// About me

I am a research scientist at the Flatiron Institute Center for Computational Quantum Physics (CCQ) where I develop and apply tensor network computational methods, primarily for quantum many-body systems.
[See my notes on matrix product states and DMRG.]

Tensor networks are a technique to represent very high-order tensors by a contracted network of low-order tensors, allowing one to make an exponential reduction in the parameters needed, while still maintaining accuracy. The prototypical application is the approximation of a many-body quantum wavefunction, but tensor networks are also useful for approximating transfer matrices of classical systems and weights of models used in machine learning.


Download my CV

// Experience

Research Scientist — Flatiron Institute

Sep 2017-Present


Research Scientist — UC Irvine

Feb 2016-Aug 2017


Postdoctoral Researcher — Perimeter Institute for Theoretical Physics

2013-Jan 2016
  • Showed an isotropic, nearest-neighbor 2d parafermion lattice model hosts a non-trivial phase supporting Fibonacci anyons
  • Performed highly cited calculations of universal entanglement terms in critical systems arising from sharp corners
  • Significantly expanded the user base of the ITensor library and developed an ambitious new version 2.0 design


Postdoctoral Researcher — UC Irvine

Groups of Steven R. White and Kieron Burke

2010-2013
  • Discovered a method for parallelizing the density matrix renormalization group in real space.
  • Co-developed an open source library for tensor product wavefunction algorithms. Website: http://itensor.org/.


Graduate Student Researcher — UC Santa Barbara

Advisor: Leon Balents

2005-2010
  • Applied a variety of analytical methods (bosonization, mean-field, spin wave, high temperature series, Monte Carlo) to study frustrated magnets.
  • Collaborated with Steven R. White on a new method for simulating finite temperature quantum systems (the METTS algorithm).

// Education

2005-2010 Ph.D. in Physics, UC Santa Barbara. Advisor: Leon Balents
2000-2005 BS in Physics, Georgia Institute of Technology. Highest honors.
2000-2005 BS in Mathematics, Georgia Institute of Technology. Highest honors.

