README.md

Learning Quantum States with Neural Networks

Introduction

Quantum mechanics emerges from classical mechanics by the relaxation of the requirement of commutativity among the observables as assumed by classical probability theory. The most immediate and striking consequence of this relaxation is the insufficiency of real-valued probability distributions to encode the interference phenomena observed by experiment. In response, we generalize the notion of probability distributions from real-valued distributions over the set of possible outcomes which combine convexly, to complex-valued distributions over the set of possible outcomes which combine linearly. The complex-valued probabilities are able to encode the observed interference patterns in their relative phases. Such quantum probability distributions do not describe mutually exclusive outcomes in which only one outcome exists prior to measurement, but rather describes outcomes in which all possible outcomes simultaneously exist prior to measurement and which interfere in a wave-like manner.

The increase in predictive power offered by quantum mechanics came with the price of computational difficulties. Unlike the classical world, whose dimensionality scales additively with the number of subsystems, the dimensionality scaling of quantum systems is multiplicative. Thus, even small systems quickly become intractable without approximation techniques. Luckily, it is rarely the case that knowledge of the full state space is required to accurately model a given system, as most information may be contained in a relatively small subspace. Many of the most successful approximation techniques of the last century, such as Born–Oppenheimer and variational techniques like Density Functional Theory, rely on this convenient notion for their success. With the rapid development of machine learning, a field which specializes in dimensionality reduction and feature extraction of very large datasets, it is natural to apply these novel techniques for dealing with the canonical large data problem of the physical sciences.

Restricted Boltzmann Machines

The Universal Approximation Theorems are a collection of results concerning the ability of artificial neural networks to arbitrarily approximate different classes of functions. In particular, the Restricted Boltzmann Machine (RBM) is a shallow two-layer network consisting of $n$ input nodes or visible units, and $m$ output nodes or hidden units such that each input $v \in \{0,1\}^n$ and output $h \in \{0,1\}^m$ are Boolean vectors of respective length. Letting $\mathcal{M}$ be a parameter manifold with points $\alpha \in \mathcal{M}$, the RBM is characterized by such parameters $\alpha = \{a,b,w\}$ where $a \in \mathbb{R}^n$ are the visible layer biases, $b \in \mathbb{R}^m$ are the hidden layer biases, and $w \in \mathbb{R}^{m \times n}$ are the weights which fully connect the layers. The network is "restricted" in the sense that there are no intra-layers connections.

Let $V = \{0,1\}^n$ be the set of inputs, let $H = \{0,1\}^m$ be the set of outputs, and let $X = V \times H$ be the set of pairs. Then the RBM is a universal approximator of Boltzmann probability distributions $p(\alpha):X \to [0,1]$ at each $\alpha \in \mathcal{M}$ defined by $$X \ni (v,h) \mapsto p(v,h,\alpha) = \frac{\exp[-E(v,h,\alpha)]}{Z(\alpha)} = \frac{\exp(a^\perp v + b^\perp h + h^\perp wv)}{\sum_{(v',h') \in X} \exp(a^\perp v' + b^\perp h' + {h'}^\perp wv')} \in [0,1]$$ where $E(v,h,\alpha) = -a^\perp v - b^\perp h - h^\perp wv$ is the parametrized Boltzmann energy and $Z(\alpha) = \sum_{(v',h') \in X} \exp(a^\perp v' + b^\perp h' + {h'}^\perp wv')$ is the partition function which normalizes the probabilities, with $\perp$ denoting the matrix transpose. From the joint probability distribution $p(\alpha)$, we may construct the marginal distributions as the restrictions $p_V(\alpha):V \to [0,1]$ and $p_H(\alpha): H \to [0,1]$ at each $\alpha \in \mathcal{M}$, given by the partial sums $$p_V(v,\alpha) = \sum_{h \in H} p(v,h,\alpha)~~,~~p_H(h,\alpha) = \sum_{v \in V} p(v,h,\alpha)$$ over $H$ and $V$ respectively. Due to the restricted nature of the RBM, the activation probabilities $p(h_i=1|v,\alpha)$ and $p(v_j=1|h,\alpha)$ of each layer are mutually exclusive for all $i \in [1,m]$ and $j \in [1,n]$ such that the conditional probabilities are the products $$p(h|v,\alpha) = \prod_{i=1}^m p(h_i=1|v,\alpha)~~,~~p(v|h,\alpha) = \prod_{j=1}^n p(v_j=1|h,\alpha)$$ of activation probabilities. The traditional method for training an RBM involves Hinton's Contrastive Divergence technique, which will not be covered here.

