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.
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.
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.
Machine Learning for Experiment and Protocol Development: We have been applying machine learning to the analysis of experimental data; we have been working with Pinshane and her group analyzing scanning tunneling electron microscopy images of atomic defects. Separately, we have been developing ways to use differential programming and reinforcment learning to develop new protocols for moving around Majorana's.
Inverse Methods: We have pioneered a new inverse approach to condensed matter systems. While the canonical approach to condensed matter is to start with a Hamiltonian and determine the ground states and its properties, we have developed techniques which start with a ground state or targeted symmetries and instead find the Hamiltonians. We have recently used this technique to find a new set of spin-liquid Hamiltonians.
Tensor Networks: Tensor networks are a powerful class of algorithms that use low entanglement states made from contracting tensors as ansatz for the ground state of a quantum many-body problem. Our recent work in this area has included a novel algorithm which converts projected Slater Determinants into matrix product states as well as a practical approach to the parallelization of the DMRG algorithm (for spins, we get a 10x speedup in wall-clock time at no additional cost in total node-hours.
Quantum Monte Carlo: QMC techniques simulate the quantum many-body problem by generate a stochastic sample of a targeted wave-function or density matrix. Algorithms we've developed include VAFT, a way to compute finite-temperature properties of quantum systems starting with a manifold of variational wave-functions; a new approach to finding a simnple basis to minimize the effects of the sign problem; and SWO, a novel way to optimize variational wave-functions inspired by machine learning.
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.