// Publications

2018
"Towards Quantum Machine Learning with Tensor Networks"
William Huggins, Piyush Patel, K. Birgitta Whaley, E.M. Stoudenmire 1803.11537
2018
"Learning Relevant Features of Data with Multi-scale Tensor Networks"
E.M. Stoudenmire, 1801.00315
2017
"Monte Carlo Tensor Network Renormalization"
William Huggins, C. Daniel Freeman, E.M. Stoudenmire, Norm M. Tubman, K. Birgitta Whaley 1710.03757
2017
"Matrix product state techniques for two-dimensional systems at finite temperature"
Benedikt Bruognolo, Zhenyue Zhu, Steven R. White, E.M. Stoudenmire 1705.05578
2017
"Towards the solution of the many-electron problem in real materials: equation of state of the hydrogen chain with state-of-the-art many-body methods"
Mario Motta, David M. Ceperley, Garnet Kin-Lic Chan, John A. Gomez, Emanuel Gull, Sheng Guo, Carlos Jimenez-Hoyos, Tran Nguyen Lan, Jia Li, Fengjie Ma, Andrew J. Millis, Nikolay V. Prokof'ev, Ushnish Ray, Gustavo E. Scuseria, Sandro Sorella, Edwin M. Stoudenmire, Qiming Sun, Igor S. Tupitsyn, Steven R. White, Dominika Zgid, Shiwei Zhang 1705.01608
2017
"Sliced Basis Density Matrix Renormalization Group for Electronic Structure"
E.M. Stoudenmire and Steven R. White Phys. Rev. Lett., 119: 046401
2016
"Supervised Learning with Quantum-Inspired Tensor Networks"
E.M. Stoudenmire and David J. Schwab Advances in Neural Information Processing Systems, 29 4799
2016
"Unusual Corrections to Scaling and Convergence of Universal Renyi Properties at Quantum Critical Points"
Sharmistha Sahoo, E.M. Stoudenmire, Jean-Marie Stephan, Trithep Devakul, Rajiv R.P. Singh, and Roger Melko, Phys. Rev. B, 93: 085120
2015
"One Dimensional Mimicking of Electronic Structure: The Case for Exponentials"
Thomas E. Baker, E.M. Stoudenmire, Lucas O. Wagner, Kieron Burke, and Steven R. White, Phys. Rev. B, 91: 235141
2015
"Many-body localization in disorder-free systems: The importance of finite-size constraints"
Z. Papic, E.M. Stoudenmire, and Dmitry A. Abanin, Ann. Phys. 362 714
2015
"Assembling Fibonacci Anyons From a Z3 Parafermion Lattice Model"
E.M. Stoudenmire, David J. Clarke, Roger S. K. Mong, and Jason Alicea, Phys. Rev. B, 91: 235112  [Editor's suggestion]
2014
"Corner Contributions to the Entanglement Entropy of Strongly-Interacting O(2) Quantum Critical Systems in 2+1 Dimensions"
E.M. Stoudenmire, Peter Gustainis, Ravi Johal, Stefan Wessel, and Roger G. Melko, Phys. Rev. B, 90: 235106
2014
"Kohn-Sham Calculations with the Exact Functional"
Lucas O. Wagner, Thomas E. Baker, E.M. Stoudenmire, Kieron Burke, and Steven R. White, Phys. Rev. B, 90: 045109  [Editor's suggestion]
2014
"Corner contribution to the entanglement entropy of an O(3) quantum critical point in 2+1 dimensions"
Ann Kallin, E.M. Stoudenmire, Paul Fendley, Rajiv R.P. Singh, Roger G. Melko, J. Stat. Mech. P06009
2013
"Guaranteed Convergence of the Kohn-Sham Equations"
Lucas O. Wagner, E.M. Stoudenmire, Kieron Burke, and Steven R. White, Phys. Rev. Lett. 111: 093003  [Editor's suggestion]
2013
"Real-Space Parallel Density Matrix Renormalization Group"
E.M. Stoudenmire and Steven R. White, Phys. Rev. B 87: 115137
2013
"Topological phases in polar-molecule quantum magnets"
Salvatore R. Manmana, E.M. Stoudenmire, Kaden R.A. Hazzard, Ana Maria Rey and Alexey Gorshkov, Phys. Rev. B 87: 081106(R)
2012
"One-dimensional continuum electronic structure with the density matrix renormalization group and its implications for density functional theory"
E.M. Stoudenmire, Lucas O. Wagner, Steven R. White and Kieron Burke,
Phys. Rev. Lett. 109: 056402
2012
"Reference electronic structure calculations in one dimension"
Lucas O. Wagner, E.M. Stoudenmire, Steven R. White and Kieron Burke,
Phys. Chem. Chem. Phys. 14: 8581
2012
"Studying two-dimensional systems with the density matrix renormalization group"
E.M. Stoudenmire and Steven R. White, Annual Reviews of Condensed Matter Physics 3: 111
2011
"Interaction effects in topological superconducting wires supporting majorana fermions"
E.M. Stoudenmire, Jason Alicea, Oleg A. Starykh and Matthew P.A. Fisher, Phys. Rev. B 84: 014503 
[Editor's suggestion, Synopsis Article]
2010
"Minimally entangled typical thermal state algorithms"
E.M. Stoudenmire and Steven R. White, New J. Phys. 12: 055026
2009
"Quadrupolar correlations and spin freezing in S=1 triangular lattice antiferromagnets"
E.M. Stoudenmire, Simon Trebst and Leon Balents, Phys. Rev. B 79: 214436
2008
"Ordered phases of the anisotropic kagome lattice antiferromagnet in a field"
E.M. Stoudenmire and Leon Balents, Phys. Rev. B 77: 174414
2005
"Magnetoresistive effects in ferromagnet-superconductor multilayers"
E.M. Stoudenmire and C.A.R. Sá de Melo, J. Appl. Phys. 97: 10J108

DMRG for 2D Quantum Systems

The density matrix renormalization group (DMRG), originally designed to find ground states of one-dimensional (1D) quantum systems, is surprisingly effective for two-dimensional (2D) systems and can give very accurate results for systems untreatable by few, if any other methods. But applying DMRG to 2D systems requires great care, since the cost grows exponentially with the linear size of one of the two directions (the "y" direction, say).

To use DMRG effectively for 2D systems, one should use information not only from the largest attainable system size (typically only of the order of 10 sites in the y direction), but from also from smaller systems where DMRG has greater control. In one recent project, we studied the scaling of entanglement at 2D quantum critical points using the numerical linked cluster expansion, which uses information from smaller clusters to reduce finite-size effects on the largest 2D cluster.

In another recent study of a 2D lattice model of coupled "anyons", insight from a 1D field-theoretic picture of our model allowed us to see fingerprints of interesting 2D phases on systems as small as two sites wide!