Neural Network Quantum States

The RBM is a natural choice for representing wave-functions of systems of spin $\frac{1}{2}$ fermions where each input vector represents a configuration of $n$ spins. Ultimately, we seek to solve the time-independent Schrödinger equation $H\ket{\psi_0} = E_0\ket{\psi_0}$ for the ground state $\ket{\psi_0}$ and its corresponding energy $E_0$ for a given system having Hamiltonian $H$. We take a variational approach by proposing a parametrized trial state as a vector-valued mapping $\ket{\psi}:\mathcal{M} \to \mathcal{H}$ on a low-dimensional parameter manifold $\mathcal{M}$ of points $\alpha \in \mathcal{M}$ to a $2^n$-dimensional state space $\mathcal{H}$ and propose infinitesimal variations to $\alpha \in \mathcal{M}$ until $\ket{\psi(\alpha)} \approx \ket{\psi_0}$.

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 $$S \ni s \mapsto \psi(s,\alpha) = \sum_{h \in H} \Psi(s,h,\alpha) = \sum_{h \in H} \exp(a^\dagger s + b^\dagger h + h^\dagger ws) = \exp(a^\dagger s) \sum_{h \in H} \exp(b^\dagger h + h^\dagger ws) = \exp\bigg(\sum_{j=1}^n a_j^*s_j\bigg) \sum_{h \in H} \exp\bigg(\sum_{i=1}^m b_i^*h_i + \sum_{i=1}^m h_i \sum_{j=1}^n w_{ij} s_j\bigg) = \exp\bigg(\sum_{j=1}^n a_j^*s_j\bigg) \sum_{h \in H} \prod_{i=1}^m \exp\bigg(b_i^*h_i + h_i \sum_{j=1}^n w_{ij} s_j\bigg) = \exp\bigg(\sum_{j=1}^n a_j^* s_j\bigg) \prod_{i=1}^m \sum_{h_i=0}^1 \exp\bigg(b_i^*h_i + h_i \sum_{j=1}^n w_{ij} s_j\bigg) = \exp\bigg(\sum_{j=1}^n a_j^* s_j\bigg) \prod_{i=1}^m \bigg[ 1 + \exp\bigg(b_i^* + \sum_{j=1}^n w_{ij} s_j\bigg)\bigg] \in \mathbb{C}$$ 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 $S \ni s \mapsto p(s,\alpha) = |\psi(s,\alpha)|^2/\sum_{s' \in S} |\psi(s',\alpha)|^2 \in [0,1]$.

The variational energy functional $E[\psi(\alpha)]$ associated to the variational state $\ket{\psi(\alpha)}$ at each $\alpha \in \mathcal{M}$ is the statistical expectation $E[\psi(\alpha)] = \langle H \rangle_{\psi(\alpha)}$ of the Hamiltonian $H$ in the state $\ket{\psi(\alpha)}$, given by $$E[\psi(\alpha)] = \frac{\langle \psi(\alpha), H\psi(\alpha) \rangle}{\langle \psi(\alpha), \psi(\alpha) \rangle} = \frac{\sum_{s,s' \in S} \psi^* (s,\alpha) H_{ss'} \psi(s',\alpha)}{\sum_{s' \in S} |\psi(s',\alpha)|^2} = \frac{\sum_{s \in S} |\psi(s,\alpha)|^2 \left(\sum_{s' \in S} H_{ss'} \frac{\psi(s',\alpha)}{\psi(s,\alpha)}\right)}{\sum_{s' \in S} |\psi(s',\alpha)|^2} = \sum_{s \in S} p(s,\alpha) E_{\text{loc}}(s,\alpha)$$ where we define the variational local energies $E_{\text{loc}}(s,\alpha) = \sum_{s' \in S} H_{ss'} \frac{\psi(s',\alpha)}{\psi(s,\alpha)}$, with $H_{ss'}$ being the matrix element of $H$ in between the states $\ket{s}$ and $\ket{s'}$. Thus $E[\psi(\alpha)] = \sum_{s \in S} p(s,\alpha) E_{\text{loc}}(s,\alpha)$ is the statistical expectation of the local energies weighted by the real probability distribution $p(\alpha):S \to [0,1]$.

