_i-E_k)/2},}& i=j,\\ 0,& \text{otherwise}\end{array}$$

• $${\overline{h}}_{ij}^{(0)}=w_{ij}=w_{ji}={\overline{h}}_{ji}^{(0)}((ij)\in L)$$

• $$\delta {\dot{\mathit{\psi }}}^{(\alpha )}={\overline{H}}^{(-\alpha )}\delta {\mathit{\psi }}^{(\alpha )}$$

• $$\delta r_0{e}^{-\beta N(d+d^{\prime })/2}{t}^{-\gamma }>C{e}^{-\beta g/2}\dot{\beta }$$

• $$T(t)=\beta (t{)}^{-1}>\frac{N(d+d^{\prime })}{2\log t}$$

補遺: 量子アニーリングのシミュレーション手法

• $$\hat{H}(s)=A(s){\hat{H}}_{\mathrm{D}}+B(s){\hat{H}}_{\mathrm{P}}$$

• $$\mathrm{T}\mathrm{T}\mathrm{S}=t_{\mathrm{r}\mathrm{u}\mathrm{n}}\frac{\log (1-p_{\mathrm{t}\mathrm{a}\mathrm{r}\mathrm{g}})}{\log (1-p_{\mathrm{s}})}$$

• $$Z_M=\sum _{\{{\sigma }_{i,k}\}}\exp [-\frac{\beta }{M}\sum _{k=1}^M{E}_{\mathrm{P}}({\mathit{\sigma }}_k)+K\sum _{i,k}{\sigma }_{i,k}{\sigma }_{i,k+1}]$$

• $$K=\frac{1}{2}\log \coth \left(\frac{\beta \mathrm{\Gamma }}{M}\right)$$

• $$Z=\sum _{\mathcal{W}}{\mathrm{\Gamma }}^{K(\mathcal{W})}\exp [-{\int }_0^{\beta }{E}_{\mathrm{P}}(\mathit{\sigma }(\tau ))d\tau ]$$

• $$w_i[{\sigma }_i(\tau )\mid \{{\sigma }_j(\tau ){\}}_{j\ne i}]\propto {\mathrm{\Gamma }}^{K_i}\exp [-{\int }_0^{\beta }{b}_i(\tau ){\sigma }_i(\tau )d\tau ]$$

• $$\begin{array}{rl}{P}_{\mathrm{a}\mathrm{c}\mathrm{c}}(\mathcal{W}\to {\mathcal{W}}^{\prime })& =min\{1,R(\mathcal{W}\to {\mathcal{W}}^{\prime })\},\\ R(\mathcal{W}\to {\mathcal{W}}^{\prime })& ={\mathrm{\Gamma }}^{\mathrm{\Delta }K}\exp [-\mathrm{\Delta }{S}_{\mathrm{P}}]\frac{q({\mathcal{W}}^{\prime }\to \mathcal{W})}{q(\mathcal{W}\to {\mathcal{W}}^{\prime })}\end{array}$$

• $$⟨{\sigma }_i^z⟩=⟨\frac{1}{\beta }{\int }_0^{\beta }{\sigma }_i(\tau )d\tau ⟩,⟨\sum _i{\sigma }_i^x⟩=⟨\frac{K(\mathcal{W})}{\beta \mathrm{\Gamma }}⟩$$

• $$E_{\mathrm{S}\mathrm{V}\mathrm{M}\mathrm{C}}=-A(s)\sum _i\sin {\theta }_i+B(s)(\sum _i{h}_i\cos {\theta }_i+\sum _{i<j}{J}_{ij}\cos {\theta }_i\cos {\theta }_j)$$

• $$\mathrm{\Delta }{E}_i=-A(s)(\sin {\theta }_i^{\prime }-\sin {\theta }_i)+B(s)(h_i+\sum _{j(\ne i)}{J}_{ij}\cos {\theta }_j)(\cos {\theta }_i^{\prime }-\cos {\theta }_i)$$

• $$Pr({\sigma }_i^{\mathrm{o}\mathrm{u}\mathrm{t}}=+1\mid {\theta }_i)=\frac{1+\cos {\theta }_i}{2},Pr({\sigma }_i^{\mathrm{o}\mathrm{u}\mathrm{t}}=-1\mid {\theta }_i)=\frac{1-\cos {\theta }_i}{2}$$

補遺: 適応的量子モンテカルロ法

1. 標準的な横磁場QMCを思い出す

• $${\hat{m}}_x=\frac{1}{N}\sum _{i=1}^N{\hat{\sigma }}_i^x$$

• $$\hat{H}=H_0({\hat{\mathit{\sigma }}}^z)-\mathrm{\Gamma }\sum _{i=1}^N{\hat{\sigma }}_i^x$$

• $$Z=\mathrm{Tr}\exp (-\beta \hat{H})$$

• $$Z\simeq \sum _{\{{\sigma }_{i,t}\}}\exp [-\frac{\beta }{M}\sum _{t=1}^M{H}_0({\mathit{\sigma }}_t)+B(\mathrm{\Gamma })\sum _{i=1}^N\sum _{t=1}^M{\sigma }_{i,t}{\sigma }_{i,t+1}]$$

• $$B(\mathrm{\Gamma })=\frac{1}{2}\log \coth \left(\frac{\beta \mathrm{\Gamma }}{M}\right)$$

2. non-stoquasticな量子揺らぎ

• $$\hat{H}=H_0({\hat{\mathit{\sigma }}}^z)-Nf({\hat{m}}_x)$$

• $$f(m_x)=\mathrm{\Gamma }{m}_x$$

• $$f(m_x)=\mathrm{\Gamma }{m}_x-\frac{\gamma }{2}{m}_x^2$$

• $$-Nf({\hat{m}}_x)=-\mathrm{\Gamma }\sum _i{\hat{\sigma }}_i^x+\frac{\gamma }{2N}{\left(\sum _i{\hat{\sigma }}_i^x\right)}^2$$

3. 補助変数で横磁化を取り出す

• $$1=\int d{m}_{x,t}\delta (m_{x,t}-\frac{1}{N}\sum _{i=1}^N{\hat{\sigma }}_{i,t}^x)$$

• $$\delta (y)={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\frac{d\tilde{m}}{2\pi }\exp \left(i\tilde{m}y\right)$$

• $$\frac{\beta N}{M}[f(m_{x,t})-{\tilde{m}}_{x,t}{m}_{x,t}]+\frac{\beta {\tilde{m}}_{x,t}}{M}\sum _i{\hat{\sigma }}_{i,t}^x$$

• $${\tilde{m}}_{x,t}=f^{\prime }(m_{x,t})$$

• $$m_{x,t}=m_x,{\tilde{m}}_{x,t}={\tilde{m}}_x$$

• $${\hat{H}}_{\mathrm{e}\mathrm{f}\mathrm{f}}(m_x)=H_0({\hat{\mathit{\sigma }}}^z)-f^{\prime }(m_x)\sum _{i=1}^N{\hat{\sigma }}_i^x$$

