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 in nuclear physics. And why this could (or should) be of interest. And how this can be linked with standard many-body theories. And perhaps some quantum computing.
I will also try to highlight educational initiatives and strategies where the ECT* has played and will/can play an important role in educating the next generation of nuclear scientists
Parts of this talk are based on Artificial Intelligence and Machine Learning in Nuclear Physics, Amber Boehnlein et al., Reviews Modern of Physics 94, 031003 (2022)
Jane Kim (MSU), Julie Butler (MSU), Patrick Cook (MSU), Danny Jammooa (MSU), Daniel Bazin (MSU), Dean Lee (MSU), Witek Nazarewicz (MSU), Michelle Kuchera (Davidson College), Even Nordhagen (UiO), Robert Solli (UiO, Expert Analytics), Bryce Fore (ANL), Alessandro Lovato (ANL), Stefano Gandolfi (LANL), Francesco Pederiva (UniTN), and Giuseppe Carleo (EPFL). Niyaz Beysengulov and Johannes Pollanen (experiment, MSU); Zachary Stewart, Jared Weidman, and Angela Wilson (quantum chemistry, MSU) Jonas Flaten, Oskar, Leinonen, Øyvind Sigmundson Schøyen, Stian Dysthe Bilek, and Håkon Emil Kristiansen (UiO). Marianne Bathen and Lasse Vines (experiments (UiO). Excuses to those I have omitted.
Training in Advanced Low Energy Nuclear Theory, aims at providing an advanced and comprehensive training to graduate students and young researchers in low-energy nuclear theory. The initiative is a multinational network between several Asian, European and Northern American institutions and aims at developing a broad curriculum that will provide the platform for a cutting-edge theory for understanding nuclei and nuclear reactions.
These objectives are met by offering series of lectures, commissioned from experienced teachers in nuclear theory. The educational material generated under this program will be collected in the form of WEB-based courses, textbooks, and a variety of modern educational resources. No such all-encompassing material is available at present; its development will allow dispersed university groups to profit from the best expertise available.
The ECT* and the INT have played central roles here.
Many lectures from DTPs and Talent courses were collected in a Lecture Notes in Physics (volume 936, see https://link.springer.com/book/10.1007/978-3-319-53336-0)
The ECT* should continue to play a key role in educating the next generation of nuclear scientists.
Introducing the Lattice Virtual Academy (LaVA) by Claudio Bonanno et al, see for example https://indico.fnal.gov/event/57249/contributions/271752/attachments/169437/227392/talk_LaVA_Bonanno.pdf.
Under development within STRONG-2020 EU-funded project and technical and financial support by FBK/ECT and by INFN.
If you wish to have a critical read on AI/ML from a societal point of view, see Kate Crawford's recent text Atlas of AI
Here: with AI/ML we intend a collection of machine learning methods with an emphasis on statistical learning and data analysisAn important and emerging field is what has been dubbed as scientific ML, see the article by Deiana et al Applications and Techniques for Fast Machine Learning in Science, arXiv:2110.13041
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
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:
The large amount of degrees of freedom pertain to both theory and experiment in nuclear physics. With increasingly complicated experiments that produce large amounts data, automated classification of events becomes increasingly important. Here, deep learning methods offer a plethora of interesting research avenues.
Almost every problem in ML and data science starts with the same ingredients:
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.
Representations of two events from the Argon-46 experiment. Each row is one event in two projections, where the color intensity of each point indicates higher charge values recorded by the detector. The bottom row illustrates a carbon event with a large fraction of noise, while the top row shows a proton event almost free of noise.
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. Basic philosophy: Let a neural network find the optimal wave function
Machine Learning and the Deuteron by Kebble and Rios and Variational Monte Carlo calculations of \( A\le 4 \) nuclei with an artificial neural-network correlator ansatz by Adams et al.
