Tag Archives: Physics

Google at APS 2024

Today the 2024 March Meeting of the American Physical Society (APS) kicks off in Minneapolis, MN. A premier conference on topics ranging across physics and related fields, APS 2024 brings together researchers, students, and industry professionals to share their discoveries and build partnerships with the goal of realizing fundamental advances in physics-related sciences and technology.

This year, Google has a strong presence at APS with a booth hosted by the Google Quantum AI team, 50+ talks throughout the conference, and participation in conference organizing activities, special sessions and events. Attending APS 2024 in person? Come visit Google’s Quantum AI booth to learn more about the exciting work we’re doing to solve some of the field’s most interesting challenges.

You can learn more about the latest cutting edge work we are presenting at the conference along with our schedule of booth events below (Googlers listed in bold).

Organizing Committee

Session Chairs include: Aaron Szasz

Booth Activities

This schedule is subject to change. Please visit the Google Quantum AI booth for more information.

Presenter: Matt McEwen
Tue, Mar 5 | 11:00 AM CST

Presenter: Tanuj Khattar
Tue, Mar 5 | 2:30 PM CST

Presenter: Tanuj Khattar
Thu, Mar 7 | 11:00 AM CST

$5M XPRIZE / Google Quantum AI competition to accelerate quantum applications Q&A
Presenter: Ryan Babbush
Thu, Mar 7 | 11:00 AM CST



Certifying highly-entangled states from few single-qubit measurements
Presenter: Hsin-Yuan Huang
Author: Hsin-Yuan Huang
Session A45: New Frontiers in Machine Learning Quantum Physics

Toward high-fidelity analog quantum simulation with superconducting qubits
Presenter: Trond Andersen
Authors: Trond I Andersen, Xiao Mi, Amir H Karamlou, Nikita Astrakhantsev, Andrey Klots, Julia Berndtsson, Andre Petukhov, Dmitry Abanin, Lev B Ioffe, Yu Chen, Vadim Smelyanskiy, Pedram Roushan
Session A51: Applications on Noisy Quantum Hardware I

Measuring circuit errors in context for surface code circuits
Presenter: Dripto M Debroy
Authors: Dripto M Debroy, Jonathan A Gross, Élie Genois, Zhang Jiang
Session B50: Characterizing Noise with QCVV Techniques

Quantum computation of stopping power for inertial fusion target design I: Physics overview and the limits of classical algorithms
Presenter: Andrew D. Baczewski
Authors: Nicholas C. Rubin, Dominic W. Berry, Alina Kononov, Fionn D. Malone, Tanuj Khattar, Alec White, Joonho Lee, Hartmut Neven, Ryan Babbush, Andrew D. Baczewski
Session B51: Heterogeneous Design for Quantum Applications
Link to Paper

Quantum computation of stopping power for inertial fusion target design II: Physics overview and the limits of classical algorithms
Presenter: Nicholas C. Rubin
Authors: Nicholas C. Rubin, Dominic W. Berry, Alina Kononov, Fionn D. Malone, Tanuj Khattar, Alec White, Joonho Lee, Hartmut Neven, Ryan Babbush, Andrew D. Baczewski
Session B51: Heterogeneous Design for Quantum Applications
Link to Paper

Calibrating Superconducting Qubits: From NISQ to Fault Tolerance
Presenter: Sabrina S Hong
Author: Sabrina S Hong
Session B56: From NISQ to Fault Tolerance

Measurement and feedforward induced entanglement negativity transition
Presenter: Ramis Movassagh
Authors: Alireza Seif, Yu-Xin Wang, Ramis Movassagh, Aashish A. Clerk
Session B31: Measurement Induced Criticality in Many-Body Systems
Link to Paper

Effective quantum volume, fidelity and computational cost of noisy quantum processing experiments
Presenter: Salvatore Mandra
Authors: Kostyantyn Kechedzhi, Sergei V Isakov, Salvatore Mandra, Benjamin Villalonga, X. Mi, Sergio Boixo, Vadim Smelyanskiy
Session B52: Quantum Algorithms and Complexity
Link to Paper

Accurate thermodynamic tables for solids using Machine Learning Interaction Potentials and Covariance of Atomic Positions
Presenter: Mgcini K Phuthi
Authors: Mgcini K Phuthi, Yang Huang, Michael Widom, Ekin D Cubuk, Venkat Viswanathan
Session D60: Machine Learning of Molecules and Materials: Chemical Space and Dynamics


IN-Situ Pulse Envelope Characterization Technique (INSPECT)
Presenter: Zhang Jiang
Authors: Zhang Jiang, Jonathan A Gross, Élie Genois
Session F50: Advanced Randomized Benchmarking and Gate Calibration

Characterizing two-qubit gates with dynamical decoupling
Presenter: Jonathan A Gross
Authors: Jonathan A Gross, Zhang Jiang, Élie Genois, Dripto M Debroy, Ze-Pei Cian*, Wojciech Mruczkiewicz
Session F50: Advanced Randomized Benchmarking and Gate Calibration

Statistical physics of regression with quadratic models
Presenter: Blake Bordelon
Authors: Blake Bordelon, Cengiz Pehlevan, Yasaman Bahri
Session EE01: V: Statistical and Nonlinear Physics II

Improved state preparation for first-quantized simulation of electronic structure
Presenter: William J Huggins
Authors: William J Huggins, Oskar Leimkuhler, Torin F Stetina, Birgitta Whaley
Session G51: Hamiltonian Simulation

Controlling large superconducting quantum processors
Presenter: Paul V. Klimov
Authors: Paul V. Klimov, Andreas Bengtsson, Chris Quintana, Alexandre Bourassa, Sabrina Hong, Andrew Dunsworth, Kevin J. Satzinger, William P. Livingston, Volodymyr Sivak, Murphy Y. Niu, Trond I. Andersen, Yaxing Zhang, Desmond Chik, Zijun Chen, Charles Neill, Catherine Erickson, Alejandro Grajales Dau, Anthony Megrant, Pedram Roushan, Alexander N. Korotkov, Julian Kelly, Vadim Smelyanskiy, Yu Chen, Hartmut Neven
Session G30: Commercial Applications of Quantum Computing)
Link to Paper

Gaussian boson sampling: Determining quantum advantage
Presenter: Peter D Drummond
Authors: Peter D Drummond, Alex Dellios, Ned Goodman, Margaret D Reid, Ben Villalonga
Session G50: Quantum Characterization, Verification, and Validation II

Attention to complexity III: learning the complexity of random quantum circuit states
Presenter: Hyejin Kim
Authors: Hyejin Kim, Yiqing Zhou, Yichen Xu, Chao Wan, Jin Zhou, Yuri D Lensky, Jesse Hoke, Pedram Roushan, Kilian Q Weinberger, Eun-Ah Kim
Session G50: Quantum Characterization, Verification, and Validation II

Balanced coupling in superconducting circuits
Presenter: Daniel T Sank
Authors: Daniel T Sank, Sergei V Isakov, Mostafa Khezri, Juan Atalaya
Session K48: Strongly Driven Superconducting Systems

Resource estimation of Fault Tolerant algorithms using Qᴜᴀʟᴛʀᴀɴ
Presenter: Tanuj Khattar
Author: Tanuj Khattar
Session K49: Algorithms and Implementations on Near-Term Quantum Computers


Discovering novel quantum dynamics with superconducting qubits
Presenter: Pedram Roushan
Author: Pedram Roushan
Session M24: Analog Quantum Simulations Across Platforms

Deciphering Tumor Heterogeneity in Triple-Negative Breast Cancer: The Crucial Role of Dynamic Cell-Cell and Cell-Matrix Interactions
Presenter: Susan Leggett
Authors: Susan Leggett, Ian Wong, Celeste Nelson, Molly Brennan, Mohak Patel, Christian Franck, Sophia Martinez, Joe Tien, Lena Gamboa, Thomas Valentin, Amanda Khoo, Evelyn K Williams
Session M27: Mechanics of Cells and Tissues II