4. 条件付き分布として普通のQMCを走らせる

• $${\mathrm{\Gamma }}_{\mathrm{e}\mathrm{f}\mathrm{f}}=f^{\prime }(m_x)$$

• $$B(m_x)=\frac{1}{2}\log \coth \left(\frac{\beta f^{\prime }(m_x)}{M}\right)$$

• $$P(\{{\mathit{\sigma }}_t\}\mid m_x)\propto \exp [-\frac{\beta }{M}\sum _{t=1}^M{H}_0({\mathit{\sigma }}_t)+B(m_x)\sum _{i,t}{\sigma }_{i,t}{\sigma }_{i,t+1}]$$

• $$P(m_x)\propto Z(m_x)\exp [-\beta N\{m_x{f}^{\prime }(m_x)-f(m_x)\}]$$

5. 横磁化を測り、有効横磁場を更新する

• $${\hat{m}}_x=\frac{1}{NM}\sum _{i=1}^N\sum _{t=1}^M{[\tanh \left(\frac{\beta {\mathrm{\Gamma }}_{\mathrm{e}\mathrm{f}\mathrm{f}}}{M}\right)]}^{{\sigma }_{i,t}{\sigma }_{i,t+1}}$$

• $$m_x={⟨{\hat{m}}_x⟩}_{P(\mathit{\sigma }\mid m_x)}$$

• $$m_x^{(s+1)}=(1-\alpha )m_x^{(s)}+\alpha {\hat{m}}_x^{(s)},0<\alpha \le 1$$

6. 反強磁性XX相互作用で見る

• $$f(m_x)=\mathrm{\Gamma }{m}_x-\frac{\gamma }{2}{m}_x^2$$

• $$f^{\prime }(m_x)=\mathrm{\Gamma }-\gamma m_x$$

• $${\mathrm{\Gamma }}_{\mathrm{e}\mathrm{f}\mathrm{f}}=\mathrm{\Gamma }-\gamma m_x$$

• $$H_0(\mathit{\sigma })=-N{\left(\frac{1}{N}\sum _{i=1}^N{\sigma }_i^z\right)}^2-h\sum _{i=1}^N{\sigma }_i^z$$

7. データ解析としての読み方

• $$m_x=m_x({\mathrm{\Gamma }}_{\mathrm{e}\mathrm{f}\mathrm{f}})$$

• $${\mathrm{\Gamma }}_{\mathrm{e}\mathrm{f}\mathrm{f}}=f^{\prime }(m_x)$$

• $$m_x=(f^{\prime }{)}^{-1}({\mathrm{\Gamma }}_{\mathrm{e}\mathrm{f}\mathrm{f}})$$

• $$m_x=\frac{\mathrm{\Gamma }-{\mathrm{\Gamma }}_{\mathrm{e}\mathrm{f}\mathrm{f}}}{\gamma }$$

補遺: 量子力学の基礎

• $$i\hslash \frac{\partial }{\partial t}\psi (x,t)=[-\frac{{\hslash }^2}{2m}\frac{{\partial }^2}{\partial x^2}+V(x,t)]\psi (x,t)$$

• $$\psi (x)=\frac{1}{\sqrt{2\pi \hslash }}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{e}^{ipx/\hslash }\psi (p)dp$$

• $$[\hat{x},\hat{p}]=i\hslash$$

• $$\mathrm{\Delta }x\mathrm{\Delta }p\ge \frac{\hslash }{2}$$

補遺: 量子コンピュータの要素

• $$|\psi ⟩=\alpha |0⟩+\beta |1⟩,|\alpha {|}^2+|\beta {|}^2=1$$

• $${\hat{H}}_{\mathrm{q}\mathrm{b}}=-\frac{ϵ}{2}{\hat{\sigma }}^z-\frac{\mathrm{\Delta }}{2}{\hat{\sigma }}^x$$

• $$\hat{H}(t)=A(t){\hat{H}}_{\mathrm{T}\mathrm{F}}+B(t){\hat{H}}_0,{\hat{H}}_{\mathrm{T}\mathrm{F}}=-\sum _i{\hat{\sigma }}_i^x$$

補遺: Landau-Zener遷移

• $$\hat{H}(t)=\hslash \left(\begin{array}{cc}-\alpha t& -g\\ -g& \alpha t\end{array}\right)$$

• $$\mathrm{\Delta }(t)=2\hslash \sqrt{(\alpha t{)}^2+g^2}$$

• $$a(t)=e^{i\alpha t^2/2}c(t),b(t)=e^{-i\alpha t^2/2}d(t)$$

• $$d(t)\simeq \exp [-g^2{\int }_{-\mathrm{\infty }}^t{e}^{i\alpha t_1^2}\left({\int }_{-\mathrm{\infty }}^{t_1}{e}^{-i\alpha t_2^2}d{t}_2\right)d{t}_1]$$

• $${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{e}^{\pm i\alpha t^2}dt=\sqrt{\frac{\pi }{\alpha }}{e}^{\pm i\pi /4}$$

• $$d(\mathrm{\infty })\simeq \exp (-\frac{\pi g^2}{2\alpha }),|d(\mathrm{\infty }){|}^2\simeq \exp (-\frac{\pi g^2}{\alpha })$$

• $$P_{\mathrm{L}\mathrm{Z}}=\exp (-\frac{\pi g^2}{\alpha })$$

補遺: グローバー探索問題

• $$\hat{H}(q)=(1-q){\hat{H}}_s+q{\hat{H}}_m,0\le q\le 1$$

• $$\mathrm{\Delta }(q)=\sqrt{1-4(1-\frac{1}{N})q(1-q)}$$

• $$T\simeq \frac{\pi }{2ϵ}\sqrt{N}$$

補遺: QAの収束定理

• $$\frac{|⟨n(t)|\dot{\hat{H}}(t)|0(t)⟩|}{{(E_n(t)-E_0(t))}^2}\ll 1$$

• $${\mathrm{\Delta }}_n(t)\ge A\mathrm{\Gamma }(t{)}^N$$

• $$\mathrm{\Gamma }(t)\sim t^{-1/(2N-1)}$$

• $$|{\mathrm{\Psi }}_{\beta }⟩=\sum _{\sigma }\sqrt{{\pi }_{\beta }(\sigma )}|\sigma ⟩$$

補遺: AQCと量子アニーリングの広がり

• $$\hat{H}(s)=(1-s){\hat{H}}_0+s{\hat{H}}_1$$

• $$\hat{H}(t)=A(t){\hat{H}}_X+B(t){\hat{H}}_Z,{\hat{H}}_X=-\sum _{i=1}^N{\hat{X}}_i$$

• $$t_f\gg \underset{s}{max}\frac{|⟨{\epsilon }_m(s)|{\partial }_s\hat{H}(s)|{\epsilon }_0(s)⟩|}{{\mathrm{\Delta }}_{m0}(s{)}^2}$$

