Update README.md

README.md

@@ -24,7 +24,7 @@ The RBM is a natural choice for representing wave-functions of systems of spin $
 
 Letting $S = \\{0,1\\}^n$ be the set of inputs of the RBM, we may choose an orthonormal basis $\\{\ket{s}\\} \subset \mathcal{H}$ labeled by the configurations $s \in S$ such that the trial state at $\alpha \in \mathcal{M}$ is a linear combination $\ket{\psi(\alpha)} = \sum_{s \in S} \psi(s,\alpha) \ket{s} \in \mathcal{H}$, where the components $\psi(s,\alpha) \in \mathbb{C}$ are wave-functions of the configurations $s \in S$ at each $\alpha \in \mathcal{M}$.
 
-The trial state wave-functions $\psi(\alpha):S \to \mathbb{C}$ are represented as a Restricted Boltzmann Machine with complex parameters $\alpha = \{a,b,w\}$, constructed as the marginal distribution on the inputs of the RBM. With inputs $S = \\{0,1\\}^n$ and outputs $H = \\{0,1\\}^m$, the RBM with complex parameters is a universal approximator of complex probability distributions $\Psi(\alpha):S \times H \to \mathbb{C}$ at each $\alpha \in \mathcal{M}$ such that the trial state wave-functions $\psi(\alpha):S \to \mathbb{C}$ at each $\alpha \in \mathcal{M}$ are the marginal distribution defined by
+The trial state wave-functions $\psi(\alpha):S \to \mathbb{C}$ are represented as a Restricted Boltzmann Machine with complex parameters $\alpha = \\{a,b,w\\}$, constructed as the marginal distribution on the inputs of the RBM. With inputs $S = \\{0,1\\}^n$ and outputs $H = \\{0,1\\}^m$, the RBM with complex parameters is a universal approximator of complex probability distributions $\Psi(\alpha):S \times H \to \mathbb{C}$ at each $\alpha \in \mathcal{M}$ such that the trial state wave-functions $\psi(\alpha):S \to \mathbb{C}$ at each $\alpha \in \mathcal{M}$ are the marginal distribution defined by
  $$S \ni s \mapsto \psi(s,\alpha) = \sum_{h \in H} \Psi(s,h,\alpha) = \sum_{h \in H} \exp(a^\dagger s + b^\dagger h + h^\dagger ws) = \exp(a^\dagger s) \sum_{h \in H} \exp(b^\dagger h + h^\dagger ws) = \exp\bigg(\sum_{j=1}^n a_j^\*s_j\bigg) \sum_{h \in H} \exp\bigg(\sum_{i=1}^m b_i^\*h_i + \sum_{i=1}^m h_i \sum_{j=1}^n w_{ij} s_j\bigg) = \exp\bigg(\sum_{j=1}^n a_j^\*s_j\bigg) \sum_{h \in H} \prod_{i=1}^m \exp\bigg(b_i^\*h_i + h_i \sum_{j=1}^n w_{ij} s_j\bigg) = \exp\bigg(\sum_{j=1}^n a_j^\* s_j\bigg) \prod_{i=1}^m \sum_{h_i=0}^1 \exp\bigg(b_i^\*h_i + h_i \sum_{j=1}^n w_{ij} s_j\bigg) = \exp\bigg(\sum_{j=1}^n a_j^\* s_j\bigg) \prod_{i=1}^m \bigg[ 1 + \exp\bigg(b_i^\* + \sum_{j=1}^n w_{ij} s_j\bigg)\bigg] \in \mathbb{C}$$
 where we ignore the normalization factor of the wave-function, and where $\dagger$ represents the matrix conjugate transpose. By the Born rule, the real, normalized probability distribution $p(\alpha):S \to [0,1]$ associated to the wave-function $\psi(\alpha)$ at each $\alpha \in \mathcal{M}$ is defined by $S \ni s \mapsto p(s,\alpha) = |\psi(s,\alpha)|^2/\sum_{s' \in S} |\psi(s',\alpha)|^2 \in [0,1]$.