Machine Learning meets Quantum Physics

The main aim is to give you a short and pedestrian introduction to how we can use Machine Learning methods to solve quantum mechanical many-body problems and how we can use such techniques. And why this could be of interest.

The hope is that after this talk you have gotten the basic ideas to get you started. Peeping into https://github.com/mhjensenseminars/MachineLearningTalk, you'll find a Jupyter notebook, slides, codes etc that will allow you to reproduce the simulations discussed here, and perhaps run your own very first calculations.

More in depth notebooks and lecture notes are at

1. Making a professional Monte Carlo code for quantum mechanical simulations https://github.com/CompPhysics/ComputationalPhysics2/blob/gh-pages/doc/pub/notebook1/ipynb/notebook1.ipynb
2. From Variational Monte Carlo to Boltzmann Machines https://github.com/CompPhysics/ComputationalPhysics2/blob/gh-pages/doc/pub/notebook2/ipynb/notebook2.ipynb
3. Nuclear Talent course on Machine Learning in Nuclear Experiment and Theory, June 22 - July 3, 2020

Feel free to try them out and please don't hesitate to ask if something is unclear.

How can we avoid the dimensionality curse? Many possibilities

1. smarter basis functions
2. resummation of specific correlations
3. stochastic sampling of high-lying states (stochastic FCI, CC and SRG/IMSRG)
4. many more

Machine Learning and Quantum Computing hold also great promise in tackling the ever increasing dimensionalities. Here we will focus on Machine Learning, with some links to recent research to quantum machine learning.

• Short intro to Machine Learning
• Variational Monte Carlo (Markov Chain Monte Carlo, \( \mathrm{MC}^2 \)) and many-body problems, solving quantum mechanical problems in a stochastic way. It will serve as our motivation for switching to Machine Learning.
• From Variational Monte Carlo to Boltzmann Machines and Deep Learning
• And then to Quantum Computing and Machine Learning

Artificial intelligence-based techniques, particularly in machine learning and optimization, are increasingly being used in many areas of experimental and theoretical physics to facilitate discovery, accelerate data analysis and modeling efforts, and bridge different physical and temporal scales in numerical models.

These techniques are proving to be powerful tools for advancing our understanding; however, they are not without significant challenges. The theoretical foundations of many tools, such as deep learning, are poorly understood, resulting in the use of techniques whose behavior (and misbehavior) is difficult to predict and understand. Similarly, physicists typically use general AI techniques that are not tailored to the needs of the experimental and theoretical work being done. Thus, many opportunities exist for major advances both in physical discovery using AI and in the theory of AI. Furthermore, there are tremendous opportunities for these fields to inform each other, for example, in creating machine learning- based methods that must obey certain constraints by design, such as the conservation of mass, momentum and energy.

Machine learning is an extremely rich field, in spite of its young age. The increases we have seen during the last three decades in computational capabilities have been followed by developments of methods and techniques for analyzing and handling large date sets, relying heavily on statistics, computer science and mathematics. The field is rather new and developing rapidly.

Popular software packages written in Python for ML are

• Scikit-learn,
• Tensorflow,
• PyTorch
• Keras,

and more. These are all freely available at their respective GitHub sites. They encompass communities of developers in the thousands or more. And the number of code developers and contributors keeps increasing.

Not all the algorithms and methods can be given a rigorous mathematical justification, opening up thereby for experimenting and trial and error and thereby exciting new developments.

A solid command of linear algebra, multivariate theory, probability theory, statistical data analysis, optimization algorithms, understanding errors and Monte Carlo methods is important in order to understand many of the various algorithms and methods.

The approaches to machine learning are many, but are often split into two main categories. In supervised learning we know the answer to a problem, and let the computer deduce the logic behind it. On the other hand, unsupervised learning is a method for finding patterns and relationship in data sets without any prior knowledge of the system. Some authours also operate with a third category, namely reinforcement learning. This is a paradigm of learning inspired by behavioural psychology, where learning is achieved by trial-and-error, solely from rewards and punishment.