• $$\Vert {\hat{P}}_{t_f}(s)-\hat{P}(s)\Vert \le ϵ$$

• $$⟨x|\hat{H}|x^{\prime }⟩\le 0(x\ne x^{\prime })$$

• $${\hat{H}}_{XX}=\frac{\gamma }{N}{\left(\sum _i{\hat{X}}_i\right)}^2$$

• $${\mathrm{\Delta }}_{min}(N)\sim e^{-cN}⟹t_f\sim e^{2cN}$$

• $$\hat{H}(s)=(1-s){\hat{H}}_0+s{\hat{H}}_1+s(1-s){\hat{H}}_C$$

• $$\mathcal{L}(s)=\mathrm{span}\{|{\epsilon }_0(s)⟩,|{\epsilon }_1(s)⟩,\dots ,|{\epsilon }_{d-1}(s)⟩\}$$

• $$\mathrm{TTS}(t_f)=t_f\frac{\log (1-p_d)}{\log [1-p_S(t_f)]}$$

補遺: これからの量子アニーリング

• $$\hat{H}(s)=-\mathrm{\Gamma }(s)\sum _i{\hat{\sigma }}_i^x+\mathcal{J}(s)(\sum _i{h}_i{\hat{\sigma }}_i^z+\sum _{⟨i,j⟩}{J}_{ij}{\hat{\sigma }}_i^z{\hat{\sigma }}_j^z)$$

• $$E(\mathit{z})=\sum _i{h}_i{z}_i+\sum _{i<j}{J}_{ij}{z}_i{z}_j$$

• $$N_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}=\sum _{i=1}^{N_{\mathrm{l}\mathrm{o}\mathrm{g}\mathrm{i}\mathrm{c}\mathrm{a}\mathrm{l}}}{\ell }_i$$

• $$\mathrm{TTS}(t_a)=t_a\frac{\log (1-p_d)}{\log (1-p_S(t_a))}$$

• $$T_{\mathrm{f}\mathrm{u}\mathrm{l}\mathrm{l}}=T_{\mathrm{e}\mathrm{m}\mathrm{b}\mathrm{e}\mathrm{d}}+T_{\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{m}}+R(T_{\mathrm{a}\mathrm{n}\mathrm{n}\mathrm{e}\mathrm{a}\mathrm{l}}+T_{\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{d}})+T_{\mathrm{p}\mathrm{o}\mathrm{s}\mathrm{t}}$$

• $$S(n)=\frac{T_{\mathrm{c}\mathrm{l}\mathrm{a}\mathrm{s}\mathrm{s}\mathrm{i}\mathrm{c}\mathrm{a}\mathrm{l}}(n)}{T_{\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{n}\mathrm{t}\mathrm{u}\mathrm{m}}(n)}$$

• $$\mathcal{B}=(I,E_{\mathrm{b}\mathrm{e}\mathrm{s}\mathrm{t}},g,T_{\mathrm{w}\mathrm{a}\mathrm{l}\mathrm{l}},T_{\mathrm{Q}\mathrm{P}\mathrm{U}},N_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}},R,{\mathcal{A}}_{\mathrm{c}\mathrm{l}\mathrm{a}\mathrm{s}\mathrm{s}\mathrm{i}\mathrm{c}\mathrm{a}\mathrm{l}},{\mathcal{A}}_{\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{n}\mathrm{t}\mathrm{u}\mathrm{m}})$$

補遺: 量子アニーリング応用事例とQUBO設計

• $$E(\mathit{x})={\mathit{x}}^{\mathsf{T}}Q\mathit{x}=\sum _i{Q}_{ii}{x}_i+\sum _{i<j}{Q}_{ij}{x}_i{x}_j$$

• $$E_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{g}}(\mathit{x})=\sum _e{\{C_e(\mathit{x})-\frac{1}{2}\}}^2+\frac{\lambda }{2}\sum _i{(\sum _{\mu \in M_i}{x}_{\mu i}-1)}^2$$

• $$f_{\mathrm{A}\mathrm{G}\mathrm{V}}(\mathit{q})=-\sum _i\sum _{\mu \in M_i}{d}_{\mu }{q}_{\mu i}+{\lambda }_1\sum _i{(\sum _{\mu \in M_i}{q}_{\mu i}-1)}^2+{\lambda }_2\sum _{e,t}{(n_{e,t}(\mathit{q})-\frac{1}{2})}^2$$

• $$E_j(\mathit{h})={\mathit{h}}^{\mathsf{T}}{W}^{\mathsf{T}}W\mathit{h}-2{\mathit{v}}_j^{\mathsf{T}}W\mathit{h}$$

• $$H=H_{A0}+H_{A1}+B_0{H}_{B0}+B_1{H}_{B1}$$

補遺: 制約つきQUBOの分解法

• $${(\sum _i{y}_i-\frac{1}{2})}^2=2\sum _{i<j}{y}_i{y}_j+\frac{1}{4}$$

• $${\nu }_k^{(t+1)}={\nu }_k^{(t)}+{\eta }_t\{C_k-⟨F_k(\mathit{x}){⟩}_{{\mathit{\nu }}^{(t)}}\}$$

• $${\mu }_k^{(t+1)}={[{\mu }_k^{(t)}+{\eta }_t\{⟨F_k(\mathit{x}){⟩}_{{\mathit{\mu }}^{(t)}}-C_k\}]}_{+}$$

• $$\underset{\mathit{x}\in \{0,1{\}}^N}{min}\sum _{i,j}(Q_{ij}-\sum _k{\rho }_k^{\ast }{A}_{kij})x_i{x}_j-{\pi }_0^{\ast }$$

• $$H_{\mathrm{m}\mathrm{a}\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{r}}={\mathit{x}}^{\mathsf{T}}C\mathit{x}+t(\mathit{w})+\sum _k{P}_k{(t(\mathit{w})+{\mathit{u}}_k^{\mathsf{T}}A\mathit{x}+s_k-{\mathit{b}}^{\mathsf{T}}{\mathit{u}}_k)}^2+\cdots$$

補遺: Hopfield模型とレプリカ法

• $$J_{ij}=\frac{1}{N}\sum _{\mu =1}^P{\xi }_i^{\mu }{\xi }_j^{\mu }$$

• $$m_{\mu }(\mathit{\sigma })=\frac{1}{N}\sum _{i=1}^N{\xi }_i^{\mu }{\sigma }_i$$

• $$\exp \left(\frac{a}{2}{x}^2\right)=\sqrt{\frac{a}{2\pi }}\int dz\exp [-\frac{a}{2}{z}^2+azx]$$

• $$z_{\mu }=\frac{1}{N}\sum _{i=1}^N{\xi }_i^{\mu }\tanh \left[\beta \sum _{\nu =1}^P{z}_{\nu }{\xi }_i^{\nu }\right]$$

• $${\mathbb{E}}_{\xi }[\log Z]=\underset{n\to 0}{lim}\frac{{\mathbb{E}}_{\xi }[Z^n]-1}{n}$$