Transverse Field Ising Model

In this demonstration, we assume the prototypical Ising spin model for a one-dimensional lattice of spin $\frac{1}{2}$ particles, whose Hamiltonian is given by $$H = -J \sum_{j=1}^{n-1} \sigma_j^z \sigma_{j+1}^z - B \sum_{j=1}^n \sigma_x$$ where we use the shorthand notation $\sigma_j^z \sigma_{j+1}^z = I^{(1)} \otimes \cdots \otimes \sigma_z^{(j)} \otimes \sigma_z^{(j+1)} \otimes \cdots \otimes I^{(n)}$ to denote the tensor product of the $2 \times 2$ identity matrix with the Pauli matrix $\sigma_z$ located at positions $j$ and $j+1$, and where $\sigma_x = I^{(1)} \otimes \cdots \otimes \sigma_x^{(j)} \otimes \cdots \otimes I^{(n)}$ denotes $\sigma_x$ at position $j$. The size of $H$ is $2^n \times 2^n$, and so it is impossible to directly diagonalize even for relatively few particles. The constant $J$ represents the nearest neighbor coupling strength, and $B$ represents the strength of the transverse field. When $J>0$, nearest neighbors tend to align parallel (ferromagnetic), and tend to align anti-parallel when $J<0$ (anti-ferromagnetic). The local energy of a configuration $s \in S$ in the Ising model can is seen to be $$E_{\text{loc}}(s,\alpha) =-J \sum_{j=1}^{n-1} \sigma_j \sigma_{j+1} - B \sum_{s' \in S_f} \frac{\psi(s',\alpha)}{\psi(s,\alpha)}$$ where $\sigma_j = -2s_j + 1 \in \{1,-1\}$ and $S_f$ consists of $n$ configurations in which a single spin of $s$ has been inverted.

Stochastic Optimization

Evaluating the energy functional $E[\psi(\alpha)] = \sum_{s \in S} p(s,\alpha) E_{\text{loc}}(s,\alpha)$ at each $\alpha \in \mathcal{M}$ involves an explicit calculation of the distribution $p(\alpha):S \to [0,1]$ for each $s \in S$ and a sum over $2^n$ states. Using Monte Carlo methods, we may draw $N$ possibly overlapping samples $\tilde{S}$ from $S$ according to the distribution $p(\alpha)$ such that the drawn samples $\tilde{S}$ are represented in proportion to their contribution to $p(\alpha)$. In other words, the samples drawn will tend to be from regions of $S$ associated with the highest probabilities and which therefore contribute the most to $E[\psi(\alpha)]$. We may then make the reasonable approximation $$E[\psi(\alpha)] \approx \frac{1}{N} \sum_{s \in \tilde{S}} E_{\text{loc}}(s,\alpha)$$ of the energy functional as a simple average of the local energies over the drawn samples $\tilde{S}$ weighted by equal probabilities $1/N$. The samples can be drawn using the well-known Metropolis-Hastings algorithm, a Markov Chain Monte Carlo algorithm of the following form:

SUBROUTINE metropolis_hastings:
    markov_chain(1) ← random_sample(n)
    FOR i ∈ [2,N] DO
        s ← markov_chain(i-1)
        rind ← random_index(lo=1, hi=n)
        s_prop ← invert_spin(config=s, at=rind)
        IF r_unif(0,1) < |ψ(s_prop)/ψ(s)|^2 THEN
            markov_chain(i) ← s_prop
        ELSE
            markov_chain(i) ← s
        END IF
    END FOR
    RETURN markov_chain
END SUBROUTINE metropolis_hastings

In practice, we allow for a thermalization period, or "burn-in" period, during which the sampling process moves the initial random sample into the stationary distribution before we can begin recording samples. As we can see, the acceptance probabilities in the Metropolis-Hastings algorithm and the form of the local energy involve only ratios of the wave-functions $\psi(s,\alpha)$ for different configurations, and therefore we are justified in ignoring the normalization factor in our derivation of $\psi(s,\alpha)$. Once all samples are drawn, we may estimate the energy functional as an average of the local energies over the drawn samples.

The stochastic optimization algorithm is a first order optimization that involves infinitesimal variations to the parameters $\alpha \in \mathcal{M}$ according to the update rule $$\alpha \leftarrow \alpha + \delta\alpha$$ where the variation $\delta\alpha$ is in the direction opposite the generalized forces $F(\alpha)$ having components $$F_l(\alpha) = \langle \partial_l^\dagger H \rangle_{\psi(\alpha)} - \langle \partial_l^\dagger \rangle_{\psi(\alpha)} \langle H \rangle_{\psi(\alpha)} = \frac{\langle \partial_l \psi(\alpha), H \psi(\alpha) \rangle}{\langle \psi(\alpha), \psi(\alpha) \rangle} - \frac{\langle \partial_l \psi(\alpha), \psi(\alpha) \rangle}{\langle \psi(\alpha), \psi(\alpha) \rangle} \frac{\langle \psi(\alpha), H \psi(\alpha) \rangle}{\langle \psi(\alpha), \psi(\alpha) \rangle} = \sum_{s \in S} p(s,\alpha) O_l^*(s,\alpha) E_{\text{loc}}(s, \alpha) - \bigg[ \sum_{s \in S} p(s,\alpha) O_l^*(s,\alpha) \bigg] \bigg[ \sum_{s \in S} p(s,\alpha) E_{\text{loc}}(s,\alpha) \bigg] \approx \frac{1}{N} \sum_{s \in \tilde{S}} O_l^*(s,\alpha) E_{\text{loc}}(s, \alpha) - \bigg[ \frac{1}{N} \sum_{s \in \tilde{S}} O_l^*(s,\alpha) \bigg] \bigg[ \frac{1}{N} \sum_{s \in \tilde{S}} E_{\text{loc}}(s, \alpha) \bigg]$$ where we define the logarithmic derivatives $$O_l(s,\alpha) = \frac{\partial}{\partial \alpha_l} \ln \psi(s, \alpha) = \frac{1}{\psi(s, \alpha)} \frac{\partial}{\partial \alpha_l} \psi(s, \alpha)$$ of the variational wave-functions $\psi(s, \alpha)$ in terms of diagonal operators $O_l$ given by $O_l \psi(s, \alpha) = O_l(s,\alpha)$. In statistical terms, the forces $F_l$ are the expected values of the product of deviation operators $\Delta \partial_l^\dagger = \partial_l^\dagger - \langle \partial_l^\dagger \rangle_{\psi(\alpha)}$ and $\Delta H = H - \langle H \rangle_{\psi(\alpha)}$ in the variational state $\ket{\psi(\alpha)}$. i.e. $$F_l(\alpha) = \langle \Delta \partial_l^\dagger \Delta H \rangle_{\psi(\alpha)} = \langle \partial_l^\dagger H - \langle \partial_l^\dagger \rangle_{\psi(\alpha)} H - \partial_l^\dagger \langle H \rangle_{\psi(\alpha)} + \langle \partial_l^\dagger \rangle_{\psi(\alpha)} \langle H \rangle_{\psi(\alpha)} \rangle_{\psi(\alpha)} = \langle \partial_l^\dagger H \rangle_{\psi(\alpha)} - \langle \partial_l^\dagger \rangle_{\psi(\alpha)} \langle H \rangle_{\psi(\alpha)}$$ as a correlation function. We then pre-condition the forces $F(\alpha)$ with a Hermitian positive-definite matrix $S^{-1}(\alpha)$ prior to updating the parameters $\alpha \in \mathcal{M}$, such that the update rule is $$\alpha \leftarrow \alpha + \delta\alpha = \alpha - \delta\tau S^{-1}(\alpha) F(\alpha)$$ for some small time step $\delta \tau > 0$, where the matrix $S(\alpha)$ is known as the stochastic reconfiguration matrix whose components are the correlation functions $$S_{kl}(\alpha) = \langle \Delta \partial_k^\dagger \Delta \partial_l \rangle_{\psi(\alpha)} = \langle \partial_k^\dagger \partial_l \rangle_{\psi(\alpha)} - \langle \partial_k^\dagger \rangle_{\psi(\alpha)} \langle \partial_l \rangle_{\psi(\alpha)} = \frac{\langle \partial_k \psi(\alpha), \partial_l \psi(\alpha) \rangle}{\langle \psi(\alpha), \psi(\alpha) \rangle} - \frac{\langle \partial_k \psi(\alpha), \psi(\alpha) \rangle}{\langle \psi(\alpha), \psi(\alpha) \rangle} \frac{\langle \psi(\alpha), \partial_l \psi(\alpha) \rangle}{\langle \psi(\alpha), \psi(\alpha) \rangle} = \sum_{s \in S} p(s,\alpha) O_k^*(s,\alpha) O_l(s,\alpha) - \bigg[ \sum_{s \in S} p(s,\alpha) O_k^*(s,\alpha) \bigg] \bigg[ \sum_{s \in S} p(s,\alpha) O_l(s,\alpha) \bigg] \approx \frac{1}{N} \sum_{s \in \tilde{S}} O_k^*(s,\alpha) O_l(s,\alpha) - \bigg[ \frac{1}{N} \sum_{s \in \tilde{S}} O_k^*(s,\alpha) \bigg] \bigg[ \frac{1}{N} \sum_{s \in \tilde{S}} O_l(s,\alpha) \bigg]$$ of the derivative deviations.

