Clark Research Group

Computational Condensed Matter

Our group works at the intersection of quantum information, condensed matter, and computing. We probe the emergent phenomena in quantum materials through simulations; develop new algorithms to be used on quantum computers; and determine the nature of entanglement phase transititions. The simple rules of quantum mechanics are responsible both for a rich array of dynamics and properties seen in quantum materials and devices; as well as a computational power that goes beyond that of classical machines.

We use a largerly computational approach or perspective to these problems. Simulation techniques in our toolbox include all forms of quantum Monte Carlo (PIMC, AFQMC, VMC,DMC), tensor networks (MERA,MPS,PEPS) and machine learning. Not only do we use known techniques, we regularly develop new algorithms broadening the horizon of what's possible.

This semester I'm excited to be teaching 498CMP.You will learn how to simulate a quantum computer, do numerical RG, build a restricted Boltzmann machine and computer Chern numbers.

Di Luo and Zhou Chen are presenting a poster and talk at the Neurips 2020 workshop on Machine Learning and the Physical Sciences.

Bryan will be (virtually) presenting at DIAS on our new work on what numerics can tell us about the many-body localization transition.

We've posted our new paper on using gauge equivariant neural networks for quantum lattice gauge theories. When writing a neural network wave-function for a gauge theory, you want your wave-function to be gauge-invariant. You could do this by giving your neural network only information that is gauge invariant but this severly limits the states you can represent. Instead if you make it gauge equivariant then it is much more flexible. That's what we do in this work.

We have an extended abstract in the Neurips workshop for simulating physical systems extending some of our work on doing quantum dynamics using the POVM formalism.

Di will be virtually presenting his work on quantum computing today at MIT.

Bryan will be (virtually) presenting at DIAS on our new work on what numerics can tell us about the many-body localization transition.

We've posted our new paper on using gauge equivariant neural networks for quantum lattice gauge theories. When writing a neural network wave-function for a gauge theory, you want your wave-function to be gauge-invariant. You could do this by giving your neural network only information that is gauge invariant but this severly limits the states you can represent. Instead if you make it gauge equivariant then it is much more flexible. That's what we do in this work.

We have an extended abstract in the Neurips workshop for simulating physical systems extending some of our work on doing quantum dynamics using the POVM formalism.

Di will be virtually presenting his work on quantum computing today at MIT.

Ryan will be presenting his work on parallelizing DMRG at Supercomputing 2020.

Bryan is giving the seminar today at NCSA's new Center for Artificial Intelligence Innovation. It will discuss the application of artificial intelligence to
improve simulations in quantum physics and quantum computing.

Our paper on combining Transformers and the POVM formalism to do simulations of open quantum systems has now
been posted. By mapping quantum states to probability distributions, we are
able to take advantage of state-of-the-art machine learning architectures.

We wish Eli luck this semester on his internship at Honeywell working on quantum computing.

Bryan received tenure and has been promoted to Associate Professor!

We're excited to be part of the newly funded (thanks DOE!) quantISED project for quantum computing applied to high energy physics.

I'll be teaching Introduction to Computing in Physics. We will be (among other things) simulating fluid dynamics; doing N-body gravity simulations; show chaotic behavior in pendula; simulate ecosystems; and running on quantum computers. Come join us!

We wish Eli luck this semester on his internship at Honeywell working on quantum computing.

Bryan received tenure and has been promoted to Associate Professor!

We're excited to be part of the newly funded (thanks DOE!) quantISED project for quantum computing applied to high energy physics.

I'll be teaching Introduction to Computing in Physics. We will be (among other things) simulating fluid dynamics; doing N-body gravity simulations; show chaotic behavior in pendula; simulate ecosystems; and running on quantum computers. Come join us!

We've posted our new paper on using reinforcement learning to discover a new class of
protocols (the *jump-move-jump* approach) for moving around Majoranas with high fidelity.

We're excited to be part of the newly funded (thanks DOE!) quantum center, Q-Next.

I will be virtually presenting at the CTC PI conference today on our work on algorithms for quantum simulations. Stop by and chat if you're around!