• $$m=\int Dz\tanh \left[\beta (m+\sqrt{\alpha r}z)\right]$$

• $$\hat{H}=-\frac{1}{2}\sum _{i,j}{J}_{ij}{\hat{\sigma }}_i^z{\hat{\sigma }}_j^z-\mathrm{\Gamma }\sum _i{\hat{\sigma }}_i^x$$

• $$-\beta f_{\mathrm{\Gamma }}(\mathit{z})=-\frac{\beta }{2}\sum _{\mu }{z}_{\mu }^2+\frac{1}{N}\sum _i\log 2\cosh \left[\beta \sqrt{h_i(\mathit{z}{)}^2+{\mathrm{\Gamma }}^2}\right]$$

• $$\hat{H}(s,\lambda )=-s\lambda N\sum _{\mu =1}^P{\left(\frac{1}{N}\sum _i{\xi }_i^{\mu }{\hat{\sigma }}_i^z\right)}^k+s(1-\lambda )N{\left(\frac{1}{N}\sum _i{\hat{\sigma }}_i^x\right)}^2-(1-s)\sum _i{\hat{\sigma }}_i^x$$

• $$\begin{array}{rl}f& =(k-1)s\lambda \sum _{\mu }(m_{\mu }{)}^k-s(1-\lambda )(m^x{)}^2\\ & -{\left[\sqrt{{\{ks\lambda \sum _{\mu }(m_{\mu }{)}^{k-1}{\xi }^{\mu }\}}^2+h_x^2}\right]}_{\xi }\end{array}$$

補遺: リバースアニーリング

• $$\hat{H}(s,\lambda )=s{\hat{H}}_0+(1-s)(1-\lambda ){\hat{H}}_{\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}}+(1-s)\lambda {\hat{V}}_{\mathrm{T}\mathrm{F}}$$

• $$\begin{array}{rl}f(s,\lambda ;m)& =s(p-1)m^p-c\sqrt{\{sp{m}^{p-1}+(1-s)(1-\lambda ){\}}^2+(1-s{)}^2{\lambda }^2}\\ & -(1-c)\sqrt{\{sp{m}^{p-1}-(1-s)(1-\lambda ){\}}^2+(1-s{)}^2{\lambda }^2}\end{array}$$

• $$-\beta f_{\mathrm{R}\mathrm{S}}=\underset{m,q,R,\tilde{m},\tilde{q},\tilde{R}}{\mathrm{extr}}[\cdots ]$$

補遺: 不均一横磁場駆動

• $${\hat{H}}_{\mathrm{D}\mathrm{W}}(s)=-\frac{1}{2}A(s)\sum _i{\hat{\sigma }}_i^x+\frac{1}{2}B(s)(\sum _i{h}_i{\hat{\sigma }}_i^z+\sum _{i<j}{J}_{ij}{\hat{\sigma }}_i^z{\hat{\sigma }}_j^z)$$

• $$f(\beta ,s;m)=s(p-1)m^p-\frac{1}{\beta }{\int }_0^{1}dx\log 2\cosh \beta \sqrt{\{sp{m}^{p-1}{\}}^2+\mathrm{\Gamma }(x,s{)}^2}$$

• $${\mathrm{\Gamma }}_i(s)⟶\mathrm{\Gamma }(x,s)⟶f(s;m)⟶\frac{\partial f}{\partial m}=0$$

補遺: 静的近似の破れ

• $$m_t=m,R(t,t^{\prime })=R,q_{ab}(t,t^{\prime })=q$$

• $$\begin{array}{rl}-\beta f_{\mathrm{R}\mathrm{S}}& =\frac{1}{4}{\beta }^2{J}^2(R^p-q^p)+\frac{1}{2}\tilde{q}q-\tilde{R}R+\beta j_0{m}^p-\tilde{m}m\\ & +\int D{z}_1\log \int D{z}_{2}2\cosh \omega \end{array}$$

• $$\alpha {\eta }^2\frac{{\mathbb{E}}_{\mathsf{z}}[C_k(\mathsf{z}{)}^2]}{(1-\eta {\chi }_k{)}^2}<1(k\ge 1)$$

原資料への導線

PDF画像で確認したい場合は、PDFビューアから科学大2026 p.2-25、20260903 p.2-15、大学院講義2024ノート p.1-23、大学院講義2024板書 p.1-50 を開けます。

補遺: BOCSとFMによるブラックボックス最適化

• $$y^{(a)}=f({\mathit{x}}^{(a)})+{ϵ}^{(a)},{\mathcal{D}}_t=\{({\mathit{x}}^{(a)},y^{(a)}){\}}_{a=1}^t$$

• $$\hat{f}(\mathit{x})=b+\sum _i{a}_i{x}_i+\sum _{i<j}{b}_{ij}{x}_i{x}_j$$

• $$\mathit{y}=\mathrm{\Phi }\mathit{\theta }+\mathit{ϵ},\mathit{ϵ}\sim \mathcal{N}(0,{\sigma }^{2}I)$$

• $${\theta }_j\mid {\lambda }_j,\tau ,{\sigma }^2\sim \mathcal{N}(0,{\sigma }^2{\tau }^2{\lambda }_j^2),{\lambda }_j,\tau \sim C^{+}(0,1)$$

• $$\begin{array}{rl}V& =({\mathrm{\Phi }}^{\mathsf{T}}\mathrm{\Phi }+{\mathrm{\Lambda }}^{-1}{)}^{-1},\\ \mathit{m}& =V{\mathrm{\Phi }}^{\mathsf{T}}\mathit{y},\\ \mathit{\theta }\mid \mathit{y},\mathrm{\Lambda },{\sigma }^2& \sim \mathcal{N}(\mathit{m},{\sigma }^{2}V)\end{array}$$

• $${\tilde{\mathit{\theta }}}_t\sim p(\mathit{\theta }\mid {\mathcal{D}}_t),{\mathit{x}}_{t+1}\approx \underset{\mathit{x}\in \mathcal{X}}{\mathrm{argmin}}\{\mathit{\varphi }(\mathit{x}{)}^{\mathsf{T}}{\tilde{\mathit{\theta }}}_t+\lambda \sum _i{x}_i\}.$$

• $${\hat{f}}_{\mathrm{F}\mathrm{M}}(\mathit{x})=w_0+\sum _i{w}_i{x}_i+\sum _{i<j}⟨{\mathit{v}}_i,{\mathit{v}}_j⟩x_i{x}_j,⟨{\mathit{v}}_i,{\mathit{v}}_j⟩=\sum _{r=1}^k{v}_{ir}{v}_{jr}.$$

• $$\mathcal{L}=\frac{1}{t}\sum _{a=1}^t{[y^{(a)}-{\hat{f}}_{\mathrm{F}\mathrm{M}}({\mathit{x}}^{(a)})]}^2+{\lambda }_w\sum _i{w}_i^2+{\lambda }_v\sum _{i,r}{v}_{ir}^2.$$