Derivation

Let $V$ be a neighborhood of a point $\alpha \in \mathcal{M}$ with a local chart of coordinate functions $\alpha_l:\mathcal{M} \to \mathbb{C}$ such that $\alpha_l(\alpha) = 0$ is the origin of the local coordinate system. To derive the stochastic optimization update rule, we first expand the function $\ket{\psi}:\mathcal{M} \to \mathcal{H}$ in a Taylor series $$\ket{\psi} = \ket{\psi(\alpha)} + \sum_l \frac{\partial}{\partial \alpha_l} \ket{\psi(\alpha)} \alpha_l + \frac{1}{2} \sum_{kl} \frac{\partial^2}{\partial \alpha_k \partial \alpha_l} \ket{\psi(\alpha)} \alpha_k \alpha_l + \cdots$$ about $\alpha \in \mathcal{M}$. At some nearby point $\alpha + \delta\alpha \in V$, we have $$\ket{\psi(\alpha + \delta\alpha)} = \ket{\psi(\alpha)} + \sum_l \delta\alpha_l \frac{\partial}{\partial \alpha_l} \ket{\psi(\alpha)} + \frac{1}{2} \sum_{kl} \delta\alpha_k \delta\alpha_l \frac{\partial^2}{\partial \alpha_k \partial \alpha_l} \ket{\psi(\alpha)} + \cdots \in \mathcal{H}$$ where $\delta\alpha_l \in \mathbb{C}$ is an infinitesimal variation in the $l$-th direction, which is approximated to first order as an affine function $$\ket{\psi(\alpha + \delta\alpha)} \approx \ket{\psi(\alpha)} + \sum_l \delta\alpha_l \frac{\partial}{\partial \alpha_l} \ket{\psi(\alpha)} \in \mathcal{H}$$ on $V$. Here, the $\delta\alpha_l$ represent the amount of change in each direction needed to linearly approximate the function $\ket{\psi}$ at $\alpha + \delta\alpha \in V$ from the neighboring point $\alpha \in \mathcal{M}$, so that the affine approximation becomes exact in the limit $\delta\alpha_l \to 0$.