Our new paper on converting (projected or powers of) Slater Determinants (and other arbitrary mean field eigenstates) to infinite matrix product states has now been posted. Our first application of this has been to compute the entanglement spectra of Slave-Fermion solutions to the BLBQ model and compare them to their exact states.

We have posted our open-source parallel tensor-tools code. DMRG is now ready for high-performance computing.

We're excited to be part of the newly funded (thanks DOE!) quantum center, Q-Next.

I will be virtually presenting at the CTC PI conference today on our work on algorithms for quantum simulations. Stop by and chat if you're around!

Our new paper on converting (projected or powers of) Slater Determinants (and other arbitrary mean field eigenstates) to infinite matrix product states has now been posted. Our first application of this has been to compute the entanglement spectra of Slave-Fermion solutions to the BLBQ model and compare them to their exact states.

We have posted our open-source parallel tensor-tools code. DMRG is now ready for high-performance computing.

I've always said that DMRG is the most important algorithm in physics which (in practice) no one runs in parallel. We've taken a key step in
resolving this problem: see our paper where we get a factor of 10 decrease in wall-clock time
at no additional node-cost for large bond-dimension J1-J2 spin-models.

We're excited to be part of the newly funded (thanks NSF!) HQAN initiative. We look forward to generating a lot of exciting science under this program.

Eli has posted his new code to find l-bits in MBL phases. It works in arbitrary dimensions and arbitrary geometries letting the community tackle MBL in previously inaccessible regimes.

There are more then 4000 papers on many-body localization and only 6 attempt to theoretically address dimensions greater then one. We have posted our new paper which gives a novel approach for solving two and three dimensional MBL systems determining the critical point between the MBL and ergodic phases in two and three dimensions.

Check out our new paper which shows (among many other things) that the MBL-ergodic transition has typical correlations which decay with a stretched exponential with exponent 1/2. Interestingly, this is the same behavior for typical correlations seen in the random singlet phase.

We're excited to be part of the newly funded (thanks NSF!) HQAN initiative. We look forward to generating a lot of exciting science under this program.

Eli has posted his new code to find l-bits in MBL phases. It works in arbitrary dimensions and arbitrary geometries letting the community tackle MBL in previously inaccessible regimes.

There are more then 4000 papers on many-body localization and only 6 attempt to theoretically address dimensions greater then one. We have posted our new paper which gives a novel approach for solving two and three dimensional MBL systems determining the critical point between the MBL and ergodic phases in two and three dimensions.

Check out our new paper which shows (among many other things) that the MBL-ergodic transition has typical correlations which decay with a stretched exponential with exponent 1/2. Interestingly, this is the same behavior for typical correlations seen in the random singlet phase.

I'm excited to be attending and giving a talk at the CCQ Machine learning conference.
I'm speaking about our recent work to overcome the obstacle of using machine learning
to simulate Fermions.

Eli has posted his QOSY code to find
all Hamiltonians which commute with a given symmetry.

One of the outstanding questions about the MBL phase is understanding the phase transition. Benjamin has posted a new paper which clarifies how hybridization happens at the transition. Consider tuning from the MBL to ergodic phase. States are hybridizing with each other during this process. In the MBL phase, this hybridization is local. In the ergodic phase, it is dominated by states which differ non-locally (there are just more then this). There is a delicate balance exactly at the transition where collisions become sufficiently delocalized to exactly cancel out this combinatorial factor leading to a breakdown of the MBL phase!

One of the outstanding questions about the MBL phase is understanding the phase transition. Benjamin has posted a new paper which clarifies how hybridization happens at the transition. Consider tuning from the MBL to ergodic phase. States are hybridizing with each other during this process. In the MBL phase, this hybridization is local. In the ergodic phase, it is dominated by states which differ non-locally (there are just more then this). There is a delicate balance exactly at the transition where collisions become sufficiently delocalized to exactly cancel out this combinatorial factor leading to a breakdown of the MBL phase!

Topology and the one-dimensional
Kondo-Heisenberg model is now out in Physical Review B.