Toward implementation of protected charge-parity qubits
Presenter: Abigail Shearrow
Authors: Abigail Shearrow, Matthew Snyder, Bradley G Cole, Kenneth R Dodge, Yebin Liu, Andrey Klots, Lev B Ioffe, Britton L Plourde, Robert McDermott
Session N48: Unconventional Superconducting Qubits

Electronic capacitance in tunnel junctions for protected charge-parity qubits
Presenter: Bradley G Cole
Authors: Bradley G Cole, Kenneth R Dodge, Yebin Liu, Abigail Shearrow, Matthew Snyder, Andrey Klots, Lev B Ioffe, Robert McDermott, B.L.T. Plourde
Session N48: Unconventional Superconducting Qubits

Overcoming leakage in quantum error correction
Presenter: Kevin C. Miao
Authors: Kevin C. Miao, Matt McEwen, Juan Atalaya, Dvir Kafri, Leonid P. Pryadko, Andreas Bengtsson, Alex Opremcak, Kevin J. Satzinger, Zijun Chen, Paul V. Klimov, Chris Quintana, Rajeev Acharya, Kyle Anderson, Markus Ansmann, Frank Arute, Kunal Arya, Abraham Asfaw, Joseph C. Bardin, Alexandre Bourassa, Jenna Bovaird, Leon Brill, Bob B. Buckley, David A. Buell, Tim Burger, Brian Burkett, Nicholas Bushnell, Juan Campero, Ben Chiaro, Roberto Collins, Paul Conner, Alexander L. Crook, Ben Curtin, Dripto M. Debroy, Sean Demura, Andrew Dunsworth, Catherine Erickson, Reza Fatemi, Vinicius S. Ferreira, Leslie Flores Burgos, Ebrahim Forati, Austin G. Fowler, Brooks Foxen, Gonzalo Garcia, William Giang, Craig Gidney, Marissa Giustina, Raja Gosula, Alejandro Grajales Dau, Jonathan A. Gross, Michael C. Hamilton, Sean D. Harrington, Paula Heu, Jeremy Hilton, Markus R. Hoffmann, Sabrina Hong, Trent Huang, Ashley Huff, Justin Iveland, Evan Jeffrey, Zhang Jiang, Cody Jones, Julian Kelly, Seon Kim, Fedor Kostritsa, John Mark Kreikebaum, David Landhuis, Pavel Laptev, Lily Laws, Kenny Lee, Brian J. Lester, Alexander T. Lill, Wayne Liu, Aditya Locharla, Erik Lucero, Steven Martin, Anthony Megrant, Xiao Mi, Shirin Montazeri, Alexis Morvan, Ofer Naaman, Matthew Neeley, Charles Neill, Ani Nersisyan, Michael Newman, Jiun How Ng, Anthony Nguyen, Murray Nguyen, Rebecca Potter, Charles Rocque, Pedram Roushan, Kannan Sankaragomathi, Christopher Schuster, Michael J. Shearn, Aaron Shorter, Noah Shutty, Vladimir Shvarts, Jindra Skruzny, W. Clarke Smith, George Sterling, Marco Szalay, Douglas Thor, Alfredo Torres, Theodore White, Bryan W. K. Woo, Z. Jamie Yao, Ping Yeh, Juhwan Yoo, Grayson Young, Adam Zalcman, Ningfeng Zhu, Nicholas Zobrist, Hartmut Neven, Vadim Smelyanskiy, Andre Petukhov, Alexander N. Korotkov, Daniel Sank, Yu Chen
Session N51: Quantum Error Correction Code Performance and Implementation I
Link to Paper

Modeling the performance of the surface code with non-uniform error distribution: Part 1
Presenter: Yuri D Lensky
Authors: Yuri D Lensky, Volodymyr Sivak, Kostyantyn Kechedzhi, Igor Aleiner
Session N51: Quantum Error Correction Code Performance and Implementation I

Modeling the performance of the surface code with non-uniform error distribution: Part 2
Presenter: Volodymyr Sivak
Authors: Volodymyr Sivak, Michael Newman, Cody Jones, Henry Schurkus, Dvir Kafri, Yuri D Lensky, Paul Klimov, Kostyantyn Kechedzhi, Vadim Smelyanskiy
Session N51: Quantum Error Correction Code Performance and Implementation I

Highly optimized tensor network contractions for the simulation of classically challenging quantum computations
Presenter: Benjamin Villalonga
Author: Benjamin Villalonga
Session Q51: Co-evolution of Quantum Classical Algorithms

Teaching modern quantum computing concepts using hands-on open-source software at all levels
Presenter: Abraham Asfaw
Author: Abraham Asfaw
Session Q61: Teaching Quantum Information at All Levels II


New circuits and an open source decoder for the color code
Presenter: Craig Gidney
Authors: Craig Gidney, Cody Jones
Session S51: Quantum Error Correction Code Performance and Implementation II
Link to Paper

Performing Hartree-Fock many-body physics calculations with large language models
Presenter: Eun-Ah Kim
Authors: Eun-Ah Kim, Haining Pan, Nayantara Mudur, William Taranto, Subhashini Venugopalan, Yasaman Bahri, Michael P Brenner
Session S18: Data Science, AI and Machine Learning in Physics I

New methods for reducing resource overhead in the surface code
Presenter: Michael Newman
Authors: Craig M Gidney, Michael Newman, Peter Brooks, Cody Jones
Session S51: Quantum Error Correction Code Performance and Implementation II
Link to Paper

Challenges and opportunities for applying quantum computers to drug design
Presenter: Raffaele Santagati
Authors: Raffaele Santagati, Alan Aspuru-Guzik, Ryan Babbush, Matthias Degroote, Leticia Gonzalez, Elica Kyoseva, Nikolaj Moll, Markus Oppel, Robert M. Parrish, Nicholas C. Rubin, Michael Streif, Christofer S. Tautermann, Horst Weiss, Nathan Wiebe, Clemens Utschig-Utschig
Session S49: Advances in Quantum Algorithms for Near-Term Applications
Link to Paper

Dispatches from Google's hunt for super-quadratic quantum advantage in new applications
Presenter: Ryan Babbush
Author: Ryan Babbush
Session T45: Recent Advances in Quantum Algorithms

Qubit as a reflectometer
Presenter: Yaxing Zhang
Authors: Yaxing Zhang, Benjamin Chiaro
Session T48: Superconducting Fabrication, Packaging, & Validation

Random-matrix theory of measurement-induced phase transitions in nonlocal Floquet quantum circuits
Presenter: Aleksei Khindanov
Authors: Aleksei Khindanov, Lara Faoro, Lev Ioffe, Igor Aleiner
Session W14: Measurement-Induced Phase Transitions

Continuum limit of finite density many-body ground states with MERA
Presenter: Subhayan Sahu
Authors: Subhayan Sahu, Guifré Vidal
Session W58: Extreme-Scale Computational Science Discovery in Fluid Dynamics and Related Disciplines II