• $$\sum _{i<j}⟨{\mathit{v}}_i,{\mathit{v}}_j⟩x_i{x}_j=\frac{1}{2}\sum _{r=1}^k[{\left(\sum _i{v}_{ir}{x}_i\right)}^2-\sum _i{v}_{ir}^2{x}_i^2].$$

• $$Q_{ii}=w_i,Q_{ij}=⟨{\mathit{v}}_i,{\mathit{v}}_j⟩(i<j),Q_{ji}=0(i<j)$$

• $$\mathrm{\Delta }{E}_i=(1-2x_i)(a_i+\sum _{j<i}{b}_{ji}{x}_j+\sum _{j>i}{b}_{ij}{x}_j)$$

• $$\begin{array}{rl}E(\mathit{s})& =C+\sum _i{h}_i{s}_i+\sum _{i<j}{J}_{ij}{s}_i{s}_j,\\ J_{ij}& =b_{ij}/4,\\ h_i& =a_i/2+\frac{1}{4}(\sum _{j<i}{b}_{ji}+\sum _{j>i}{b}_{ij}),\\ C& =\frac{1}{2}\sum _i{a}_i+\frac{1}{4}\sum _{i<j}{b}_{ij}\end{array}$$

• $$E=\hat{f}+\rho (x_1+x_2+x_3-2{)}^2$$

• $$f_{\mathrm{b}\mathrm{e}\mathrm{s}\mathrm{t}}(t)=\underset{1\le a\le t}{min}f({\mathit{x}}^{(a)})$$

• $$T_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}=T_{\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{a}\mathrm{l}}+\sum _{\text{各反復}}(T_{\mathrm{l}\mathrm{e}\mathrm{a}\mathrm{r}\mathrm{n}}+T_{\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{p}\mathrm{o}\mathrm{s}\mathrm{e}}+T_{\mathrm{e}\mathrm{v}\mathrm{a}\mathrm{l}\mathrm{u}\mathrm{a}\mathrm{t}\mathrm{e}})$$

補遺: ボルツマンマシンと量子サンプリング

1. エネルギーを学ぶとはどういうことか

• $$E_{\theta }(\mathit{s})=-\sum _i{b}_i{s}_i-\sum _{i<j}{W}_{ij}{s}_i{s}_j,p_{\theta }(\mathit{s})=\frac{e^{-\beta E_{\theta }(\mathit{s})}}{Z_{\theta }},Z_{\theta }=\sum _{\mathit{s}}{e}^{-\beta E_{\theta }(\mathit{s})}.$$

• $$\ell (\theta )=\frac{1}{M}\sum _{a=1}^M\log p_{\theta }({\mathit{s}}^{(a)})=-\beta ⟨E_{\theta }{⟩}_{\mathcal{D}}-\log Z_{\theta }$$

• $$\begin{array}{rl}\frac{\partial \ell }{\partial W_{ij}}& =\beta (⟨s_i{s}_j{⟩}_{\mathcal{D}}-⟨s_i{s}_j{⟩}_{p_{\theta }}),\\ \frac{\partial \ell }{\partial b_i}& =\beta (⟨s_i{⟩}_{\mathcal{D}}-⟨s_i{⟩}_{p_{\theta }}),\\ W_{ij}& \leftarrow W_{ij}+\eta \frac{\partial \ell }{\partial W_{ij}}\end{array}$$

2. 隠れ変数と制限ボルツマンマシン

• $$E_{\theta }(\mathit{v},\mathit{h})=-\sum _i{a}_i{v}_i-\sum _a{c}_a{h}_a-\sum _{i,a}{W}_{ia}{v}_i{h}_a,p_{\theta }(\mathit{v})=\sum _{\mathit{h}}{p}_{\theta }(\mathit{v},\mathit{h}).$$

• $$p(h_a=+1\mid \mathit{v})=\frac{e^{\beta u_a}}{2\cosh (\beta u_a)}=\frac{1}{1+e^{-2\beta u_a}},⟨h_a\mid \mathit{v}⟩=\tanh (\beta u_a).$$

• $$\begin{array}{rl}{p}_{\theta }(\mathit{v})& =\frac{e^{-\beta F_{\theta }(\mathit{v})}}{Z_{\theta }},\\ F_{\theta }(\mathit{v})& =-\sum _i{a}_i{v}_i-\frac{1}{\beta }\sum _a\log [2\cosh \left(\beta (c_a+\sum _i{W}_{ia}{v}_i)\right)].\end{array}$$

• $$\frac{\partial \ell }{\partial W_{ia}}=\beta [\frac{1}{M}\sum _{m=1}^M{v}_i^{(m)}\tanh \left(\beta (c_a+\sum _j{W}_{ja}{v}_j^{(m)})\right)-⟨v_i{h}_a{⟩}_{p_{\theta }}]$$

2.1 隠れ変数を消すと、多体相互作用が現れる

• $$F(\mathit{v})=-\frac{\log 2}{\beta }-\frac{\beta }{2}{\left(\sum _i{w}_i{v}_i\right)}^2+\frac{{\beta }^3}{12}{\left(\sum _i{w}_i{v}_i\right)}^4+O({\beta }^5{u}^6).$$

• $$F=C-\sum _{i<j}{J}_{ij}^{\mathrm{e}\mathrm{f}\mathrm{f}}{v}_i{v}_j-K_{1234}^{\mathrm{e}\mathrm{f}\mathrm{f}}{v}_1{v}_2{v}_3{v}_4,J_{ij}^{\mathrm{e}\mathrm{f}\mathrm{f}}=\beta w_i{w}_j+O({\beta }^3\parallel w{\parallel }^4),K_{1234}^{\mathrm{e}\mathrm{f}\mathrm{f}}=-2{\beta }^3{w}_1{w}_2{w}_3{w}_4+O({\beta }^5\parallel w{\parallel }^6).$$

4. フリーズ現象――温度が止まるのではなく、分布が追従できなくなる

• $$\hat{H}(s)=-A(s)\sum _i{\hat{X}}_i+B(s){\hat{E}}_{\theta }(\hat{\mathit{Z}})$$

• $${\tau }_{\mathrm{r}\mathrm{e}\mathrm{l}\mathrm{a}\mathrm{x}}(s_{\ast })\sim t_a(1-s_{\ast }),{\tau }_{\mathrm{r}\mathrm{e}\mathrm{l}\mathrm{a}\mathrm{x}}(s)\gg t_a(1-s)(s>s_{\ast })$$

4.1 フリーズ点から有効温度を読む

• $$p_{\mathrm{o}\mathrm{u}\mathrm{t}}(\mathit{s})\simeq \frac{\exp [-{\beta }_{\mathrm{e}\mathrm{f}\mathrm{f}}{E}_{\theta }(\mathit{s})]}{Z({\beta }_{\mathrm{e}\mathrm{f}\mathrm{f}})},{\beta }_{\mathrm{e}\mathrm{f}\mathrm{f}}=\frac{B(s_{\ast })}{k_{\mathrm{B}}{T}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}}.$$