Similarly, we may define a path $\alpha:[t, t + \delta t] \to \mathcal{M}$ in $\mathcal{M}$ such that $\alpha(t) = \alpha \in \mathcal{M}$ and $\alpha(t + \delta t) = \alpha + \delta\alpha \in V$ along with a unitary propagator $U(\delta t) = \exp(-i \delta t H)$ such that, at some nearby point $\alpha(t + \delta t) \in V$, we have $$\ket{\psi(\alpha(t + \delta t))} = U(\delta t)\ket{\psi(\alpha(t))} = \ket{\psi(\alpha(t))} - i \delta t H \ket{\psi(\alpha(t))} + \frac{(-i \delta t)^2}{2} H^2 \ket{\psi(\alpha(t))} + \cdots \in \mathcal{H}$$ which is approximated to first order as an affine function $$\ket{\psi(\alpha(t + \delta t))} \approx \ket{\psi(\alpha(t))} - i \delta t H \ket{\psi(\alpha(t))} \in \mathcal{H}$$ on $V$. Here, the Hamiltonian $H$ is the infinitesimal generator of the one-parameter unitary group of time translations whose elements are the unitary transformations $U(t_2 - t_1):\mathcal{H} \to \mathcal{H}$ on the state space $\mathcal{H}$ for any $t_1, t_2 \in \mathbb{R}$, so that the affine approximation becomes exact in the limit $\delta t \to 0$. Note that the propagator $U(\delta t)$ is recovered (up to a constant) by solution of the time-dependent Schrödinger equation $i \frac{d}{dt} \ket{\psi(\alpha(t))} = H \ket{\psi(\alpha(t))} \Rightarrow \ket{\psi(\alpha(t_2))} = U(t_2-t_1) \ket{\psi(\alpha(t_1))}$ by taking $t_1 = t$ and $t_2 = t + \delta t$.

By performing a Wick rotation $\tau = it$ and defining a path $\alpha:[\tau, \tau + \delta \tau] \to \mathcal{M}$ in $\mathcal{M}$ such that $\alpha(\tau) = \alpha \in \mathcal{M}$ and $\alpha(\tau + \delta \tau) = \alpha + \delta\alpha \in V$, we may propagate the state $\ket{\psi(\alpha(\tau))}$ similarly with a non-unitary propagator $U(\delta \tau) = \exp(- \delta \tau H)$ such that $$\ket{\psi(\alpha(\tau + \delta \tau))} \approx \ket{\psi(\alpha(\tau))} - \delta \tau H \ket{\psi(\alpha(\tau))} \in \mathcal{H}$$ is an affine function on $V$ which becomes exact in the limit $\delta\tau \to 0$. The propagator $U(\delta \tau)$ is recovered (up to a constant) by solution of the imaginary-time Schrödinger equation $-\frac{d}{d\tau} \ket{\psi(\alpha(\tau))} = H \ket{\psi(\alpha(\tau))} \Rightarrow \ket{\psi(\alpha(\tau_2))} = U(\tau_2-\tau_1) \ket{\psi(\alpha(\tau_1))}$ by taking $\tau_1 = \tau$ and $\tau_2 = \tau + \delta \tau$. Using this fact, we may compare the first order terms of $\ket{\psi(\alpha(\tau + \delta \tau))}$ and $\ket{\psi(\alpha + \delta\alpha)}$ to find that $$\delta \tau \frac{d}{d\tau} \ket{\psi(\alpha(\tau))} = \delta \tau \sum_l \frac{\partial}{\partial \alpha_l} \ket{\psi(\alpha(\tau))} \frac{d \alpha_l}{d\tau}(\tau) = \sum_l \delta\alpha_l(\tau) \frac{\partial}{\partial \alpha_l} \ket{\psi(\alpha(\tau))}$$ is a linear combination of the tangent vectors $\frac{d \alpha_l}{d\tau}(\tau)$ at $\alpha(\tau) \in \mathcal{M}$, so that the variation $\delta\alpha(\tau) = \delta \tau \frac{d \alpha}{d\tau}(\tau)$ at time $\tau$ is in the direction of the tangent vector $\frac{d \alpha}{d\tau}(\tau)$ at $\alpha(\tau) \in \mathcal{M}$ such that $$\alpha(\tau + \delta \tau) = \alpha(\tau) + \delta\alpha(\tau) = \alpha(\tau) + \delta \tau \frac{d \alpha}{d\tau}(\tau)$$ is the change in the parameters $\alpha(\tau) \in \mathcal{M}$ due to the non-unitary, imaginary-time evolution of the state $\ket{\psi(\alpha(\tau))}$ over the interval $[\tau, \tau + \delta \tau]$.