Adams et al: $$ \begin{align} H_{LO} &=-\sum_i \frac{{\vec{\nabla}_i^2}}{2m_N} +\sum_{i < j} {\left(C_1 + C_2\, \vec{\sigma_i}\cdot\vec{\sigma_j}\right) e^{-r_{ij}^2\Lambda^2 / 4 }} \nonumber\\ &+D_0 \sum_{i < j < k} \sum_{\text{cyc}} {e^{-\left(r_{ik}^2+r_{ij}^2\right)\Lambda^2/4}}\,, \label{_auto1} \end{align} $$where \( m_N \) is the mass of the nucleon, \( \vec{\sigma_i} \) is the Pauli matrix acting on nucleon \( i \), and \( \sum_{\text{cyc}} \) stands for the cyclic permutation of \( i \), \( j \), and \( k \). The low-energy constants \( C_1 \) and \( C_2 \) are fit to the deuteron binding energy and to the neutron-neutron scattering length
An appealing feature of the neural network ansatz is that it is more general than the more conventional product of two- and three-body spin-independent Jastrow functions
$$ \begin{align} |\Psi_V^J \rangle = \prod_{i < j < k} \Big( 1-\sum_{\text{cyc}} u(r_{ij}) u(r_{jk})\Big) \prod_{i < j} f(r_{ij}) | \Phi\rangle\,, \label{_auto2} \end{align} $$which is commonly used for nuclear Hamiltonians that do not contain tensor and spin-orbit terms. The above function is replaced by a four-layer Neural Network.
The Hamiltonian of the quantum dot is given by
$$ \hat{H} = \hat{H}_0 + \hat{V}, $$where \( \hat{H}_0 \) is the many-body HO Hamiltonian, and \( \hat{V} \) is the inter-electron Coulomb interactions. In dimensionless units,
$$ \hat{V}= \sum_{i < j}^N \frac{1}{r_{ij}}, $$with \( r_{ij}=\sqrt{\mathbf{r}_i^2 - \mathbf{r}_j^2} \).
This leads to the separable Hamiltonian, with the relative motion part given by (\( r_{ij}=r \))
$$ \hat{H}_r=-\nabla^2_r + \frac{1}{4}\omega^2r^2+ \frac{1}{r}, $$plus a standard Harmonic Oscillator problem for the center-of-mass motion. This system has analytical solutions in two and three dimensions (M. Taut 1993 and 1994).
Predictions made with eleven global mass model and Bayesian model averaging
Beam time and compute cycles are expensive!
Here we describe a method for generating motional entanglement between two electrons trapped above the surface of superfluid helium. In this proposed scheme these electronic charge qubits are laterally confined via electrostatic gates to create an anharmonic trapping potential. When the system is cooled to sufficiently low temperature these in-plane charge qubit states are quantized and circuit quantum electrodynamic methods can be used to control and readout single qubit operations. Perspectives for quantum simulations with quantum dots systems will be discussed.
Electrons on superfluid helium represent (see https://www.youtube.com/watch?v=EuDuM-fe-lA&ab_channel=JoshuahHeath) a promising platform for investigating strongly-coupled qubits.
A systematic investigation of the controlled generation of entanglement between two trapped electrons under the influence of coherent microwave driving pulses, taking into account the effects of the Coulomb interaction between electrons, may be of great interest for quantum information processing using trapped electrons.
Entanglement is the fundamental characteristic that distinguishes quantum systems composed of two or more coupled objects from their classical counterparts. The study of entanglement in precisely engineered quantum systems with countably many degrees of freedom is at the forefront of modern physics and is a key resource in quantum information science (QIS). This is particularly true in the development of two-qubit logic for quantum computation.
The generation of two-qubit entanglement has been demonstrated in a wide variety of physical systems used in present-day quantum computing, including superconducting circuits, trapped ions, semiconductor quantum dots, color-center defects in diamond, and neutral atoms in optical lattices, just to name a few.
Generating an entanglement between two quantum systems rely on exploiting interactions in a controllable way. The details in the interaction Hamiltonian between two systems defines the protocol schemes for two-qubit logic.