Congratulations to Benjamin Villalonga who has passed his thesis defense today! We wish him luck on his next step at Google Quantum.

I'm giving a talk today at (virtual) Yale on variational wavefunctions in the era of AI.

Check out our paper Deep Learning Enabled Strain Mapping of Single-Atom Defects in Two-Dimensional Transition Metal Dichalcogenides with Sub-Picometer Precision which has now come out in Nano Letters.

Congratulations to Benjamin Villalonga who has passed his thesis defense today! We wish him luck on his next step at Google Quantum.

I'm giving a talk today at (virtual) Yale on variational wavefunctions in the era of AI.

Check out our paper Deep Learning Enabled Strain Mapping of Single-Atom Defects in Two-Dimensional Transition Metal Dichalcogenides with Sub-Picometer Precision which has now come out in Nano Letters.

Please come by and see my invited March Meeting talk on using machine learning to improve
variational wavefunctions (and their optimization).

Our paper on looking (and not finding)
majorana zero modes in a Kondo-Heisenberg ladder has now been posted.

Check out our new work
where we have been collaborating with Pinshane Huang and her group to develop
machine learning techniques to apply to scanning transmission electron microscopy.
In this work, we've measured strain fields in WSeTe to unparalleled accuracy.

I'm teaching Computing in Physics this semester where we will be building a quantum computer simulator from scratch; doing numerical RG; programming restricted Boltzmann machines; and computing Chern numbers.

Check out our conference proceedings (arxiv) on using dipolar molecular systems as analog quantum simulators.

I'm teaching Computing in Physics this semester where we will be building a quantum computer simulator from scratch; doing numerical RG; programming restricted Boltzmann machines; and computing Chern numbers.

Check out our conference proceedings (arxiv) on using dipolar molecular systems as analog quantum simulators.

Most physical systems of sufficient complexity can simulate each other with only a polynomial slow-down. This is known as the extended Church Turing thesis. Bouncing billiard balls compute at (roughly) the same speed as your macbook (and visa versa). Quantum computing is the only known counterexample to this thesis. Algorithms such as factoring and finding the ground state of reasonable physics systems appear to be classically difficult but efficient on a quantum computer.

We have broad interests in the area of quantum computing. One of our primary activities is to figure out how to leverage the power of quantum mechanics to solve classically intractable problems. For example, we have developed better algorithms for simulating chemistry and physical systems ranging from algorithms which improve computing observables to improvements for variational quantum eigensolvers in the NISQ era.

Another area of interest has been developing improved classical simulation of quantum computers. While tensor networks are a de-facto standard for one dimension, it is not clear how to best classically simulate quantum circuits in two dimensions. We have recently presented an approach which maps quantum states → positive probability distributions → Transformers (a State-of-the-art machine learning primitive). Then we use Monte carlo to update the Transformer upon the application of a quantum gate.

We have also been collaborating with experimentalists to develop analog simulation approaches to map dipolar molecular Hamiltonians (i.e. as found in Bryce Gadway's lab) to high-energy physics models.

Other areas within QIS we have worked in include the entanglement of quantum circuits, majorana protocol development with machine learning, Hamiltonian learning, tensor networks, and understanding the line between quantum and classical computing.

Quantum materials support a rich array of exotic phenomena such as superconductivity and heavy fermion behavior.
We've recently been interested in *frustrated magnets *- insulating materials with
magnetic spins living on lattices of triangles.

(Ferro)-magnetism has been known since the ancient Greeks. We now know that magnetism is a collective effect where a macroscopic number of spins align in the same direction. If instead, the spins are frustrated, no simple pattern of spins can can form and exotic phenomena such as spin-liquids can appear.

We have recently discovered a quantum (spin 1/2) Hamiltonian on the kagome lattice (corner sharing triangles) with exponential degneracy from which all phases on the kagome lattice seem to arise. This replaces the former lore (classical frustration in the Ising limit) for why there is rich physics in frustrated systems with a more fundamental quantum understanding in terms of proximity to an exponentially degenerate Hamiltonian.