Let $T_\alpha \mathcal{M}$ denote the tangent space of $\mathcal{M}$ at $\alpha(\tau) \in \mathcal{M}$ and let $T_{\psi(\alpha)} \mathcal{H}$ denote the tangent space of $\mathcal{H}$ at $\ket{\psi(\alpha(\tau))} \in \mathcal{H}$. To determine the actual form of the tangent vector $\frac{d \alpha}{d\tau}(\tau) \in T_\alpha \mathcal{M}$ at time $\tau$, we push forward the local vector field $\frac{d \alpha}{d\tau}$ on $V \subset \mathcal{M}$ to the local vector field $\frac{d}{d\tau} \ket{\psi(\alpha)}$ on the image $\tilde{V} = \ket{\psi(V)} \subset \mathcal{H}$ of $V$ under the mapping $\ket{\psi}:\mathcal{M} \to \mathcal{H}$ and impose the constraint that the projection $$\left\langle \frac{d}{d\tau} \psi(\alpha(\tau)), \bigg[ \Delta \frac{d}{d\tau} + \Delta H \bigg] \psi(\alpha(\tau)) \right\rangle = 0$$ onto the tangent vector $\frac{d}{d\tau} \ket{\psi(\alpha(\tau))} \in T_{\psi(\alpha)} \tilde{V}$ vanishes, a type of Galerkin condition known as the Dirac-Frenkel-McLachlan variational principle. Here, we require precisely that the tangent vector $[ \Delta \frac{d}{d\tau} + \Delta H ] \ket{\psi(\alpha(\tau))} \in T_{\psi(\alpha)} \mathcal{H}$ formed by the difference of deviations $\Delta \frac{d}{d\tau} = \frac{d}{d\tau} - \left\langle \frac{d}{d\tau} \right\rangle_{\psi(\alpha)}$ and $-\Delta H = \langle H \rangle_{\psi(\alpha)} - H$ is orthogonal to the tangent vector $\frac{d}{d\tau} \ket{\psi(\alpha(\tau))} \in T_{\psi(\alpha)} \tilde{V}$. This condition is motivated by the fact that the subspace $\tilde{V} \subset \mathcal{H}$ is typically of much smaller dimension than $\mathcal{H}$ itself since $\mathcal{M}$ is typically of much smaller dimension than $\mathcal{H}$ as manifolds, resulting in a situation where the tangent vector $\frac{d}{d\tau} \ket{\psi(\alpha(\tau))} \in T_{\psi(\alpha)} \tilde{V}$ is contained to a low dimensional subspace but $H \ket{\psi(\alpha(\tau))} \in T_{\psi(\alpha)} \mathcal{H}$ is not necessarily contained to this subspace. In order for the imaginary-time Schrödinger equation $-\frac{d}{d\tau} \ket{\psi(\alpha(\tau))} = H \ket{\psi(\alpha(\tau))}$ to hold, we must have that the norm of the state $[\frac{d}{d\tau} + H] \ket{\psi(\alpha(\tau))}$ is vanishing, or equivalently that its projection onto the subspace containing $\frac{d}{d\tau} \ket{\psi(\alpha(\tau))}$ is vanishing, i.e. that there is no overlap between $[\frac{d}{d\tau} + H] \ket{\psi(\alpha(\tau))}$ and $\frac{d}{d\tau} \ket{\psi(\alpha(\tau))}$. The