In superconducting circuits the interaction between qubits may arise from direct capacitive coupling between circuit elements or by indirect coupling of two qubits to a common resonator (virtually populating resonator mode) which results in a non-local Hamiltonian in the form of exchange interaction. This allow to implement various schemes for entanglement, such as controlled-phase gate, resonator-induced phase gate, cross-resonance gates etc.
Entanglement gates in trapped ions are produced by means of the Coulomb interaction, where shared motional modes of two or more ions, entangled to their internal states, used for transferring excitations between ion qubits. This has been experimentally demonstrated.
In photonic quantum computing schemes two-qubit entangling operations are realized by nonlinear interactions between two photons scattering from quantum dots, plasmonic nanowires, diamond vacancy centers and others embedded into waveguides. Two-qubit gates in semiconductor quantum dots are based on spin-spin exchange interactions or generated by coupling to a superconducting resonator via artificial spin-orbit interaction.
Coulomb interaction governed entanglement can be realized in the system of electrons on the surface of superfluid helium, where qubit states are formed by in-plane lateral motional or out-of plane Rydberg states. Trapped near the surface of liquid helium these states have different spatial charge configurations and the wavefunctions of different electrons do not overlap.
This results in a strong exchange free Coulomb interaction which depends on the states of the electrons. The lack of disorder in the systems also leads to slow electron decoherence, which has attracted interest to the system as a candidate for quantum information processing.
The static Coulomb interaction arises from a virtual photon exchange process between two charge particles according to quantum electrodynamics. This results in a correlated motion of two charges generating quantum entanglement.
Surface state electrons (SSE) 'floating' above liquid helium originates from quantization of electron's perpendicular to the surface motion in a trapping potential formed by attractive force from image charge and a large \( \sim \) 1 eV barrier at the liquid-vacuum interface. At low temperatures the SSE are trapped in the lowest Rydberg state for vertical motion some 11 nm above the helium surface, which is perfectly clean and has a permittivity close to that of vacuum.
The weak interaction with rthe enviroment, which is mainly governed by interaction with quantized surface capillary waves (ripplons) and bulk phonons, ensures long coherence times - a vital ingredient for any qubit platform.
Hamiltonian:
$$ \begin{align} \hat{H} &=\frac{\hat{p}_1^2}{2} + \sum_{i = 1}^7 V_i\alpha_i[\hat{x}_1] + \frac{\hat{p}_2^2}{2} + \sum_{i = 1}^7 V_i\alpha_i[\hat{x}_2] + \frac{\kappa}{\sqrt{(\hat{x}_1-\hat{x}_2)^2 + a^2}} \label{_auto3}\\ &= h[\hat{p}_1,\hat{x}_1] + h[\hat{p}_2,\hat{x}_2] + u[\hat{x}_1,\hat{x}_2] \label{_auto4} \end{align} $$
Hamiltonian:
$$ \begin{align} \hat{H} &= \frac{\hat{p}_1^2}{2} + v[\hat{x}_1] + \frac{\hat{p}_2^2}{2} + v[\hat{x}_2] + \frac{\kappa}{\sqrt{(\hat{x}_1-\hat{x}_2)^2 + a^2}} \label{_auto5}\\ &= h[\hat{p}_1,\hat{x}_1] + h[\hat{p}_2,\hat{x}_2] + u[\hat{x}_1,\hat{x}_2] \label{_auto6} \end{align} $$
Hamiltonian:
$$ \begin{align} \hat{H} &= \frac{\hat{p}_1^2}{2} + v[\hat{x}_1] + \frac{\hat{p}_2^2}{2} + v[\hat{x}_2] + \frac{\kappa}{\sqrt{(\hat{x}_1-\hat{x}_2)^2 + a^2}} \label{_auto7}\\ &\equiv h[\hat{p}_1,\hat{x}_1] + h[\hat{p}_2,\hat{x}_2] + u[\hat{x}_1,\hat{x}_2] \label{_auto8} \end{align} $$Energy states:
$$ \begin{equation} \vert\Phi_k\rangle = \sum_{i < j} C_{ij, k}\frac{\vert \varphi_i \varphi_j\rangle - \vert \varphi_j \varphi_i\rangle}{\sqrt{2}}, \label{_auto9} \end{equation} $$(Slater basis)
Hamiltonian:
$$ \begin{align} \hat{H} &= \frac{\hat{p}_1^2}{2} + v^L[\hat{x}_1] + \frac{\hat{p}_2^2}{2} + v^R[\hat{x}_2] + \frac{\kappa}{\sqrt{(\hat{x}_1-\hat{x}_2)^2 + a^2}} \label{_auto10}\\ &\equiv h^L[\hat{p}_1,\hat{x}_1] + h^R[\hat{p}_2,\hat{x}_2] + u[\hat{x}_1,\hat{x}_2] \label{_auto11} \end{align} $$Energy states:
$$ \begin{equation} \vert\Phi_k\rangle = \sum_{i} \sum_{j} C_{ij, k}\vert \varphi^L_i \varphi^R_j\rangle, \label{_auto12} \end{equation} $$(product basis)
Energy states:
$$ \begin{equation} \vert\Phi_k\rangle = \sum_{i} \sum_{j} C_{ij, k}\vert \varphi^L_i \varphi^R_j\rangle, \label{_auto13} \end{equation} $$(product basis)
Hamiltonian:
$$ \begin{align} \hat{H} &= \frac{\hat{p}_1^2}{2} + v^L[\hat{x}_1] + \frac{\hat{p}_2^2}{2} + v^R[\hat{x}_2] + \frac{\kappa}{\sqrt{(\hat{x}_1-\hat{x}_2)^2 + a^2}} \label{_auto14}\\ &\equiv h^L[\hat{p}_1,\hat{x}_1] + h^R[\hat{p}_2,\hat{x}_2] + u[\hat{x}_1,\hat{x}_2] \label{_auto15} \end{align} $$Energy states:
$$ \begin{equation} \vert\Phi_k\rangle = \sum_{i} \sum_{j} C_{ij, k}\vert \varphi^L_i \varphi^R_j\rangle, \label{_auto16} \end{equation} $$(Hartree basis)
Hamiltonian:
$$ \begin{align} \hat{H} &= \frac{\hat{p}_1^2}{2} + v^L[\hat{x}_1] + \frac{\hat{p}_2^2}{2} + v^R[\hat{x}_2] + \frac{\kappa}{\sqrt{(\hat{x}_1-\hat{x}_2)^2 + a^2}} \label{_auto17}\\ &\equiv h^L[\hat{p}_1,\hat{x}_1] + h^R[\hat{p}_2,\hat{x}_2] + u[\hat{x}_1,\hat{x}_2] \label{_auto18} \end{align} $$Energy states:
$$ \begin{equation} \vert\Phi_k\rangle = \sum_{i = 0}^{N^L} \sum_{j = 0}^{N^R} C_{ij, k}\vert \varphi^L_i \varphi^R_j\rangle, \label{_auto19} \end{equation} $$(Hartree basis)
Energy states:
$$ \begin{equation} \vert\Phi_k\rangle = \sum_{i = 0}^{N^L} \sum_{j = 0}^{N^R} C_{ij, k}\vert \varphi^L_i \varphi^R_j\rangle, \label{_auto20} \end{equation} $$(Hartree basis)
Hamiltonian:
$$ \begin{align} \hat{H} &= \frac{\hat{p}_1^2}{2} + v^L[\hat{x}_1] + \frac{\hat{p}_2^2}{2} + v^R[\hat{x}_2] + \frac{\kappa}{\sqrt{(\hat{x}_1-\hat{x}_2)^2 + a^2}} \label{_auto21}\\ &\equiv h^L[\hat{p}_1,\hat{x}_1] + h^R[\hat{p}_2,\hat{x}_2] + u[\hat{x}_1,\hat{x}_2] \label{_auto22} \end{align} $$Energy states:
$$ \begin{equation} \vert\Phi_k\rangle = \sum_{i = 0}^{N^L} \sum_{j = 0}^{N^R} C_{ij, k}\vert \varphi^L_i \varphi^R_j\rangle, \label{_auto23} \end{equation} $$(Hartree basis)
By adjusting the potential we can change the anharmonicities and detuning of the wells.