Related Work:

  • E.M. Stoudenmire, Peter Gustainis, Ravi Johal, Stefan Wessel, and Roger G. Melko, "Corner Contributions to the Entanglement Entropy of Strongly-Interacting O(2) Quantum Critical Systems in 2+1 Dimensions", Phys. Rev. B, 90: 235106 (2014)
  • E.M. Stoudenmire, David J. Clarke, Roger S. K. Mong, and Jason Alicea, "Assembling Fibonacci Anyons From a Z3 Parafermion Lattice Model", arxiv:1501.05305 (2014)
  • E.M. Stoudenmire and Steven R. White, "Real-Space Parallel Density Matrix Renormalization Group", Phys. Rev. B 87: 115137 (2013)
  • E.M. Stoudenmire and Steven R. White, "Studying two-dimensional systems with the density matrix renormalization group", Annual Reviews of Condensed Matter Physics 3: 111 (2012)

Minimally Entangled Typical Thermal States (METTS)

What does a snapshot of a thermal quantum state look like?

Quantum Monte Carlo methods have been very successful in simulating thermal quantum systems by mapping them to classical ensembles which can be efficiently sampled. However, this approach can fail dramatically for systems with mobile fermions or frustrated magnetic exchange interactions, in which case the system is said to have a "sign problem".

One way around the sign problem is to represent a thermal quantum system as an ensemble of pure-state wavefunctions instead of virtual classical states. A particularly efficient ensemble to sample is the set of minimally entangled typical thermal states or METTS. These states are descendants of classical product wavefunctions which contain just enough entanglement to capture finite-temperature quantum correlations.

METTS not only provide an efficient method for simulating thermal quantum systems in 1d and 2d, but provide an intuitive picture of thermal quantum states. There is a wealth of systems waiting to be studied with METTS including one- and two-dimensional frustrated spin-lattice models and continuum electronic wires. We are also investigating powerful new extensions to the basic METTS algorithm that will enable us to push further into two dimensions.

More Reading:

  • Steven R. White, "Minimally entangled typical quantum states at finite temperature", Phys. Rev. Lett. 102: 190601 (2009)
  • E.M. Stoudenmire and Steven R. White, "Minimally entangled typical thermal state algorithms", New J. Phys. 12: 055026 (2010)

Exact Density Functional Theory With Model 1D Systems

At the heart of the enormously successful density functional theory (DFT) method lies a functional that gives the ground state energy of any system provided only its density. Can this exact energy functional be written down? What are its key properties?

Except for a handful of small systems, computing the exact functional in three dimensions is too difficult to allow a systematic study of its properties. However, in one dimension the extreme power of the density matrix renormalization group (DMRG) allows us to calculate the exact densities of a wide variety of continuum, long-range interacting systems resembling long chains of artificial atoms. From these exact densities, it is relatively straightforward to calculate the exact density functional and a host of other quantities of interest in DFT. We can then use our 1d laboratory to study established density-functional approximations and explore improved functionals which could be applied to strongly correlated systems in 3d.

I gave a talk on this subject in November 2012 at the Perimeter Institute, and a slightly more pedagogical talk at National Taiwan University in December 2012.

Related works:

  • Lucas O. Wagner, Thomas E. Baker, E.M. Stoudenmire, Kieron Burke, and Steven R. White, "Kohn-Sham Calculations with the Exact Functional", arxiv:1405.0864
  • Lucas O. Wagner, E.M. Stoudenmire, Kieron Burke, and Steven R. White, "Guaranteed convergence of the Kohn-Sham equations", Phys. Rev. Lett. 111: 093003 (2013)
  • E.M. Stoudenmire, Lucas O. Wagner, Steven R. White and Kieron Burke, "One-dimensional continuum electronic structure with the density matrix renormalization group and its implications for density functional theory", Phys. Rev. Lett. 109: 056402 (2012)
  • Lucas O. Wagner, E.M. Stoudenmire, Steven R. White and Kieron Burke, "Reference electronic structure calculations in one dimension", Phys. Chem. Chem. Phys. 14: 8581 (2012)

The Intelligent Tensor (ITensor) C++ Library

What does a DMRG wavefunction look like?

Beginning with the work of Östlund and Rommer, it has become clear that the density matrix renormalization group (DMRG) is just one of a family of powerful methods for simulating quantum systems. The concept linking these methods is the tensor network state, a diagrammatic language for capturing the internal structure of many-body wavefunctions.

The ITensor Library is an open source C++ library for rapidly developing and applying tensor network algorithms. ITensor is competitive with publicly available codes when performing straight DMRG calculations, but its flexibility has enabled us to extend DMRG in ways that would be inconceivable with other approaches. It is especially well suited for developing next-generation tensor network algorithms such as PEPS, MERA, and METTS (see above).

See a list of papers written using ITensor.

// Contact



E.M. Stoudenmire
Flatiron Institute
Center for Computational Quantum Physics