Dynamics of magnetization at infinite temperature in a Heisenberg spin chain
Presenter: Eliott Rosenberg
Authors: Eliott Rosenberg, Trond Andersen, Rhine Samajdar, Andre Petukhov, Jesse Hoke*, Dmitry Abanin, Andreas Bengtsson, Ilya Drozdov, Catherine Erickson, Paul Klimov, Xiao Mi, Alexis Morvan, Matthew Neeley, Charles Neill, Rajeev Acharya, Richard Allen, Kyle Anderson, Markus Ansmann, Frank Arute, Kunal Arya, Abraham Asfaw, Juan Atalaya, Joseph Bardin, A. Bilmes, Gina Bortoli, Alexandre Bourassa, Jenna Bovaird, Leon Brill, Michael Broughton, Bob B. Buckley, David Buell, Tim Burger, Brian Burkett, Nicholas Bushnell, Juan Campero, Hung-Shen Chang, Zijun Chen, Benjamin Chiaro, Desmond Chik, Josh Cogan, Roberto Collins, Paul Conner, William Courtney, Alexander Crook, Ben Curtin, Dripto Debroy, Alexander Del Toro Barba, Sean Demura, Agustin Di Paolo, Andrew Dunsworth, Clint Earle, E. Farhi, Reza Fatemi, Vinicius Ferreira, Leslie Flores, Ebrahim Forati, Austin Fowler, Brooks Foxen, Gonzalo Garcia, Élie Genois, William Giang, Craig Gidney, Dar Gilboa, Marissa Giustina, Raja Gosula, Alejandro Grajales Dau, Jonathan Gross, Steve Habegger, Michael Hamilton, Monica Hansen, Matthew Harrigan, Sean Harrington, Paula Heu, Gordon Hill, Markus Hoffmann, Sabrina Hong, Trent Huang, Ashley Huff, William Huggins, Lev Ioffe, Sergei Isakov, Justin Iveland, Evan Jeffrey, Zhang Jiang, Cody Jones, Pavol Juhas, D. Kafri, Tanuj Khattar, Mostafa Khezri, Mária Kieferová, Seon Kim, Alexei Kitaev, Andrey Klots, Alexander Korotkov, Fedor Kostritsa, John Mark Kreikebaum, David Landhuis, Pavel Laptev, Kim Ming Lau, Lily Laws, Joonho Lee, Kenneth Lee, Yuri Lensky, Brian Lester, Alexander Lill, Wayne Liu, William P. Livingston, A. Locharla, Salvatore Mandrà, Orion Martin, Steven Martin, Jarrod McClean, Matthew McEwen, Seneca Meeks, Kevin Miao, Amanda Mieszala, Shirin Montazeri, Ramis Movassagh, Wojciech Mruczkiewicz, Ani Nersisyan, Michael Newman, Jiun How Ng, Anthony Nguyen, Murray Nguyen, M. Niu, Thomas O'Brien, Seun Omonije, Alex Opremcak, Rebecca Potter, Leonid Pryadko, Chris Quintana, David Rhodes, Charles Rocque, N. Rubin, Negar Saei, Daniel Sank, Kannan Sankaragomathi, Kevin Satzinger, Henry Schurkus, Christopher Schuster, Michael Shearn, Aaron Shorter, Noah Shutty, Vladimir Shvarts, Volodymyr Sivak, Jindra Skruzny, Clarke Smith, Rolando Somma, George Sterling, Doug Strain, Marco Szalay, Douglas Thor, Alfredo Torres, Guifre Vidal, Benjamin Villalonga, Catherine Vollgraff Heidweiller, Theodore White, Bryan Woo, Cheng Xing, Jamie Yao, Ping Yeh, Juhwan Yoo, Grayson Young, Adam Zalcman, Yaxing Zhang, Ningfeng Zhu, Nicholas Zobrist, Hartmut Neven, Ryan Babbush, Dave Bacon, Sergio Boixo, Jeremy Hilton, Erik Lucero, Anthony Megrant, Julian Kelly, Yu Chen, Vadim Smelyanskiy, Vedika Khemani, Sarang Gopalakrishnan, Tomaž Prosen, Pedram Roushan
Session W50: Quantum Simulation of Many-Body Physics
Link to Paper

The fast multipole method on a quantum computer
Presenter: Kianna Wan
Authors: Kianna Wan, Dominic W Berry, Ryan Babbush
Session W50: Quantum Simulation of Many-Body Physics


The quantum computing industry and protecting national security: what tools will work?
Presenter: Kate Weber
Author: Kate Weber
Session Y43: Industry, Innovation, and National Security: Finding the Right Balance

Novel charging effects in the fluxonium qubit
Presenter: Agustin Di Paolo
Authors: Agustin Di Paolo, Kyle Serniak, Andrew J Kerman, William D Oliver
Session Y46: Fluxonium-Based Superconducting Quibits

Microwave Engineering of Parametric Interactions in Superconducting Circuits
Presenter: Ofer Naaman
Author: Ofer Naaman
Session Z46: Broadband Parametric Amplifiers and Circulators

Linear spin wave theory of large magnetic unit cells using the Kernel Polynomial Method
Presenter: Harry Lane
Authors: Harry Lane, Hao Zhang, David A Dahlbom, Sam Quinn, Rolando D Somma, Martin P Mourigal, Cristian D Batista, Kipton Barros
Session Z62: Cooperative Phenomena, Theory

*Work done while at Google

Source: Google AI Blog

A new quantum algorithm for classical mechanics with an exponential speedup

Quantum computers promise to solve some problems exponentially faster than classical computers, but there are only a handful of examples with such a dramatic speedup, such as Shor’s factoring algorithm and quantum simulation. Of those few examples, the majority of them involve simulating physical systems that are inherently quantum mechanical — a natural application for quantum computers. But what about simulating systems that are not inherently quantum? Can quantum computers offer an exponential advantage for this?

In “Exponential quantum speedup in simulating coupled classical oscillators”, published in Physical Review X (PRX) and presented at the Symposium on Foundations of Computer Science (FOCS 2023), we report on the discovery of a new quantum algorithm that offers an exponential advantage for simulating coupled classical harmonic oscillators. These are some of the most fundamental, ubiquitous systems in nature and can describe the physics of countless natural systems, from electrical circuits to molecular vibrations to the mechanics of bridges. In collaboration with Dominic Berry of Macquarie University and Nathan Wiebe of the University of Toronto, we found a mapping that can transform any system involving coupled oscillators into a problem describing the time evolution of a quantum system. Given certain constraints, this problem can be solved with a quantum computer exponentially faster than it can with a classical computer. Further, we use this mapping to prove that any problem efficiently solvable by a quantum algorithm can be recast as a problem involving a network of coupled oscillators, albeit exponentially many of them. In addition to unlocking previously unknown applications of quantum computers, this result provides a new method of designing new quantum algorithms by reasoning purely about classical systems.

Simulating coupled oscillators

The systems we consider consist of classical harmonic oscillators. An example of a single harmonic oscillator is a mass (such as a ball) attached to a spring. If you displace the mass from its rest position, then the spring will induce a restoring force, pushing or pulling the mass in the opposite direction. This restoring force causes the mass to oscillate back and forth.

A simple example of a harmonic oscillator is a mass connected to a wall by a spring. [Image Source: Wikimedia]

Now consider coupled harmonic oscillators, where multiple masses are attached to one another through springs. Displace one mass, and it will induce a wave of oscillations to pulse through the system. As one might expect, simulating the oscillations of a large number of masses on a classical computer gets increasingly difficult.

An example system of masses connected by springs that can be simulated with the quantum algorithm.

To enable the simulation of a large number of coupled harmonic oscillators, we came up with a mapping that encodes the positions and velocities of all masses and springs into the quantum wavefunction of a system of qubits. Since the number of parameters describing the wavefunction of a system of qubits grows exponentially with the number of qubits, we can encode the information of N balls into a quantum mechanical system of only about log(N) qubits. As long as there is a compact description of the system (i.e., the properties of the masses and the springs), we can evolve the wavefunction to learn coordinates of the balls and springs at a later time with far fewer resources than if we had used a naïve classical approach to simulate the balls and springs.

We showed that a certain class of coupled-classical oscillator systems can be efficiently simulated on a quantum computer. But this alone does not rule out the possibility that there exists some as-yet-unknown clever classical algorithm that is similarly efficient in its use of resources. To show that our quantum algorithm achieves an exponential speedup over any possible classical algorithm, we provide two additional pieces of evidence.

The glued-trees problem and the quantum oracle

For the first piece of evidence, we use our mapping to show that the quantum algorithm can efficiently solve a famous problem about graphs known to be difficult to solve classically, called the glued-trees problem. The problem takes two branching trees — a graph whose nodes each branch to two more nodes, resembling the branching paths of a tree — and glues their branches together through a random set of edges, as shown in the figure below.

A visual representation of the glued trees problem. Here we start at the node labeled ENTRANCE and are allowed to locally explore the graph, which is obtained by randomly gluing together two binary trees. The goal is to find the node labeled EXIT.

The goal of the glued-trees problem is to find the exit node — the “root” of the second tree — as efficiently as possible. But the exact configuration of the nodes and edges of the glued trees are initially hidden from us. To learn about the system, we must query an oracle, which can answer specific questions about the setup. This oracle allows us to explore the trees, but only locally. Decades ago, it was shown that the number of queries required to find the exit node on a classical computer is proportional to a polynomial factor of N, the total number of nodes.