Another way to categorize machine learning tasks is to consider the desired output of a system. Some of the most common tasks are:

• Classification: Outputs are divided into two or more classes. The goal is to produce a model that assigns inputs into one of these classes. An example is to identify digits based on pictures of hand-written ones. Classification is typically supervised learning.
• Regression: Finding a functional relationship between an input data set and a reference data set. The goal is to construct a function that maps input data to continuous output values.
• Clustering: Data are divided into groups with certain common traits, without knowing the different groups beforehand. It is thus a form of unsupervised learning.

• An excellent reference, Mehta et al. and Physics Reports (2019).
• Machine Learning and the Physical Sciences by Carleo et al
• Ab initio solution of the many-electron Schrödinger equation with deep neural networks by Pfau et al..
• Machine Learning and the Deuteron by Kebble and Rios
• Variational Monte Carlo calculations of \( A\le 4 \) nuclei with an artificial neural-network correlator ansatz by Adams et al.
• Unsupervised Learning for Identifying Events in Active Target Experiments by Solli et al.
• Report from the A.I. For Nuclear Physics Workshop by Bedaque et al.

This work is inspired by the idea of representing the wave function with a restricted Boltzmann machine (RBM), presented recently by G. Carleo and M. Troyer, Science 355, Issue 6325, pp. 602-606 (2017). They named such a wave function/network a neural network quantum state (NQS). In their article they apply it to the quantum mechanical spin lattice systems of the Ising model and Heisenberg model, with encouraging results. See also the recent work by Adams et al..

Thanks to Daniel Bazin (MSU), Jane Kim (MSU), Julie Butler (MSU), Sean Liddick (MSU), Michelle Kuchera (Davidson College), Vilde Flugsrud (UiO), Even Nordhagen (UiO), Bendik Samseth (UiO) and Robert Solli (UiO) for many discussions and interpretations.

Almost every problem in ML and data science starts with the same ingredients:

• The dataset \( \mathbf{x} \) (could be some observable quantity of the system we are studying)
• A model which is a function of a set of parameters \( \mathbf{\alpha} \) that relates to the dataset, say a likelihood function \( p(\mathbf{x}\vert \mathbf{\alpha}) \) or just a simple model \( f(\mathbf{\alpha}) \)
• A so-called loss/cost/risk function \( \mathcal{C} (\mathbf{x}, f(\mathbf{\alpha})) \) which allows us to decide how well our model represents the dataset.

We seek to minimize the function \( \mathcal{C} (\mathbf{x}, f(\mathbf{\alpha})) \) by finding the parameter values which minimize \( \mathcal{C} \). This leads to various minimization algorithms. It may surprise many, but at the heart of all machine learning algortihms there is an optimization problem.

The field of artificial neural networks has a long history of development, and is closely connected with the advancement of computer science and computers in general. A model of artificial neurons was first developed by McCulloch and Pitts in 1943 to study signal processing in the brain and has later been refined by others. The general idea is to mimic neural networks in the human brain, which is composed of billions of neurons that communicate with each other by sending electrical signals. Each neuron accumulates its incoming signals, which must exceed an activation threshold to yield an output. If the threshold is not overcome, the neuron remains inactive, i.e. has zero output.

This behaviour has inspired a simple mathematical model for an artificial neuron.

 
$$ y = f\left(\sum_{i=1}^n w_ix_i\right) = f(u) $$

 
Here, the output \( y \) of the neuron is the value of its activation function, which have as input a weighted sum of signals \( x_i, \dots ,x_n \) received by \( n \) other neurons.

An artificial neural network (NN), is a computational model that consists of layers of connected neurons, or nodes. It is supposed to mimic a biological nervous system by letting each neuron interact with other neurons by sending signals in the form of mathematical functions between layers. A wide variety of different NNs have been developed, but most of them consist of an input layer, an output layer and eventual layers in-between, called hidden layers. All layers can contain an arbitrary number of nodes, and each connection between two nodes is associated with a weight variable.