same argument holds for the state $[\Delta\frac{d}{d\tau} + \Delta H] \ket{\psi(\alpha(\tau))}$ formed by the deviation operators, which we choose for their advantageous statistical properties. From the overlap condition, we have explicitly $$0 = \sum_k \frac{d \alpha_k^*}{d\tau}(\tau) \frac{\partial}{\partial \alpha_k} \bra{\psi(\alpha(\tau))} \bigg[ \sum_l \frac{d \alpha_l}{d\tau}(\tau) \frac{\partial}{\partial \alpha_l} \ket{\psi(\alpha(\tau))} - \sum_l \frac{d \alpha_l}{d\tau}(\tau) \langle \partial_l \rangle_{\psi(\alpha)} \ket{\psi(\alpha(\tau))} + H \ket{\psi(\alpha(\tau))} - \langle H \rangle_{\psi(\alpha)} \ket{\psi(\alpha(\tau))} \bigg] = \sum_k \frac{d \alpha_k^*}{d\tau}(\tau) \bigg[ \sum_l \frac{d \alpha_l}{d\tau}(\tau) \langle \partial_k^\dagger \partial_l \rangle_{\psi(\alpha)} - \sum_l \frac{d \alpha_l}{d\tau}(\tau) \langle \partial_k^\dagger \rangle_{\psi(\alpha)} \langle \partial_l \rangle_{\psi(\alpha)} + \langle \partial_k^\dagger H \rangle_{\psi(\alpha)} - \langle \partial_k^\dagger \rangle_{\psi(\alpha)} \langle H \rangle_{\psi(\alpha)} \bigg] = \sum_k \frac{d \alpha_k^*}{d\tau}(\tau) \bigg[ \sum_l \frac{d \alpha_l}{d\tau}(\tau) S_{kl}(\alpha) + F_k(\alpha) \bigg]$$ which is true when each term is identically zero, i.e. when $$\sum_l \frac{d \alpha_l}{d\tau}(\tau) S_{kl}(\alpha) = - F_k(\alpha)$$ is true. This system of linear equations can be written in matrix form as $$S(\alpha) \frac{d \alpha}{d\tau}(\tau) = -F(\alpha)$$ whose formal solution is $$\frac{d \alpha}{d\tau}(\tau) = - S^{-1}(\alpha) F(\alpha)$$ such that $$\alpha(\tau + \delta \tau) = \alpha(\tau) + \delta\alpha(\tau) = \alpha(\tau) + \delta \tau \frac{d \alpha}{d\tau}(\tau) = \alpha(\tau) - \delta \tau S^{-1}(\alpha) F(\alpha)$$ is the change in the parameters $\alpha(\tau) \in \mathcal{M}$ due to the non-unitary, imaginary-time evolution of the state $\ket{\psi(\alpha(\tau))}$ over the interval $[\tau, \tau + \delta \tau]$.

It must be noted that the initialization of the parameters can have a dramatic effect on the performance of the algorithm. The initial state $\ket{\psi(\alpha(0))}$ must be chosen such that $\langle \psi_0, \psi(\alpha(0)) \rangle \neq 0$, or else learning is not possible. The more overlap there is with the ground state, the more efficient the algorithm will be. With at least some overlap, we will expect that $\ket{\psi(\alpha(\tau))} \to \ket{\psi_0}$ as $\tau \to \infty$ for a sufficiently small time step $\delta\tau$.