But recasting this as a problem with balls and springs, we can imagine each node as a ball and each connection between two nodes as a spring. Pluck the entrance node (the root of the first tree), and the oscillations will pulse through the trees. It only takes a time that scales with the depth of the tree — which is exponentially smaller than N — to reach the exit node. So, by mapping the glued-trees ball-and-spring system to a quantum system and evolving it for that time, we can detect the vibrations of the exit node and determine it exponentially faster than we could using a classical computer.


The second and strongest piece of evidence that our algorithm is exponentially more efficient than any possible classical algorithm is revealed by examination of the set of problems a quantum computer can solve efficiently (i.e., solvable in polynomial time), referred to as bounded-error quantum polynomial time or BQP. The hardest problems in BQP are called “BQP-complete”.

While it is generally accepted that there exist some problems that a quantum algorithm can solve efficiently and a classical algorithm cannot, this has not yet been proven. So, the best evidence we can provide is that our problem is BQP-complete, that is, it is among the hardest problems in BQP. If someone were to find an efficient classical algorithm for solving our problem, then every problem solved by a quantum computer efficiently would be classically solvable! Not even the factoring problem (finding the prime factors of a given large number), which forms the basis of modern encryption and was famously solved by Shor’s algorithm, is expected to be BQP-complete.

A diagram showing the believed relationships of the classes BPP and BQP, which are the set of problems that can be efficiently solved on a classical computer and quantum computer, respectively. BQP-complete problems are the hardest problems in BQP.

To show that our problem of simulating balls and springs is indeed BQP-complete, we start with a standard BQP-complete problem of simulating universal quantum circuits, and show that every quantum circuit can be expressed as a system of many balls coupled with springs. Therefore, our problem is also BQP-complete.

Implications and future work

This effort also sheds light on work from 2002, when theoretical computer scientist Lov K. Grover and his colleague, Anirvan M. Sengupta, used an analogy to coupled pendulums to illustrate how Grover’s famous quantum search algorithm could find the correct element in an unsorted database quadratically faster than could be done classically. With the proper setup and initial conditions, it would be possible to tell whether one of N pendulums was different from the others — the analogue of finding the correct element in a database — after the system had evolved for time that was only ~√(N). While this hints at a connection between certain classical oscillating systems and quantum algorithms, it falls short of explaining why Grover’s quantum algorithm achieves a quantum advantage.

Our results make that connection precise. We showed that the dynamics of any classical system of harmonic oscillators can indeed be equivalently understood as the dynamics of a corresponding quantum system of exponentially smaller size. In this way we can simulate Grover and Sengupta’s system of pendulums on a quantum computer of log(N) qubits, and find a different quantum algorithm that can find the correct element in time ~√(N). The analogy we discovered between classical and quantum systems can be used to construct other quantum algorithms offering exponential speedups, where the reason for the speedups is now more evident from the way that classical waves propagate.

Our work also reveals that every quantum algorithm can be equivalently understood as the propagation of a classical wave in a system of coupled oscillators. This would imply that, for example, we can in principle build a classical system that solves the factoring problem after it has evolved for time that is exponentially smaller than the runtime of any known classical algorithm that solves factoring. This may look like an efficient classical algorithm for factoring, but the catch is that the number of oscillators is exponentially large, making it an impractical way to solve factoring.

Coupled harmonic oscillators are ubiquitous in nature, describing a broad range of systems from electrical circuits to chains of molecules to structures such as bridges. While our work here focuses on the fundamental complexity of this broad class of problems, we expect that it will guide us in searching for real-world examples of harmonic oscillator problems in which a quantum computer could offer an exponential advantage.


We would like to thank our Quantum Computing Science Communicator, Katie McCormick, for helping to write this blog post.

Source: Google AI Blog

Improving simulations of clouds and their effects on climate

Today’s climate models successfully capture broad global warming trends. However, because of uncertainties about processes that are small in scale yet globally important, such as clouds and ocean turbulence, these models’ predictions of upcoming climate changes are not very accurate in detail. For example, predictions of the time by which the global mean surface temperature of Earth will have warmed 2℃, relative to preindustrial times, vary by 40–50 years (a full human generation) among today’s models. As a result, we do not have the accurate and geographically granular predictions we need to plan resilient infrastructure, adapt supply chains to climate disruption, and assess the risks of climate-related hazards to vulnerable communities.

In large part this is because clouds dominate errors and uncertainties in climate predictions for the coming decades [1, 2, 3]. Clouds reflect sunlight and exert a greenhouse effect, making them crucial for regulating Earth's energy balance and mediating the response of the climate system to changes in greenhouse gas concentrations. However, they are too small in scale to be directly resolvable in today’s climate models. Current climate models resolve motions at scales of tens to a hundred kilometers, with a few pushing toward the kilometer-scale. However, the turbulent air motions that sustain, for example, the low clouds that cover large swaths of tropical oceans have scales of meters to tens of meters. Because of this wide difference in scale, climate models use empirical parameterizations of clouds, rather than simulating them directly, which result in large errors and uncertainties.

While clouds cannot be directly resolved in global climate models, their turbulent dynamics can be simulated in limited areas by using high-resolution large eddy simulations (LES). However, the high computational cost of simulating clouds with LES has inhibited broad and systematic numerical experimentation, and it has held back the generation of large datasets for training parameterization schemes to represent clouds in coarser-resolution global climate models.

In “Accelerating Large-Eddy Simulations of Clouds with Tensor Processing Units”, published in Journal of Advances in Modeling Earth Systems (JAMES), and in collaboration with a Climate Modeling Alliance (CliMA) lead who is a visiting researcher at Google, we demonstrate that Tensor Processing Units (TPUs) — application-specific integrated circuits that were originally developed for machine learning (ML) applications — can be effectively used to perform LES of clouds. We show that TPUs, in conjunction with tailored software implementations, can be used to simulate particularly computationally challenging marine stratocumulus clouds in the conditions observed during the Dynamics and Chemistry of Marine Stratocumulus (DYCOMS) field study. This successful TPU-based LES code reveals the utility of TPUs, with their large computational resources and tight interconnects, for cloud simulations.

Climate model accuracy for critical metrics, like precipitation or the energy balance at the top of the atmosphere, has improved roughly 10% per decade in the last 20 years. Our goal is for this research to enable a 50% reduction in climate model errors by improving their representation of clouds.

Large-eddy simulations on TPUs

In this work, we focus on stratocumulus clouds, which cover ~20% of the tropical oceans and are the most prevalent cloud type on earth. Current climate models are not yet able to reproduce stratocumulus cloud behavior correctly, which has been one of the largest sources of errors in these models. Our work will provide a much more accurate ground truth for large-scale climate models.

Our simulations of clouds on TPUs exhibit unprecedented computational throughput and scaling, making it possible, for example, to simulate stratocumulus clouds with 10× speedup over real-time evolution across areas up to about 35 × 54 km2. Such domain sizes are close to the cross-sectional area of typical global climate model grid boxes. Our results open up new avenues for computational experiments, and for substantially enlarging the sample of LES available to train parameterizations of clouds for global climate models.

Rendering of the cloud evolution from a simulation of a 285 x 285 x 2 km3 stratocumulus cloud sheet. This is the largest cloud sheet of its kind ever simulated. Left: An oblique view of the cloud field with the camera cruising. Right: Top view of the cloud field with the camera gradually pulled away.

The LES code is written in TensorFlow, an open-source software platform developed by Google for ML applications. The code takes advantage of TensorFlow’s graph computation and Accelerated Linear Algebra (XLA) optimizations, which enable the full exploitation of TPU hardware, including the high-speed, low-latency inter-chip interconnects (ICI) that helped us achieve this unprecedented performance. At the same time, the TensorFlow code makes it easy to incorporate ML components directly within the physics-based fluid solver.

We validated the code by simulating canonical test cases for atmospheric flow solvers, such as a buoyant bubble that rises in neutral stratification, and a negatively buoyant bubble that sinks and impinges on the surface. These test cases show that the TPU-based code faithfully simulates the flows, with increasingly fine turbulent details emerging as the resolution increases. The validation tests culminate in simulations of the conditions during the DYCOMS field campaign. The TPU-based code reliably reproduces the cloud fields and turbulence characteristics observed by aircraft during a field campaign — a feat that is notoriously difficult to achieve for LES because of the rapid changes in temperature and other thermodynamic properties at the top of the stratocumulus decks.