Given a hamiltonian \( H \) and a trial wave function \( \Psi_T \), the variational principle states that the expectation value of \( \langle H \rangle \), defined through

 
$$ \langle E \rangle = \frac{\int d\boldsymbol{R}\Psi^{\ast}_T(\boldsymbol{R})H(\boldsymbol{R})\Psi_T(\boldsymbol{R})} {\int d\boldsymbol{R}\Psi^{\ast}_T(\boldsymbol{R})\Psi_T(\boldsymbol{R})}, $$

 
is an upper bound to the ground state energy \( E_0 \) of the hamiltonian \( H \), that is

 
$$ E_0 \le \langle E \rangle. $$

 
In general, the integrals involved in the calculation of various expectation values are multi-dimensional ones. Traditional integration methods such as the Gauss-Legendre will not be adequate for say the computation of the energy of a many-body system.

Choose a trial wave function \( \psi_T(\boldsymbol{R}) \).

 
$$ P(\boldsymbol{R},\boldsymbol{\alpha})= \frac{\left|\psi_T(\boldsymbol{R},\boldsymbol{\alpha})\right|^2}{\int \left|\psi_T(\boldsymbol{R},\boldsymbol{\alpha})\right|^2d\boldsymbol{R}}. $$

 
This is our model, or likelihood/probability distribution function (PDF). It depends on some variational parameters \( \boldsymbol{\alpha} \). The approximation to the expectation value of the Hamiltonian is now

 
$$ \langle E[\boldsymbol{\alpha}] \rangle = \frac{\int d\boldsymbol{R}\Psi^{\ast}_T(\boldsymbol{R},\boldsymbol{\alpha})H(\boldsymbol{R})\Psi_T(\boldsymbol{R},\boldsymbol{\alpha})} {\int d\boldsymbol{R}\Psi^{\ast}_T(\boldsymbol{R},\boldsymbol{\alpha})\Psi_T(\boldsymbol{R},\boldsymbol{\alpha})}. $$

 
$$ E_L(\boldsymbol{R},\boldsymbol{\alpha})=\frac{1}{\psi_T(\boldsymbol{R},\boldsymbol{\alpha})}H\psi_T(\boldsymbol{R},\boldsymbol{\alpha}), $$

 
called the local energy, which, together with our trial PDF yields

 
$$ \langle E[\boldsymbol{\alpha}] \rangle=\int P(\boldsymbol{R})E_L(\boldsymbol{R},\boldsymbol{\alpha}) d\boldsymbol{R}\approx \frac{1}{N}\sum_{i=1}^NE_L(\boldsymbol{R_i},\boldsymbol{\alpha}) $$

 
with \( N \) being the number of Monte Carlo samples.

The Algorithm for performing a variational Monte Carlo calculations runs thus as this

• Initialisation: Fix the number of Monte Carlo steps. Choose an initial \( \boldsymbol{R} \) and variational parameters \( \alpha \) and calculate \( \left|\psi_T(\boldsymbol{R},\boldsymbol{\alpha})\right|^2 \).
• Initialise the energy and the variance and start the Monte Carlo calculation by looping over trials.

• Calculate a trial position \( \boldsymbol{R}_p=\boldsymbol{R}+r*step \) where \( r \) is a random variable \( r \in [0,1] \).
• Metropolis algorithm to accept or reject this move \( w = P(\boldsymbol{R}_p,\boldsymbol{\alpha})/P(\boldsymbol{R},\boldsymbol{\alpha}) \).
• If the step is accepted, then we set \( \boldsymbol{R}=\boldsymbol{R}_p \).
• Update averages
• Finish and compute final averages.

Observe that the jumping in space is governed by the variable step. This is often called brute-force sampling. Need importance sampling to get more relevant sampling.

We want to perform a Variational Monte Carlo calculation of the ground state of two electrons in a quantum dot well with different oscillator energies, assuming total spin \( S=0 \). Our trial wave function has the following form

 
$$ \begin{equation} \psi_{T}(\boldsymbol{r}_1,\boldsymbol{r}_2) = C\exp{\left(-\alpha_1\omega(r_1^2+r_2^2)/2\right)} \exp{\left(\frac{r_{12}}{(1+\alpha_2 r_{12})}\right)}, \tag{1} \end{equation} $$

 
where the variables \( \alpha_1 \) and \( \alpha_2 \) represent our variational parameters.

Why does the trial function look like this? How did we get there? This is one of our main motivations for switching to Machine Learning.

To find the derivatives of the local energy expectation value as function of the variational parameters, we can use the chain rule and the hermiticity of the Hamiltonian.

