Machine learning and quantum computing in physics research and education

The main emphasis is to give you a short and pedestrian introduction to the whys and hows we can use (with several examples) machine learning methods and quantum technologies in physics. And why this could (or should) be of interest. I will also try to link to educational activities.

These slides are at https://mhjensenseminars.github.io/MachineLearningTalk/doc/pub/Tsukubasymposium/html/Tsukubasymposium-reveal.html

• Parts of the talk based on Artificial Intelligence and Machine Learning in Nuclear Physics, Amber Boehnlein et al., Reviews Modern of Physics 94, (2022)
• See also Predicting solid state material platforms for quantum technologies Oliver Lerstøl Hebnes et al, Nature Materials Communications, (2002).

• Machine Learning applied to classical and quantum mechanical systems and analysis of physics experiments
• Quantum Engineering
• Quantum algorithms
• Quantum Machine Learning
• Educational initiatives

• MSU: Ben Hall, Jane Kim, Julie Butler, Danny Jammoa, Nicholas Cariello, Johannes Pollanen (Expt), Niyaz Beysengulov (Expt), Dean Lee, Scott Bogner, Heiko Hergert, Matt Hirn, Huey-Wen Lin, Alexei Bazavov, Andrea Shindler and Angela Wilson
• UiO: Stian Bilek, Heine Åbø (OsloMet), Håkon Emil Kristiansen, Øyvind Schøyen Sigmundsson, Jonas Boym Flaten, Linus Ekstrøm, Oskar Leinoen, Kristian Wold (OsloMet), Lasse Vines (Expt) Andrej Kuznetsov (Expt), David Rivas Gongora (Expt) and Marianne Bathen (Expt, now PD at ETH)

This work is supported by the U.S. Department of Energy, Office of Science, office of Nuclear Physics under grant No. DE-SC0021152 and U.S. National Science Foundation Grants No. PHY-1404159 and PHY-2013047 and the Norwegian ministry of Education and Research for PhD fellowships.

The authors discuss applications and techniques for fast machine learning (ML) in science – the concept of integrating power ML methods into the real-time experimental data processing loop to accelerate scientific discovery. The report covers three main areas

1. applications for fast ML across a number of scientific domains;
2. techniques for training and implementing performant and resource-efficient ML algorithms;
3. and computing architectures, platforms, and technologies for deploying these algorithms.

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.

• Machine learning for data mining: Oftentimes, it is necessary to be able to accurately calculate observables that have not been measured, to supplement the existing databases.
• Density functional theory: Energy density functional calibration involving Bayesian optimization and deep learning ML. A promising avenue for ML applications is the emulation of DFT results.
• Nuclear physics properties with ML: Improving predictive power of nuclear models by emulating model residuals.
• Effective field theory and nuclear many-body systems: Truncation errors and low-energy coupling constant calibration, nucleon-nucleon scattering calculations, variational calculations with ANN for light nuclei, NN extrapolation of nuclear structure observables
• Nuclear shell model UQ: ML methods have been used to provide UQ of configuration interaction calculations.

• Low-energy nuclear reactions UQ: Bayesian optimization studies of the nucleon-nucleus optical potential, R-matrix analyses, and statistical spatial networks to study patterns in nuclear reaction networks.
• Neutron star properties and nuclear matter equation of state: constraining the equation of state by properties on neutron stars and selected properties of finite nuclei
• Experimental design: Bayesian ML provides a framework to maximize the success of on experiment based on the best information available on existing data, experimental conditions, and theoretical models.

• Reconstruction of particle trajectories or classification of events are typical examples where ML methods are being used. However, since these data can often be extremely noisy, the precision necessary for discovery in physics requires algorithmic improvements. Research along such directions, interfacing nuclear physics with AI/ML is expected to play a significant role in physics discoveries related to new facilities. The treatment of corrupted data in imaging and image processing is also a relevant topic.
• Design of detectors represents an important area of applications for ML/AI methods in nuclear physics.

• An important application of AI/ML methods is to improve the estimation of bias or uncertainty due to the introduction of or lack of physical constraints in various theoretical models.
• In theory, we expect to use AI/ML algorithms and methods to improve our knowledge about correlations of physical model parameters in data for quantum many-body systems. Deep learning methods show great promise in circumventing the exploding dimensionalities encountered in quantum mechanical many-body studies.
• Merging a frequentist approach (the standard path in ML theory) with a Bayesian approach, has the potential to infer better probabilitity distributions and error estimates. As an example, methods for fast Monte-Carlo- based Bayesian computation of nuclear density functionals show great promise.
• Machine Learning and Quantum Computing is a very interesting avenue to explore. See for example talk of Sofia Vallecorsa.