One of the test cases used to validate our TPU Cloud simulator. The fine structures from the density current generated by the negatively buoyant bubble impinging on the surface are much better resolved with a high resolution grid (10m, bottom row) compared to a low resolution grid (200 m, top row).


With this foundation established, our next goal is to substantially enlarge existing databases of high-resolution cloud simulations that researchers building climate models can use to develop better cloud parameterizations — whether these are for physics-based models, ML models, or hybrids of the two. This requires additional physical processes beyond that described in the paper; for example, the need to integrate radiative transfer processes into the code. Our goal is to generate data across a variety of cloud types, e.g., thunderstorm clouds.

Rendering of a thunderstorm simulation using the same simulator as the stratocumulus simulation work. Rainfall can also be observed near the ground.

This work illustrates how advances in hardware for ML can be surprisingly effective when repurposed in other research areas — in this case, climate modeling. These simulations provide detailed training data for processes such as in-cloud turbulence, which are not directly observable, yet are crucially important for climate modeling and prediction.


We would like to thank the co-authors of the paper: Sheide Chammas, Qing Wang, Matthias Ihme, and John Anderson. We’d also like to thank Carla Bromberg, Rob Carver, Fei Sha, and Tyler Russell for their insights and contributions to the work.

Source: Google AI Blog

Formation of Robust Bound States of Interacting Photons

When quantum computers were first proposed, they were hoped to be a way to better understand the quantum world. With a so-called “quantum simulator,” one could engineer a quantum computer to investigate how various quantum phenomena arise, including those that are intractable to simulate with a classical computer.

But making a useful quantum simulator has been a challenge. Until now, quantum simulations with superconducting qubits have predominantly been used to verify pre-existing theoretical predictions and have rarely explored or discovered new phenomena. Only a few experiments with trapped ions or cold atoms have revealed new insights. Superconducting qubits, even though they are one of the main candidates for universal quantum computing and have demonstrated computational capabilities beyond classical reach, have so far not delivered on their potential for discovery.

In “Formation of Robust Bound States of Interacting Photons”, published in Nature, we describe a previously unpredicted phenomenon first discovered through experimental investigation. First, we present the experimental confirmation of the theoretical prediction of the existence of a composite particle of interacting photons, or a bound state, using the Google Sycamore quantum processor. Second, while studying this system, we discovered that even though one might guess the bound states to be fragile, they remain robust to perturbations that we expected to have otherwise destroyed them. Not only does this open the possibility of designing systems that leverage interactions between photons, it also marks a step forward in the use of superconducting quantum processors to make new scientific discoveries by simulating non-equilibrium quantum dynamics.


Photons, or quanta of electromagnetic radiation like light and microwaves, typically don’t interact. For example, two intersecting flashlight beams will pass through one another undisturbed. In many applications, like telecommunications, the weak interactions of photons is a valuable feature. For other applications, such as computers based on light, the lack of interactions between photons is a shortcoming.

In a quantum processor, the qubits host microwave photons, which can be made to interact through two-qubit operations. This allows us to simulate the XXZ model, which describes the behavior of interacting photons. Importantly, this is one of the few examples of integrable models, i.e., one with a high degree of symmetry, which greatly reduces its complexity. When we implement the XXZ model on the Sycamore processor, we observe something striking: the interactions force the photons into bundles known as bound states.

Using this well-understood model as a starting point, we then push the study into a less-understood regime. We break the high level of symmetries displayed in the XXZ model by adding extra sites that can be occupied by the photons, making the system no longer integrable. While this nonintegrable regime is expected to exhibit chaotic behavior where bound states dissolve into their usual, solitary selves, we instead find that they survive!

Bound Photons

To engineer a system that can support the formation of bound states, we study a ring of superconducting qubits that host microwave photons. If a photon is present, the value of the qubit is “1”, and if not, the value is “0”. Through the so-called “fSim” quantum gate, we connect neighboring sites, allowing the photons to hop around and interact with other photons on the nearest-neighboring sites.

Superconducting qubits can be occupied or unoccupied with microwave photons. The “fSim” gate operation allows photons to hop and interact with each other. The corresponding unitary evolution has a hopping term between two sites (orange) and an interaction term corresponding to an added phase when two adjacent sites are occupied by a photon.
We implement the fSim gate between neighboring qubits (left) to effectively form a ring of 24 interconnected qubits on which we simulate the behavior of the interacting photons (right).

The interactions between the photons affect their so-called “phase.” This phase keeps track of the oscillation of the photon’s wavefunction. When the photons are non-interacting, their phase accumulation is rather uninteresting. Like a well-rehearsed choir, they’re all in sync with one another. In this case, a photon that was initially next to another photon can hop away from its neighbor without getting out of sync. Just as every person in the choir contributes to the song, every possible path the photon can take contributes to the photon’s overall wavefunction. A group of photons initially clustered on neighboring sites will evolve into a superposition of all possible paths each photon might have taken.

When photons interact with their neighbors, this is no longer the case. If one photon hops away from its neighbor, its rate of phase accumulation changes, becoming out of sync with its neighbors. All paths in which the photons split apart overlap, leading to destructive interference. It would be like each choir member singing at their own pace — the song itself gets washed out, becoming impossible to discern through the din of the individual singers. Among all the possible configuration paths, the only possible scenario that survives is the configuration in which all photons remain clustered together in a bound state. This is why interaction can enhance and lead to the formation of a bound state: by suppressing all other possibilities in which photons are not bound together.

Left: Evolution of interacting photons forming a bound state. Right: Time goes from left to right, each path represents one of the paths that can break the 2-photon bonded state. Due to interactions, these paths interfere destructively, preventing the photons from splitting apart.
Occupation probability versus gate cycle, or discrete time step, for n-photon bound states. We prepare bound states of varying sizes and watch them evolve. We observe that the majority of the photons (darker colors) remain bound together.

In our processor, we start by putting two to five photons on adjacent sites (i.e., initializing two to five adjacent qubits in “1”, and the remaining qubits in “0”), and then study how they propagate. First, we notice that in the theoretically predicted parameter regime, they remain stuck together. Next, we find that the larger bound states move more slowly around the ring, consistent with the fact that they are “heavier”. This can be seen in the plot above where the lattice sites closest to Site 12, the initial position of the photons, remain darker than the others with increasing number of photons (nph) in the bound state, indicating that with more photons bound together there is less propagation around the ring.

Bound States Behave Like Single Composite Particles

To more rigorously show that the bound states indeed behave as single particles with well-defined physical properties, we devise a method to measure how the energy of the particles changes with momentum, i.e., the energy-momentum dispersion relation.

To measure the energy of the bound state, we use the fact that the energy difference between two states determines how fast their relative phase grows with time. Hence, we prepare the bound state in a superposition with the state that has no photons, and measure their phase difference as a function of time and space. Then, to convert the result of this measurement to a dispersion relation, we utilize a Fourier transform, which translates position and time into momentum and energy, respectively. We’re left with the familiar energy-momentum relationship of excitations in a lattice.

Spectroscopy of bound states. We compare the phase accumulation of an n-photon bound state with that of the vacuum (no photons) as a function of lattice site and time. A 2D Fourier transform yields the dispersion relation of the bound-state quasiparticle.

Breaking Integrability

The above system is “integrable,” meaning that it has a sufficient number of conserved quantities that its dynamics are constrained to a small part of the available computational space. In such integrable regimes, the appearance of bound states is not that surprising. In fact, bound states in similar systems were predicted in 2012, then observed in 2013. However, these bound states are fragile and their existence is usually thought to derive from integrability. For more complex systems, there is less symmetry and integrability is quickly lost. Our initial idea was to probe how these bound states disappear as we break integrability to better understand their rigidity.

To break integrability, we modify which qubits are connected with fSim gates. We add qubits so that at alternating sites, in addition to hopping to each of its two nearest-neighboring sites, a photon can also hop to a third site oriented radially outward from the ring.