4.2 横磁場が残ったまま凍る場合

• $${\rho }_{\ast }\simeq \frac{e^{-\hat{H}(s_{\ast })/(k_{\mathrm{B}}{T}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}})}}{\mathrm{Tr}{e}^{-\hat{H}(s_{\ast })/(k_{\mathrm{B}}{T}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}})}}$$

5. 温度を推定する基本――縮退度とエネルギーの尺度

• $$\frac{P_0}{P_1}=\frac{g_0}{g_1}{e}^{\beta (E_1-E_0)},\hat{\beta }=\frac{\log (P_0/P_1)-\log (g_0/g_1)}{E_1-E_0}$$

• $$\log \frac{P_{{\beta }_1}(E)}{P_{{\beta }_2}(E)}=-({\beta }_1-{\beta }_2)E+\log \frac{Z_{{\beta }_2}}{Z_{{\beta }_1}}.$$

6. 急停止極限からの非断熱摂動

• $$p_0(\mathit{s})=|⟨\mathit{s}|+{⟩}^{\otimes N}{|}^2=2^{-N}.$$

• $$p_{\beta }(\mathit{s})=2^{-N}[1-\beta (E(\mathit{s})-\bar{E})+O({\beta }^2)].$$

7. 時間依存摂動論を、一段ずつ計算する

• $$\begin{array}{rl}\hat{H}(t)& ={\hat{H}}_0(t)+\lambda B(t){\hat{E}}^{\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{g}},{\hat{H}}_0(t)=-A(t)\sum _i{X}_i,\\ {\hat{E}}^{\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{g}}& =\sum _{n\ge 1}{\hat{E}}_n^{\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{g}},{\hat{E}}_n^{\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{g}}=-\sum _{|S|=n}{J}_S^{\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{g}}\prod _{i\in S}{Z}_i,\\ \mathrm{\Phi }(t)& ={\int }_t^{\tau }A(u)du.\end{array}$$

7.1 横磁場だけの時間発展

• $$U_0(\tau ,t)Z_i{U}_0(\tau ,t{)}^{†}=Z_i\cos [2\mathrm{\Phi }(t)]+Y_i\sin [2\mathrm{\Phi }(t)]$$

7.2 Dyson展開の一次項

• $$⟨\mathit{s}|\psi (\tau )⟩=\frac{e^{i{\phi }_0}}{2^{N/2}}[1-i\lambda \sum _n{E}_n^{\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{g}}(\mathit{s}){\int }_0^{\tau }B(t)e^{-2in\mathrm{\Phi }(t)}dt+O({\lambda }^2)].$$

• $$\begin{array}{rl}{p}_{\mathrm{D}\mathrm{Q}\mathrm{A}}(\mathit{s})& =2^{-N}[1-\lambda \sum _n{K}_n{E}_n^{\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{g}}(\mathit{s})+O({\lambda }^2)],\\ K_n& =2{\int }_0^{\tau }B(t)\sin [2n{\int }_t^{\tau }A(u)du]dt.\end{array}$$

8. 一体項と二体項の温度をそろえる

• $$E_n^{\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{g}}=\frac{{\beta }_{\ast }}{K_n}{E}_{\ast ,n},b_i^{\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{g}}=\frac{{\beta }_{\ast }}{K_1}{b}_i^{\ast },W_{ij}^{\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{g}}=\frac{{\beta }_{\ast }}{K_2}{W}_{ij}^{\ast }.$$

• $$c_n=\frac{K_n}{K_1},E_n^{\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{g}}=\frac{E_{\ast ,n}}{c_n},c_2=\frac{{\int }_0^{\tau }B(t)\sin [4\mathrm{\Phi }(t)]dt}{{\int }_0^{\tau }B(t)\sin [2\mathrm{\Phi }(t)]dt}.$$

8.1 線形に横磁場を切るとFresnel積分が出る

• $$K_n=2{\int }_0^{\tau }\sin (nr{u}^2)du\simeq \sqrt{\frac{\pi }{2nr}},K_1\simeq \sqrt{\frac{\pi }{2r}},c_2\simeq \frac{1}{\sqrt{2}}.$$

9. 理論から予測する係数と、実機で測る補正係数

• $${\alpha }_{\mathrm{c}\mathrm{a}\mathrm{l}}=\frac{{\beta }_{\mathrm{d}\mathrm{e}\mathrm{v}\mathrm{i}\mathrm{c}\mathrm{e}}}{{\beta }_{\mathrm{u}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{a}\mathrm{r}\mathrm{y}}},E^{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{r}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{e}\mathrm{d}}=\frac{E^{\mathrm{d}\mathrm{e}\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}\mathrm{e}\mathrm{d}}}{{\alpha }_{\mathrm{c}\mathrm{a}\mathrm{l}}}.$$

10. 「摂動」と「パラメータ推定」を別の角度から眺める

• $$\frac{\partial ⟨T_{\mu }⟩}{\partial {\theta }_{\nu }}=\beta (⟨T_{\mu }{T}_{\nu }⟩-⟨T_{\mu }⟩⟨T_{\nu }⟩)=\beta C_{\mu \nu }.$$

11. 小さな例で、温度と補正方向を検算する

• $$Z=4\cosh (\beta W),⟨s_1{s}_2⟩=\tanh (\beta W),W=\frac{\mathrm{artanh}⟨s_1{s}_2⟩}{\beta }.$$

• $$K_n=2{\int }_0^{\tau }\sin [2na(\tau -t)]dt=\frac{1-\cos (2na\tau )}{na}.$$

11.1 温度がずれると、何を学んでしまうのか

• $$g_{\ast }(W)={\beta }_{\ast }[c_{\mathcal{D}}-\tanh ({\beta }_{\ast }W)],\tilde{g}(W)={\beta }_{\ast }[c_{\mathcal{D}}-\tanh ({\beta }_{\mathrm{d}}W)].$$

• $$W_{\mathrm{s}\mathrm{t}\mathrm{o}\mathrm{p}}=\frac{\mathrm{artanh}{c}_{\mathcal{D}}}{{\beta }_{\mathrm{d}}},⟨s_1{s}_2{⟩}_{{\beta }_{\ast }}=\tanh \left(\frac{{\beta }_{\ast }}{{\beta }_{\mathrm{d}}}\mathrm{artanh}{c}_{\mathcal{D}}\right).$$

11.2 四状態の頻度から、学習を最後まで追う

• $$W_{k+1}=W_k+0.5(0.6-\tanh W_k),p_k(++)=p_k(--)=\frac{1+\tanh W_k}{4},p_k(+-)=p_k(-+)=\frac{1-\tanh W_k}{4}.$$

12. 実際の学習の手順と、分布を確かめる指標

• $$D_{\mathrm{T}\mathrm{V}}(p,q)=\frac{1}{2}\sum _{\mathit{s}}|p(\mathit{s})-q(\mathit{s})|,D_{\mathrm{K}\mathrm{L}}(p\parallel q)=\sum _{\mathit{s}}p(\mathit{s})\log \frac{p(\mathit{s})}{q(\mathit{s})}$$