• 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.
• Report from the A.I. For Nuclear Physics Workshop by Bedaque et al., Eur J. Phys. A 57, (2021)
• Particle Data Group summary on ML methods
• And the BAND collaboration at https://bandframework.github.io/

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 Gaussian quadrature will not be adequate for say the computation of the energy of a many-body system. Basic philosophy: Let a neural network find the optimal wave function

1. be scalable
2. have qubits that can be entangled
3. have reliable initializations protocols to a standard state
4. have a set of universal quantum gates to control the quantum evolution
5. have a coherence time much longer than the gate operation time
6. have a reliable read-out mechanism for measuring the qubit states
7. and many more

1. Time-Dependent full configuration interaction theory
2. Time-dependent Coupled-Cluster theory
3. Designing quantum circuits

1. General university course on quantum mech and quantum technologies
2. Information Systems
3. From Classical Information theory to Quantum Information theory
4. Classical vs. Quantum Logic
5. Classical and Quantum Laboratory
6. Discipline-Based Quantum Mechanics
7. Quantum Software
8. Quantum Hardware
9. more

• Need for AI/Machine Learning in physics, lots of ongoing activities
• To solve many complex problems and facilitate discoveries, multidisciplinary efforts efforts are required involving scientists in physics, statistics, computational science, applied math and other fields.
• There is a need for focused AI/ML learning efforts that will benefit accelerator science and experimental and theoretical programs
• How do we develop insights, competences, knowledge in statistical learning that can advance a given field?

• For example: Can we use ML to find out which correlations are relevant and thereby diminish the dimensionality problem in standard many-body theories?
• Can we use AI/ML in detector analysis, accelerator design, analysis of experimental data and more?
• Can we use AL/ML to carry out reliable extrapolations by using current experimental knowledge and current theoretical models?
• The community needs to invest in relevant educational efforts and training of scientists with knowledge in AI/ML. These are great challenges to the CS and DS communities
• Quantum technologies require directed focus on educational efforts
• Most likely tons of things I have forgotten

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.

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

 
$$ \begin{equation} \psi_{T}(\boldsymbol{r}_1,\boldsymbol{r}_2) = C\exp{\left(-\alpha_1\omega(r_1^2+r_2^2)/2\right)} \exp{\left(\frac{r_{12}}{(1+\alpha_2 r_{12})}\right)}, \tag{3} \end{equation} $$

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).

We have a model, our likelihood function.

How should we define the cost function?

Suppose the trial function (our model) is the exact wave function. The action of the hamiltionan on the wave function

 
$$ H\Psi = \mathrm{constant}\times \Psi, $$

The integral which defines various expectation values involving moments of the hamiltonian becomes then

 
$$ \langle E^n \rangle = \langle H^n \rangle = \frac{\int d\boldsymbol{R}\Psi^{\ast}(\boldsymbol{R})H^n(\boldsymbol{R})\Psi(\boldsymbol{R})} {\int d\boldsymbol{R}\Psi^{\ast}(\boldsymbol{R})\Psi(\boldsymbol{R})}= \mathrm{constant}\times\frac{\int d\boldsymbol{R}\Psi^{\ast}(\boldsymbol{R})\Psi(\boldsymbol{R})} {\int d\boldsymbol{R}\Psi^{\ast}(\boldsymbol{R})\Psi(\boldsymbol{R})}=\mathrm{constant}. $$

 
This gives an important information: If I want the variance, the exact wave function leads to zero variance!

The variance is defined as

 
$$ \sigma_E = \langle E^2\rangle - \langle E\rangle^2. $$

Variation is then performed by minimizing both the energy and the variance.

We can then take the derivatives of

 
$$ \sigma_E = \langle E^2\rangle - \langle E\rangle^2, $$

with respect to the variational parameters. The derivatives of the variance can then be used to defined the so-called Hessian matrix, which in turn allows us to use minimization methods like Newton's method or standard gradient methods.

This leads to however a more complicated expression, with obvious errors when evaluating integrals by Monte Carlo integration. Less used, see however Filippi and Umrigar. The expression becomes complicated

 
$$ \begin{align} \bar{E}_{ij} &= 2\left[ \langle (\frac{\bar{\Psi}_{ij}}{\Psi}+\frac{\bar{\Psi}_{j}}{\Psi}\frac{\bar{\Psi}_{i}}{\Psi})(E_L-\langle E\rangle)\rangle -\langle \frac{\bar{\Psi}_{i}}{\Psi}\rangle\bar{E}_j-\langle \frac{\bar{\Psi}_{j}}{\Psi}\rangle\bar{E}_i\right] \tag{4}\\ \nonumber &+\langle \frac{\bar{\Psi}_{i}}{\Psi}E_L{_j}\rangle +\langle \frac{\bar{\Psi}_{j}}{\Psi}E_L{_i}\rangle -\langle \frac{\bar{\Psi}_{i}}{\Psi}\rangle\langle E_L{_j}\rangle \langle \frac{\bar{\Psi}_{j}}{\Psi}\rangle\langle E_L{_i}\rangle. \end{align} $$

Evaluating the cost function means having to evaluate the above second derivative of the energy.

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:

 
$$ \begin{align} E(\mathbf{x}, \mathbf{h}) = - \sum_i^M x_i a_i- \sum_j^N b_j h_j - \sum_{i,j}^{M,N} x_i w_{ij} h_j, \tag{7} \end{align} $$

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:

 
$$ \begin{align} E(\mathbf{x}, \mathbf{h}) = \sum_i^M \frac{(x_i - a_i)^2}{2\sigma_i^2} - \sum_j^N b_j h_j - \sum_{i,j}^{M,N} \frac{x_i w_{ij} h_j}{\sigma_i^2}. \tag{8} \end{align} $$