While a bound state is constrained to a very small part of phase space, we expected that the chaotic behavior associated with integrability breaking would allow the system to explore the phase space more freely. This would cause the bound states to break apart. We find that this is not the case. Even when the integrability breaking is so strong that the photons are equally likely to hop to the third site as they are to hop to either of the two adjacent ring sites, the bound state remains intact, up to the decoherence effect that makes them slowly decay (see paper for details).

Top: New geometry to break integrability. Alternating sites are connected to a third site oriented radially outward. This increases the complexity of the system, and allows for potentially chaotic behavior. Bottom: Despite this added complexity pushing the system beyond integrability, we find that the 3-photon bound state remains stable even for a relatively large perturbation. The probability of remaining bound decreases slowly due to decoherence (see paper).


We don’t yet have a satisfying explanation for this unexpected resilience. We speculate that it may be related to a phenomenon called prethermalization, where incommensurate energy scales in the system can prevent a system from reaching thermal equilibrium as quickly as it otherwise would. We believe further investigations will hopefully lead to new insights into many-body quantum physics, including the interplay of prethermalization and integrability.


We would like to thank our Quantum Science Communicator Katherine McCormick for her help writing this blog post.

Source: Google AI Blog

Finding Complex Metal Oxides for Technology Advancement

A crystalline material has atoms systematically arranged in repeating units, with this structure and the elements it contains determining the material’s properties. For example, silicon’s crystal structure allows it to be widely used in the semiconductor industry, whereas graphite’s soft, layered structure makes for great pencils. One class of crystalline materials that are critical for a wide range of applications, ranging from battery technology to electrolysis of water (i.e., splitting H2O into its component hydrogen and oxygen), are crystalline metal oxides, which have repeating units of oxygen and metals. Researchers suspect that there is a significant number of crystalline metal oxides that could prove to be useful, but their number and the extent of their useful properties is unknown.

In “Discovery of complex oxides via automated experiments and data science”, a collaborative effort with partners at the Joint Center for Artificial Photosynthesis (JCAP), a Department of Energy (DOE) Energy Innovation Hub at Caltech, we present a systematic search for new complex crystalline metal oxides using a novel approach for rapid materials synthesis and characterization. Using a customized inkjet printer to print samples with different ratios of metals, we were able to generate more than 350k distinct compositions, a number of which we discovered had interesting properties. One example, based on cobalt, tantalum and tin, exhibited tunable transparency, catalytic activity, and stability in strong acid electrolytes, a rare combination of properties of importance for renewable energy technologies. To stimulate continued research in this field, we are releasing a database consisting of nine channels of optical absorption measurements, which can be used as an indicator of interesting properties, across 376,752 distinct compositions of 108 3-metal oxide systems, along with model results that identify the most promising compositions for a variety of technical applications.

There are on the order of 100 properties of interest in materials science that are relevant to enhancing existing technologies and to creating new ones, ranging from electrical, optical, and magnetic to thermal and mechanical. Traditionally, exploring materials for a target technology involves considering only one or a few such properties at a time, resulting in many parallel efforts where the same materials are being evaluated. Machine learning (ML) for material properties prediction has been successfully deployed in many of these parallel efforts, but the models are inherently specialized and fail to capture the universality of the prediction problem. Instead of asking traditional questions of how ML can help find a suitable material for a particular property, we instead apply ML to find a short-list of materials that may be exceptional for any given property. This strategy combines high throughput materials experiments with a physics-aware data science workflow.

A challenge in realizing this strategy is that the search space for new crystalline metal oxides is enormous. For example, the Inorganic Crystal Structure Database (ICSD) lists 73 metals that exist in oxides composed of a single metal and oxygen. Generating novel compounds simply by making various combinations of these metals would yield 62,196 possible 3-metal oxide systems, some of which will contain several unique structures. If, in addition, one were to vary the relative quantities of each metal, the set of possible combinations would be orders of magnitude larger.

However, while this search space is large, only a small fraction of these novel compositions will form new crystalline structures, with the majority simply resulting in combinations of existing structures. While these combinations of structures may be interesting for some applications, the goal is to find the core single-structure compositions. Of the possible 3-metal oxide systems, the ICSD reports only 2,205 with experimentally confirmed compositions, indicating that the vast majority of possible compositions either have not been explored or have yielded negative results and have not been published. In the present work we do not directly measure the crystal structures of new materials, but instead use high throughput experiments to enable ML-based inferences of where new structures can be found.

Our goal was to explore a large swath of chemical space as quickly as possible. Whereas traditional synthesis techniques like physical vapor deposition can create high quality thin films, we decided to reuse an existing technology that was already optimized to mix and deposit small amounts of material very quickly: an inkjet printer. We made each metal element printable by dissolving a metal nitrate or metal chloride into an ink solution. We then printed a series of lines on glass plates, where the ratios of the elements used in the printing varied along each line according to our experiment design so that we could generate thousands of unique compositions per plate. Several such plates were then dried and baked together in a series of ovens to oxidize the metals. Due to the inherent variability in the printing, drying, and baking of the plates, we opted to print 10 duplicates of each composition. Even with this level of replication, we still were able to generate novel compositions 100x faster than traditional vapor deposition techniques.

The modified professional grade inkjet printer.
Top: A printed and baked plate that is 10 x 15 cm. Bottom: A close-up of a portion of the plate. Since the optical properties vary with composition, the gradient in composition appears as a color gradient along each line.

When making samples at this rate, it is hard to find a characterization technique that can keep up. A traditional approach to design a material for a specific purpose would require significant time to measure the pertinent properties of each combination, but for the analysis to keep up with our high-throughput printing method, we needed something faster. So, we built a custom microscope capable of taking pictures at nine discrete wavelengths ranging from the ultraviolet (385 nm), through the visible, to the infrared (850 nm). This microscope produced over 20 TB of image data over the course of the project, which we used to calculate the optical absorption coefficients of each sample at each wavelength. While optical absorption itself is important for technologies such as solar energy harvesting, in our work we are interested in optical absorption vs. wavelength as a fingerprint of each material.

After generating 376,752 distinct compositions, we needed to know which ones were actually interesting. We hypothesized that since the structure of a material determines its properties, when a material property (in this case, the optical absorption spectrum) changes in a nontrivial way, that could indicate a structural change. To test this, we built two ML models to identify potentially interesting compositions.

As the composition of metals changes in a metal oxide, the crystal structure of the resulting material may change. The map of the compositions that crystallize into the same structure, which we call the phase, is the “phase diagram”. The first model, the ‘phase diagram’ model, is a physics-based model that assumes thermodynamic equilibrium, which imposes limits on the number of phases that can coexist. Assuming that the optical properties of a combination of crystalline phases vary linearly with the ratio of each crystalline phase, the model generates a set of phases that best fit the optical absorption spectra. The phase diagram model involved a comprehensive search through the space of thermodynamically allowed phase diagrams. The second model seeks to identify “emergent properties” by identifying 3-metal oxide absorption spectra that can not be explained by a linear combination of 1-metal or 2-metal oxide signals.

Phase analysis of compounds with different relative fractions of the metals iron (Fe), tin (Sn) and yttrium (Y). Left: Panels showing the absorption coefficient at different wavelengths: a) 375 nm; b) 530 nm; c) 660 nm, d) 850 nm. Right: Based on the absorption, the phase diagram model identifies the boundaries at which changes in the relative composition in the compound lead to different optical properties and hence suggest compositions with potentially interesting behavior. In panels e), f) and g), red points are candidate phases, and vertices where blue lines meet indicate interesting phase behavior. Panel h) shows the emergent property model, where compositions are colored by the log-likelihood of their properties being explainable by lower-order compositions (darker colors are more likely to represent more interesting compounds).