Let us define (with the notation \( \langle E[\boldsymbol{\alpha}]\rangle =\langle E_L\rangle \))

 
$$ \bar{E}_{\alpha_i}=\frac{d\langle E_L\rangle}{d\alpha_i}, $$

 
as the derivative of the energy with respect to the variational parameter \( \alpha_i \) We define also the derivative of the trial function (skipping the subindex \( T \)) as

 
$$ \bar{\Psi}_{i}=\frac{d\Psi}{d\alpha_i}. $$

The elements of the gradient of the local energy are then (using the chain rule and the hermiticity of the Hamiltonian)

 
$$ \bar{E}_{i}= 2\left( \langle \frac{\bar{\Psi}_{i}}{\Psi}E_L\rangle -\langle \frac{\bar{\Psi}_{i}}{\Psi}\rangle\langle E_L \rangle\right). $$

 
From a computational point of view it means that you need to compute the expectation values of

 
$$ \langle \frac{\bar{\Psi}_{i}}{\Psi}E_L\rangle, $$

 
and

 
$$ \langle \frac{\bar{\Psi}_{i}}{\Psi}\rangle\langle E_L\rangle $$

 
These integrals are evaluted using MC intergration (with all its possible error sources). We can then use methods like stochastic gradient or other minimization methods to find the optimal variational parameters (I don't discuss this topic here, but these methods are very important in ML).

A standard BM network is divided into a set of observable and visible units \( \hat{x} \) and a set of unknown hidden units/nodes \( \hat{h} \).

Additionally there can be bias nodes for the hidden and visible layers. These biases are normally set to \( 1 \).

BMs are stackable, meaning we can train a BM which serves as input to another BM. We can construct deep networks for learning complex PDFs. The layers can be trained one after another, a feature which makes them popular in deep learning

RBMs were first developed using binary units in both the visible and hidden layer. The corresponding energy function is defined as follows:

 
$$ \begin{align} E(\mathbf{x}, \mathbf{h}) = - \sum_i^M x_i a_i- \sum_j^N b_j h_j - \sum_{i,j}^{M,N} x_i w_{ij} h_j, \tag{4} \end{align} $$

 
where the binary values taken on by the nodes are most commonly 0 and 1.

Another variant is the RBM where the visible units are Gaussian while the hidden units remain binary:

 
$$ \begin{align} E(\mathbf{x}, \mathbf{h}) = \sum_i^M \frac{(x_i - a_i)^2}{2\sigma_i^2} - \sum_j^N b_j h_j - \sum_{i,j}^{M,N} \frac{x_i w_{ij} h_j}{\sigma_i^2}. \tag{5} \end{align} $$

You can find the codes for the simple two-electron case at the Github repository https://github.com/CompPhysics/ComputationalPhysics2/blob/gh-pages/doc/pub/notebook2/ipynb/notebook2.ipynb.

Performance measures for scikit-learn standard classifiers, as well as the trained QBM.

• Lots of experimental analysis coming, see for example Unsupervised Learning for Identifying Events in Active Target Experiments by Solli et al. as well references and examples in Report from the A.I. For Nuclear Physics Workshop by Bedaque et al..
• Extension of the work of G. Carleo and M. Troyer, Science 355, Issue 6325, pp. 602-606 (2017) gives excellent results for two-electron systems as well as good agreement with standard VMC calculations from \( N=2 \) to more than \( N=100 \) electrons.
• Promising results. Next step is to use trial wave function in final Green's function Monte Carlo calculations.
• Minimization problem can be tricky.
• Can we use ML to find out which correlations are relevant and thereby diminish the dimensionality problem in say CC or SRG theories?
• Quantum Boltzmann Machines (QBMs) to devise better quantum gates for quantum computations and many more exciting topics
• QBMs can be utilized to train models for data from experiments with quantum systems. They might be employed for combinatorial optimization. Classical BMs have been investigated in this context and developing and analyzing quantum algorithms for combinatorial optimization is an active area of research.
• All in all, there are many possible applications which still have to be explored with many more exciting research avenues. See for example Maria Schuld et al on evaluating analytic gradients on quantum hardware.