13. 拡散モデルへの橋――分布の勾配を学ぶ

• $${\mathit{s}}_{\theta }(\mathit{x})={\mathrm{\nabla }}_{\mathit{x}}\log p_{\theta }(\mathit{x})=-\beta {\mathrm{\nabla }}_{\mathit{x}}{E}_{\theta }(\mathit{x}).$$

13.1 未知の分布の勾配を、データから学ぶ

• $$\begin{array}{rl}J(\theta )& =\frac{1}{2}{\mathbb{E}}_q[\parallel {\mathit{s}}_{\theta }-\mathrm{\nabla }\log q{\parallel }^2]\\ & ={\mathbb{E}}_q[\frac{1}{2}\parallel {\mathit{s}}_{\theta }{\parallel }^2+\mathrm{\nabla }\cdot {\mathit{s}}_{\theta }]+C_q.\end{array}$$

13.2 ノイズ除去を学ぶと、スコアが得られる

• $${\mathrm{\nabla }}_{{\mathit{x}}_t}\log q({\mathit{x}}_t\mid {\mathit{x}}_0)=-\frac{{\mathit{x}}_t-{\alpha }_t{\mathit{x}}_0}{{\sigma }_t^2}=-\frac{\mathit{ϵ}}{{\sigma }_t},\mathcal{L}={\mathbb{E}}_{t,{\mathit{x}}_0,\mathit{ϵ}}[\lambda (t){\Vert {\mathit{s}}_{\theta }({\mathit{x}}_t,t)+\frac{\mathit{ϵ}}{{\sigma }_t}\Vert }^2].$$

• $${\mathit{s}}_{\mathrm{o}\mathrm{p}\mathrm{t}}({\mathit{x}}_t,t)=\mathbb{E}[-\frac{\mathit{ϵ}}{{\sigma }_t}|{\mathit{x}}_t]={\mathrm{\nabla }}_{{\mathit{x}}_t}\log q_t({\mathit{x}}_t).$$

補遺: 生成モデルの歴史と量子アニーリング

2. BM――分配関数の難しさを、相関の測定に移す

• $$p_{\theta }(\mathit{s})=\frac{e^{-\beta E_{\theta }(\mathit{s})}}{Z_{\theta }},\frac{\partial \ell }{\partial W_{ij}}=\beta [⟨s_i{s}_j{⟩}_{\mathrm{d}\mathrm{a}\mathrm{t}\mathrm{a}}-⟨s_i{s}_j{⟩}_{p_{\theta }}].$$

3. RBM――条件付き独立性で推論を簡単にする

• $$E(\mathit{v},\mathit{h})=-\mathit{a}\cdot \mathit{v}-\mathit{c}\cdot \mathit{h}-\sum _{i,a}{W}_{ia}{v}_i{h}_a,p(h_a=1\mid \mathit{v})=\frac{1}{1+e^{-2\beta (c_a+\sum _i{W}_{ia}{v}_i)}}.$$

4.1 生成と推論を分ける

• $$p_{\theta ,\psi }(\mathit{x})=\int p_{\theta }(\mathit{x}\mid \mathit{z})p_{\psi }(\mathit{z})d\mathit{z},p_{\theta ,\psi }(\mathit{z}\mid \mathit{x})=\frac{p_{\theta }(\mathit{x}\mid \mathit{z})p_{\psi }(\mathit{z})}{p_{\theta ,\psi }(\mathit{x})}.$$

4.2 ELBOを一段ずつ導く

• $$\begin{array}{rl}\log p_{\theta ,\psi }(\mathit{x})& =\mathcal{L}(\mathit{x})+D_{\mathrm{K}\mathrm{L}}(q_{\varphi }(\mathit{z}\mid \mathit{x})\parallel p_{\theta ,\psi }(\mathit{z}\mid \mathit{x})),\\ \mathcal{L}(\mathit{x})& ={\mathbb{E}}_{q_{\varphi }}\log p_{\theta }(\mathit{x}\mid \mathit{z})-D_{\mathrm{K}\mathrm{L}}(q_{\varphi }(\mathit{z}\mid \mathit{x})\parallel p_{\psi }(\mathit{z})).\end{array}$$

4.3 BM事前分布を置くと、負相が再び現れる

• $${\mathrm{\nabla }}_{\psi }{\mathbb{E}}_{\mathrm{d}\mathrm{a}\mathrm{t}\mathrm{a}}\mathcal{L}=-\beta {\mathbb{E}}_{{\overline{q}}_{\varphi }}{\mathrm{\nabla }}_{\psi }{E}_{\psi }+\beta {\mathbb{E}}_{p_{\psi }}{\mathrm{\nabla }}_{\psi }{E}_{\psi }.$$

• $$D_{\mathrm{K}\mathrm{L}}(q\parallel p_{\psi })=\beta (F[q]-F_{\mathrm{e}\mathrm{q}}),{\mathbb{E}}_{\mathrm{d}\mathrm{a}\mathrm{t}\mathrm{a}}{D}_{\mathrm{K}\mathrm{L}}(q_{\varphi }(z\mid x)\parallel p_{\psi }(z))=I_q(X;Z)+D_{\mathrm{K}\mathrm{L}}({\overline{q}}_{\varphi }\parallel p_{\psi }).$$

5. GAN――識別器から密度比の情報を得る

• $$\underset{\theta }{min}\underset{\omega }{max}V(D,G)={\mathbb{E}}_{p_{\mathrm{d}\mathrm{a}\mathrm{t}\mathrm{a}}}\log D_{\omega }(\mathit{x})+{\mathbb{E}}_{p_z}\log [1-D_{\omega }(G_{\theta }(\mathit{z}))].$$

• $$D^{\ast }(x)=\frac{p_{\mathrm{d}\mathrm{a}\mathrm{t}\mathrm{a}}(x)}{p_{\mathrm{d}\mathrm{a}\mathrm{t}\mathrm{a}}(x)+p_G(x)},V(D^{\ast },G)=-\log 4+2D_{\mathrm{J}\mathrm{S}}(p_{\mathrm{d}\mathrm{a}\mathrm{t}\mathrm{a}}\parallel p_G).$$

6.1 ノイズ予測とスコアの関係

• $$x_t=\sqrt{{\overline{\alpha }}_t}{x}_0+\sqrt{1-{\overline{\alpha }}_t}ϵ,ϵ\sim \mathcal{N}(0,I),{\mathcal{L}}_{\mathrm{n}\mathrm{o}\mathrm{i}\mathrm{s}\mathrm{e}}=\mathbb{E}\parallel ϵ-{ϵ}_{\theta }(x_t,t){\parallel }^2.$$