Experimental Verification
In the end our systematic, combinatorial sweep of 108 3-metal oxide systems found 51 of these systems exhibited interesting behavior. Of these 108 systems, only 1 of them has an experimentally reported entry in the ICSD. We performed an in-depth experimental study of one unexplored system, the Co-Ta-Sn oxides. With guidance from the high throughput workflow, we validated the discovery of a new family of solid solutions by x-ray diffraction, successfully resynthesized the new materials using a common technique (physical vapor deposition), validated the surprisingly high transparency in compositions with up to 30% Co, and performed follow-up electrochemical testing that demonstrated electrocatalytic activity for water oxidation (a critical step in hydrogen fuel synthesis from water). Catalyst testing for water oxidation is far more expensive than the optical screening from our high throughput workflow, and even though there is no known connection between the optical properties and the catalytic properties, we use the analysis of optical properties to select a small number of compositions for catalyst testing, demonstrating our high level concept of using one high throughput workflow to down-select materials for practically any target technology.

The Co-Ta-Sn oxide example illustrates how finding new materials quickly is an important step in developing improved technologies, such as those critical for hydrogen production. We hope this work inspires the materials community — for the experimentalists, we hope to inspire creativity in aggressively scaling high-throughput techniques, and for computationalists, we hope to provide a rich dataset with plenty of negative results to better inform ML and other data science models.

It was a pleasure and a privilege to work with John Gregoire and Joel Haber at Caltech for this complex, long-running project. Additionally, we would like to thank Zan Armstrong, Sam Yang, Kevin Kan, Lan Zhou, Matthias Richter, Chris Roat, Nick Wagner, Marc Coram, Marc Berndl, Pat Riley, and Ted Baltz for their contributions.

Source: Google AI Blog

The Technology Behind our Recent Improvements in Flood Forecasting

Flooding is the most common natural disaster on the planet, affecting the lives of hundreds of millions of people around the globe and causing around $10 billion in damages each year. Building on our work in previous years, earlier this week we announced some of our recent efforts to improve flood forecasting in India and Bangladesh, expanding coverage to more than 250 million people, and providing unprecedented lead time, accuracy and clarity.

To enable these breakthroughs, we have devised a new approach for inundation modeling, called a morphological inundation model, which combines physics-based modeling with machine learning (ML) to create more accurate and scalable inundation models in real-world settings. Additionally, our new alert-targeting model allows identifying areas at risk of flooding at unprecedented scale using end-to-end machine learning models and data that is publicly available globally. In this post, we also describe developments for the next generation of flood forecasting systems, called HydroNets (presented at ICLR AI for Earth Sciences and EGU this year), which is a new architecture specially built for hydrologic modeling across multiple basins, while still optimizing for accuracy at each location.

Forecasting Water Levels
The first step in a flood forecasting system is to identify whether a river is expected to flood. Hydrologic models (or gauge-to-gauge models) have long been used by governments and disaster management agencies to improve the accuracy and extend the lead time of their forecasts. These models receive inputs like precipitation or upstream gauge measurements of water level (i.e., the absolute elevation of the water above sea level) and output a forecast for the water level (or discharge) in the river at some time in the future.

The hydrologic model component of the flood forecasting system described in this week’s Keyword post doubled the lead time of flood alerts for areas covering more than 75 million people. These models not only increase lead time, but also provide unprecedented accuracy, achieving an R2 score of more than 99% across all basins we cover, and predicting the water level within a 15 cm error bound more than 90% of the time. Once a river is predicted to reach flood level, the next step in generating actionable warnings is to convert the river level forecast into a prediction for how the floodplain will be affected.

Morphological Inundation Modeling
In prior work, we developed high quality elevation maps based on satellite imagery, and ran physics-based models to simulate water flow across these digital terrains, which allowed warnings with unprecedented resolution and accuracy in data-scarce regions. In collaboration with our satellite partners, Airbus, Maxar and Planet, we have now expanded the elevation maps to cover hundreds of millions of square kilometers. However, in order to scale up the coverage to such a large area while still retaining high accuracy, we had to re-invent how we develop inundation models.

Inundation modeling estimates what areas will be flooded and how deep the water will be. This visualization conceptually shows how inundation could be simulated, how risk levels could be defined (represented by red and white colors), and how the model could be used to identify areas that should be warned (green dots).

Inundation modeling at scale suffers from three significant challenges. Due to the large areas involved and the resolution required for such models, they necessarily have high computational complexity. In addition, most global elevation maps don’t include riverbed bathymetry, which is important for accurate modeling. Finally, the errors in existing data, which may include gauge measurement errors, missing features in the elevation maps, and the like, need to be understood and corrected. Correcting such problems may require collecting additional high-quality data or fixing erroneous data manually, neither of which scale well.

Our new approach to inundation modeling, which we call a morphological model, addresses these issues by using several innovative tricks. Instead of modeling the complex behaviors of water flow in real time, we compute modifications to the morphology of the elevation map that allow one to simulate the inundation using simple physical principles, such as those describing hydrostatic systems.

First, we train a pure-ML model (devoid of physics-based information) to estimate the one-dimensional river profile from gauge measurements. The model takes as input the water level at a specific point on the river (the stream gauge) and outputs the river profile, which is the water level at all points in the river. We assume that if the gauge increases, the water level increases monotonically, i.e., the water level at other points in the river increases as well. We also assume that the absolute elevation of the river profile decreases downstream (i.e., the river flows downhill).

We then use this learned model and some heuristics to edit the elevation map to approximately “cancel out” the pressure gradient that would exist if that region were flooded. This new synthetic elevation map provides the foundation on which we model the flood behavior using a simple flood-fill algorithm. Finally, we match the resulting flooded map to the satellite-based flood extent with the original stream gauge measurement.

This approach abandons some of the realistic constraints of classical physics-based models, but in data scarce regions where existing methods currently struggle, its flexibility allows the model to automatically learn the correct bathymetry and fix various errors to which physics-based models are sensitive. This morphological model improves accuracy by 3%, which can significantly improve forecasts for large areas, while also allowing for much more rapid model development by reducing the need for manual modeling and correction.

Alert targeting
Many people reside in areas that are not covered by the morphological inundation models, yet access to accurate predictions are still urgently needed. To reach this population and to increase the impact of our flood forecasting models, we designed an end-to-end ML-based approach, using almost exclusively data that is globally publicly available, such as stream gauge measurements, public satellite imagery, and low resolution elevation maps. We train the model to use the data it is receiving to directly infer the inundation map in real time.

A direct ML approach from real-time measurements to inundation.

This approach works well “out of the box” when the model only needs to forecast an event that is within the range of events previously observed. Extrapolating to more extreme conditions is much more challenging. Nevertheless, proper use of existing elevation maps and real-time measurements can enable alerts that are more accurate than presently available for those in areas not covered by the more detailed morphological inundation models. Because this model is highly scalable, we were able to launch it across India after only a few months of work, and we hope to roll it out to many more countries soon.

Improving Water Levels Forecasting
In an effort to continue improving flood forecasting, we have developed HydroNets — a specialized deep neural network architecture built specifically for water levels forecasting — which allows us utilize some exciting recent advances in ML-based hydrology in a real-world operational setting. Two prominent features distinguish it from standard hydrologic models. First, it is able to differentiate between model components that generalize well between sites, such as the modeling of rainfall-runoff processes, and those that are specific to a given site, like the rating curve, which converts a predicted discharge volume into an expected water level. This enables the model to generalize well to different sites, while still fine-tuning its performance to each location. Second, HydroNets takes into account the structure of the river network being modeled, by training a large architecture that is actually a web of smaller neural networks, each representing a different location along the river. This allows neural networks that are modeling upstream sites to pass information encoded in embeddings to models of downstream sites, so that every model can know everything it needs without a drastic increase in parameters.

The animation below illustrates the structure and flow of information in HydroNets. The output from the modeling of upstream sub-basins is combined into a single representation of a given basin state. It is then processed by the shared model component, which is informed by all basins in the network, and passed on to the label prediction model, which calculates the water level (and the loss function). The output from this iteration of the network is then passed on to inform downstream models, and so on.

An illustration of the HydroNets architecture.

We’re incredibly excited about this progress, and are working hard on improving our systems further.

This work is a collaboration between many research teams at Google, and is part of our AI for Social Good efforts. We'd also like to thank our Geo and Policy teams, as well as Google.org.