We search for well configurations corresponding to three different types of interaction between the two electrons.
Both anharmonicity and detuning changes with the shape of our well. We create a voltage parameterization
$$ \begin{equation} V(\lambda) = (1-\lambda)V_\mathrm{I} + \lambda V_\mathrm{III} \label{_auto24} \end{equation} $$
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
$$ \begin{equation} \psi_{T}(\boldsymbol{r}_1,\boldsymbol{r}_2) = C\exp{\left(-\alpha_1\omega(r_1^2+r_2^2)/2\right)} \exp{\left(\frac{r_{12}}{(1+\alpha_2 r_{12})}\right)}, \label{eq:trial} \end{equation} $$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 an ansatz for the correlated part of the wave function, it is useful to rewrite the two-particle local energy in terms of the relative and center-of-mass motion. Let us denote the distance between the two electrons as \( r_{12} \). We omit the center-of-mass motion since we are only interested in the case when \( r_{12} \rightarrow 0 \). The contribution from the center-of-mass (CoM) variable \( \boldsymbol{R}_{\mathrm{CoM}} \) gives only a finite contribution. We focus only on the terms that are relevant for \( r_{12} \) and for three dimensions. The relevant local energy operator becomes then (with \( l=0 \))
$$ \lim_{r_{12} \rightarrow 0}E_L(R)= \frac{1}{{\cal R}_T(r_{12})}\left(-2\frac{d^2}{dr_{ij}^2}-\frac{4}{r_{ij}}\frac{d}{dr_{ij}}+ \frac{2}{r_{ij}}\right){\cal R}_T(r_{12}). $$In order to avoid divergencies when \( r_{12}\rightarrow 0 \) we obtain the so-called cusp condition
$$ \frac{d {\cal R}_T(r_{12})}{dr_{12}} = \frac{1}{2} {\cal R}_T(r_{12})\qquad r_{12}\to 0 $$The above results in
$$ {\cal R}_T \propto \exp{(r_{ij}/2)}, $$for anti-parallel spins and
$$ {\cal R}_T \propto \exp{(r_{ij}/4)}, $$for anti-parallel spins. This is the so-called cusp condition for the relative motion, resulting in a minimal requirement for the correlation part of the wave fuction. For general systems containing more than say two electrons, we have this condition for each electron pair \( ij \).
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
$$ \begin{align} \bar{E}_{ij} &= 2\left[ \langle (\frac{\bar{\Psi}_{ij}}{\Psi}+\frac{\bar{\Psi}_{j}}{\Psi}\frac{\bar{\Psi}_{i}}{\Psi})(E_L-\langle E\rangle)\rangle -\langle \frac{\bar{\Psi}_{i}}{\Psi}\rangle\bar{E}_j-\langle \frac{\bar{\Psi}_{j}}{\Psi}\rangle\bar{E}_i\right] \label{_auto25}\\ \nonumber &+\langle \frac{\bar{\Psi}_{i}}{\Psi}E_L{_j}\rangle +\langle \frac{\bar{\Psi}_{j}}{\Psi}E_L{_i}\rangle -\langle \frac{\bar{\Psi}_{i}}{\Psi}\rangle\langle E_L{_j}\rangle \langle \frac{\bar{\Psi}_{j}}{\Psi}\rangle\langle E_L{_i}\rangle. \end{align} $$Evaluating the cost function means having to evaluate the above second derivative of the energy.