• $$s_t(x_t)={\mathrm{\nabla }}_{x_t}\log p_t(x_t)=-\frac{\mathbb{E}[ϵ\mid x_t]}{\sqrt{1-{\overline{\alpha }}_t}},s_{\theta }(x_t,t)=-\frac{{ϵ}_{\theta }(x_t,t)}{\sqrt{1-{\overline{\alpha }}_t}}.$$

6.2 逆時間の式と、講義の物理との接続

• $$dx=[f(x,t)-g(t{)}^2{\mathrm{\nabla }}_x\log p_t(x)]dt+g(t)d{\overline{W}}_t,dt<0.$$

7. 量子アニーラをどこへ組み込むか

• $${\rho }_{\theta }=\frac{e^{-\beta {\hat{H}}_{\theta }}}{\mathrm{Tr}{e}^{-\beta {\hat{H}}_{\theta }}},p_{\theta }(v)=\mathrm{Tr}({\mathrm{\Pi }}_v{\rho }_{\theta }),{\hat{H}}_{\theta }=-\sum _i{\mathrm{\Gamma }}_i{X}_i+E_{\theta }(\mathit{Z}).$$

潜在分布のエネルギーとQUBO係数

• 半制限BMのエネルギーと学習係数
• 結合分布と可視コードの周辺分布
• 事前分布の正相・負相による勾配
• Ising係数からQUBO係数への変換

補遺: 断熱ショートカット（STA）

1. 最後を合わせることと、途中も追従すること

• 1. 最後を合わせることと、途中も追従すること：式001

• 1. 最後を合わせることと、途中も追従すること：式031

3. 瞬間固有基底で見る非断熱遷移

• 3. 瞬間固有基底で見る非断熱遷移：式002

• 3. 瞬間固有基底で見る非断熱遷移：式003

• 3. 瞬間固有基底で見る非断熱遷移：式004

• 3. 瞬間固有基底で見る非断熱遷移：式005

4. 例1：スピンを反転し、途中も基底状態に留める

• 4. 例1：スピンを反転し、途中も基底状態に留める：式006

• 4. 例1：スピンを反転し、途中も基底状態に留める：式007

• 4. 例1：スピンを反転し、途中も基底状態に留める：式008

• 4. 例1：スピンを反転し、途中も基底状態に留める：式009

• 4. 例1：スピンを反転し、途中も基底状態に留める：式010

• 4. 例1：スピンを反転し、途中も基底状態に留める：式011

• 4. 例1：スピンを反転し、途中も基底状態に留める：式012

• 4. 例1：スピンを反転し、途中も基底状態に留める：式013

5. 避けられた交差での符号とギャップ

• 5. 避けられた交差での符号とギャップ：式014

• 5. 避けられた交差での符号とギャップ：式015

6. 例2：波動関数を広げ、終点で振動を残さず止める

• 6. 例2：波動関数を広げ、終点で振動を残さず止める：式016

• 6. 例2：波動関数を広げ、終点で振動を残さず止める：式017

• 6. 例2：波動関数を広げ、終点で振動を残さず止める：式018

• 6. 例2：波動関数を広げ、終点で振動を残さず止める：式019

• 6. 例2：波動関数を広げ、終点で振動を残さず止める：式020

• 6. 例2：波動関数を広げ、終点で振動を残さず止める：式021

• 6. 例2：波動関数を広げ、終点で振動を残さず止める：式022

• 6. 例2：波動関数を広げ、終点で振動を残さず止める：式023

• 6. 例2：波動関数を広げ、終点で振動を残さず止める：式024

• 6. 例2：波動関数を広げ、終点で振動を残さず止める：式025

7. 交換関係で確かめる動的不変量

• 7. 交換関係で確かめる動的不変量：式026

• 7. 交換関係で確かめる動的不変量：式027

• 7. 交換関係で確かめる動的不変量：式028

8. 波動関数の幅とエネルギーで結果を読む

• 8. 波動関数の幅とエネルギーで結果を読む：式029

• 8. 波動関数の幅とエネルギーで結果を読む：式030

補遺: GA・SA・QAとHopfの収束評価

2. 証明するGAの更新則を決める

• 2. 証明するGAの更新則を決める：式001

• 2. 証明するGAの更新則を決める：式002

• 2. 証明するGAの更新則を決める：式003

3. 一ビット・二個体なら行列を手で書ける

• 3. 一ビット・二個体なら行列を手で書ける：式004

• 3. 一ビット・二個体なら行列を手で書ける：式005

• 3. 一ビット・二個体なら行列を手で書ける：式006

4. 正の行列とHopfの定理

• 4. 正の行列とHopfの定理：式007

• 4. 正の行列とHopfの定理：式008

• 4. 正の行列とHopfの定理：式009

• 4. 正の行列とHopfの定理：式010

5. SA・QAと同じところ、異なるところ

• 5. SA・QAと同じところ、異なるところ：式011

• 5. SA・QAと同じところ、異なるところ：式012

• 5. SA・QAと同じところ、異なるところ：式013

• 5. SA・QAと同じところ、異なるところ：式014

6. 分布の重なりを見ると、評価が鋭くなる

• 6. 分布の重なりを見ると、評価が鋭くなる：式015

• 6. 分布の重なりを見ると、評価が鋭くなる：式016

• 6. 分布の重なりを見ると、評価が鋭くなる：式017

• 6. 分布の重なりを見ると、評価が鋭くなる：式018

• 6. 分布の重なりを見ると、評価が鋭くなる：式019

• 6. 分布の重なりを見ると、評価が鋭くなる：式020

7. 突然変異を下げても定常分布についていけるか

• 7. 突然変異を下げても定常分布についていけるか：式021

• 7. 突然変異を下げても定常分布についていけるか：式022

• 7. 突然変異を下げても定常分布についていけるか：式023

• 7. 突然変異を下げても定常分布についていけるか：式024

• 7. 突然変異を下げても定常分布についていけるか：式025

• 7. 突然変異を下げても定常分布についていけるか：式026

8. 定常分布が最適解へ集中することを示す

• 8. 定常分布が最適解へ集中することを示す：式027

• 8. 定常分布が最適解へ集中することを示す：式028

• 8. 定常分布が最適解へ集中することを示す：式029

• 8. 定常分布が最適解へ集中することを示す：式030

• 8. 定常分布が最適解へ集中することを示す：式031

9. 固定した選択圧では、最適解に集中しないことがある

• 9. 固定した選択圧では、最適解に集中しないことがある：式032

• 9. 固定した選択圧では、最適解に集中しないことがある：式033

10. 最適解を含むことだけなら、さらに緩い条件で示せる

• 10. 最適解を含むことだけなら、さらに緩い条件で示せる：式034

• 10. 最適解を含むことだけなら、さらに緩い条件で示せる：式035

11. 有限時間の成功確率と比較の限界

• 11. 有限時間の成功確率と比較の限界：式036

• 11. 有限時間の成功確率と比較の限界：式037

• 11. 有限時間の成功確率と比較の限界：式038

• 11. 有限時間の成功確率と比較の限界：式039

• 11. 有限時間の成功確率と比較の限界：式040