We also are interested in finding simple Hamiltonians with exotic behavior. We've recently shown that the stuffed honeycomb lattice has a rich phase diagram supporting a multitude of classical and quantum phases including some interesting spin liquids phase(s).

We also work closely with experimentalists; for example, we have worked with Greg MacDougall finding effective Hamiltonians for a variety of spinel materials.

To learn more about frustrated magnetism, see my talk at the Perimeter Institute on variational methods applied to the honeycomb and kagome lattice.

Algorithms for the Quantum Many Body Problem

The key to an improved understanding of the quantum many-body problem is improved algorithms. Areas we have been recently working on include

* Machine Learning for wave-functions: * A wave-function is a box that takes
configurations (i.e. where are my electrons) and gives back a complex number. We have recently
been using machine-learning architectures to replace this black-box. We've figured out how
to simulate Fermions using neural networks; developed a new way to use transformers to simulate
quantum circuits; and determined how machine learning approaches can better approximate
the fixed-point dynamics of open quantum systems.

In a phase transition, there is a sharp change as we tune some parameter. In the last decade, a qualitatively new type of phase transitions has been discovered - entanglement phase transitions where the scaling of the entanglement changes sharply. A canonical example of such a transition is between the ergodic phase and the many-body localized (MBL) phase. The MBL phase is a phase of matter with area-law entanglement where thermalization breaks down. When you put a hot coffee down on your desk, it eventually cools to the temperature of the room. The MBL phase is the moral equivalent of the never-cooling cup of coffee.

Our work has focused on two primary activities: (1) developing a tensor-network conceptualization as well as algorithmic approach to the MBL phase and (2) using simulations to understand the MBL-ergodic transition.

We showed that one can characterize MBL matter as a phase which can be diagonalized
by short unitary tensor networks. This also implies a very elegant characterization of the entire
spectra of MBL eigenstates. Every MBL eigenstate can be selected by choosing, for each site,
either the ground state or excited state tensor and combining them together. These two ways of
viewing the MBL phase imply (and are implied by) the l-bit formulation of MBL. In addition
to this tensor-network characterization of MBL, we developed two additional algorithms for
working with the MBL phase: (1) *SIMPS/ES-DMRG* a DMRG-esqe approach for finding
interior MBL eigenstates and (2) *Tensorified Wegner-Wilson Flow*, an approach for diagonalizing
an MBL Hamiltonian.

More recently, we have been focused on developing an understanding of the MBL-ergodic transition. We have discovered two key properties of the transition: the entanglement is bimodal and that the typical correlation at the transition goes as a stretched exponential with exponent 1/2. In addition, we were able to show that the transition is driven by eigenstates hybridizing non-locally (but in a qualitatively different way from the ergodic phase.)

We have also recently developed a way to tackle the MBL transition in two and three dimensions.

I am currently teaching: **An Introduction to Modern Computational Physics ** [link]

I am heavily involved in the computational aspects of our physics curriculum at Illinois. I've developed from scratch two of our
computational courses: *Computing in Physics* and
*An Algorithmic Perspective on Strongly Correlated Systems* and have
done significant work improving and adding units to An Introduction to Modern Computational Physics
(i.e *fluid dynamics, predator and prey, chaos, n-body simulations, machine learning, and quantum computing*).

Other courses I've previously taught include
Graduate Quantum Mechanics (I, II),
*Atomic Scale Simulations *,
*Undergraduate Quantum Mechanics*,
among others (see here).

Besides courses, I've given various summer school lectures (see here).

You might also be interested in learning about:

*Variational Monte Carlo:* See my notes
and video from the Boulder Summer School; the
videos [part 1, part 2] from
the Cornell Summer School on Emergent Phenomena; or my tutorials.

*Diffusion Monte Carlo:* See my notes
and video from the Boulder Summer School.

*Path Integral Monte Carlo: * See my tutorials (with Ken Esler and Paul Yubo).

*Density Matrix Renormalization Group: * Work through problem set 3 and problem set 4 from my graduate course to build a simple DMRG code in Julia.