Source: Google AI Blog

New Solutions for Quantum Gravity with TensorFlow

Recent strides in machine learning (ML) research have led to the development of tools useful for research problems well beyond the realm for which they were designed. The value of these tools when applied to topics ranging from teaching robots how to throw to predicting the olfactory properties of molecules is now beginning to be realized. Inspired by advances such as these, we undertook the challenge of applying TensorFlow, a computing platform normally used for ML, to advance the understanding of fundamental physics.

Perhaps the biggest open problem in fundamental theoretical physics may be that our current understanding of quantum mechanics only includes three of the four fundamental forces — the electromagnetic, strong, and weak forces. There is currently no complete quantum theory that also includes the force of gravitation, while still matching experimental observations, i.e., an accurate model of quantum gravity.

One promising approach to a unified model that includes quantum gravity, which has survived many mathematical consistency checks, is called M-Theory, or "The Theory formerly known as Strings,” introduced in 1995 by Edward Witten. In the everyday world, we all experience four dimensions—three spatial dimensions (x, y, and z), plus time (t). M-Theory predicts that, at very short lengths, the Universe is described, instead, by eleven dimensions. But, as one can imagine, establishing the connection between the four-dimensional world that we observe and the 11-dimensional world predicted by M-theory is exceedingly difficult to solve analytically. In fact, it might require analytic manipulation of equations having more terms than there are electrons in the Universe.

This summer, we published an article in the Journal of High Energy Physics where we introduced novel ways to address such problems through creative use of ML technology. Using simplifications enabled by TensorFlow, we managed to bring the total number of known (stable or unstable) equilibrium solutions for one particular type of M-Theory spacetime geometries to 194, including a new and tachyon-free four-dimensional model universe. The geometries that we studied are special in that they are still (barely) accessible with exact calculations that do not require neglecting potentially important terms. We have also released a short instructive Google colab as well as a more powerful Python library for use in related research.

Applying TensorFlow to M-Theory
This work is predicated on a key observation that a mixed numerical and analytic approach can be more powerful than a purely analytical method. Instead of attempting to find analytic solutions with brute force, we use a numerical approach that leverages TensorFlow for the initial search for solutions to the model. This then yields hypotheses on which specific combinations can be tested and analyzed with stringent mathematical methods, ultimately proving the actual existence of a conjectured solution. This represents a novel methodology for making further progress in theoretical physics.

We hope that these results will be an important step in interpreting M-theory, and demonstrate how the research community can use new ML tools, such as TensorFlow, to approach other similarly complex problems. We are already applying the newly discovered methods in further theoretical physics research.

This research was conducted by Iulia M. Comşa, Moritz Firsching, and Thomas Fischbacher. Additional thanks go to Jyrki Alakuijala, Rahul Sukthankar, and Jay Yagnik for encouragement and support.

Source: Google AI Blog

An Inside Look at Flood Forecasting

Several years ago, we identified flood forecasts as a unique opportunity to improve people’s lives, and began looking into how Google’s infrastructure and machine learning expertise can help in this field. Last year, we started our flood forecasting pilot in the Patna region, and since then we have expanded our flood forecasting coverage, as part of our larger AI for Social Good efforts. In this post, we discuss some of the technology and methodology behind this effort.

The Inundation Model
A critical step in developing an accurate flood forecasting system is to develop inundation models, which use either a measurement or a forecast of the water level in a river as an input, and simulate the water behavior across the floodplain.
A 3D visualization of a hydraulic model simulating various river conditions.
This allows us to translate current or future river conditions, to highly spatially accurate risk maps - which tell us what areas will be flooded and what areas will be safe. Inundation models depend on four major components, each with its own challenges and innovations:

Real-time Water Level Measurements
To run these models operationally, we need to know what is happening on the ground in real-time, and thus we rely on partnerships with the relevant government agencies to receive timely and accurate information. Our first governmental partner is the Indian Central Water Commission (CWC), which measures water levels hourly in over a thousand stream gauges across all of India, aggregates this data, and produces forecasts based on upstream measurements. The CWC provides these real-time river measurements and forecasts, which are then used as inputs for our models.
CWC employees measuring water level and discharge near Lucknow, India.
Elevation Map Creation
Once we know how much water is in a river, it is critical that the models have a good map of the terrain. High-resolution digital elevation models (DEMs) are incredibly useful for a wide range of applications in the earth sciences, but are still difficult to acquire in most of the world, especially for flood forecasting. This is because meter-wide features of the ground conditions can create a critical difference in the resulting flooding (embankments are one exceptionally important example), but publicly accessible global DEMs have resolutions of tens of meters. To help address this challenge, we’ve developed a novel methodology to produce high resolution DEMs based on completely standard optical imagery.

We start with the large and varied collection of satellite images used in Google Maps. Correlating and aligning the images in large batches, we simultaneously optimize for satellite camera model corrections (for orientation errors, etc.) and for coarse terrain elevation. We then use the corrected camera models to create a depth map for each image. To make the elevation map, we optimally fuse the depth maps together at each location. Finally, we remove objects such as trees and bridges so that they don’t block water flow in our simulations. This can be done manually or by training convolutional neural networks that can identify where the terrain elevations need to be interpolated. The result is a roughly 1 meter DEM, which can be used to run hydraulic models.

Hydraulic Modeling
Once we have both these inputs - the riverine measurements and forecasts, and the elevation map - we can begin the modeling itself, which can be divided into two main components. The first and most substantial component is the physics-based hydraulic model, which updates the location and velocity of the water through time based on (an approximated) computation of the laws of physics. Specifically, we’ve implemented a solver for the 2D form of the shallow-water Saint-Venant equations. These models are suitably accurate when given accurate inputs and run at high resolutions, but their computational complexity creates challenges - it is proportional to the cube of the resolution desired. That is, if you double the resolution, you’ll need roughly 8 times as much processing time. Since we’re committed to the high-resolution required for highly accurate forecasts, this can lead to unscalable computational costs, even for Google!

To help address this problem, we’ve created a unique implementation of our hydraulic model, optimized for Tensor Processing Units (TPUs). While TPUs were optimized for neural networks (rather than differential equation solvers like our hydraulic model), their highly parallelized nature leads to the performance per TPU core being 85x times faster than the performance per CPU core. For additional efficiency improvements, we’re also looking at using machine learning to replace some of the physics-based algorithmics, extending data-driven discretization to two-dimensional hydraulic models, so we can support even larger grids and cover even more people.
A snapshot of a TPU-based simulation of flooding in Goalpara, mid-event.
As mentioned earlier, the hydraulic model is only one component of our inundation forecasts. We’ve repeatedly found locations where our hydraulic models are not sufficiently accurate - whether that’s due to inaccuracies in the DEM, breaches in embankments, or unexpected water sources. Our goal is to find effective ways to reduce these errors. For this purpose, we added a predictive inundation model, based on historical measurements. Since 2014, the European Space Agency has been operating a satellite constellation named Sentinel-1 with C-band Synthetic-Aperture Radar (SAR) instruments. SAR imagery is great at identifying inundation, and can do so regardless of weather conditions and clouds. Based on this valuable data set, we correlate historical water level measurements with historical inundations, allowing us to identify consistent corrections to our hydraulic model. Based on the outputs of both components, we can estimate which disagreements are due to genuine ground condition changes, and which are due to modeling inaccuracies.
Flood warnings across Google’s interfaces.
Looking Forward
We still have a lot to do to fully realize the benefits of our inundation models. First and foremost, we’re working hard to expand the coverage of our operational systems, both within India and to new countries. There’s also a lot more information we want to be able to provide in real time, including forecasted flood depth, temporal information and more. Additionally, we’re researching how to best convey this information to individuals to maximize clarity and encourage them to take the necessary protective actions.

Computationally, while the inundation model is a good tool for improving the spatial resolution (and therefore the accuracy and reliability) of existing flood forecasts, multiple governmental agencies and international organizations we’ve spoken to are concerned about areas that do not have access to effective flood forecasts at all, or whose forecasts don’t provide enough lead time for effective response. In parallel to our work on the inundation model, we’re working on some basic research into improved hydrologic models, which we hope will allow governments not only to produce more spatially accurate forecasts, but also achieve longer preparation time.

Hydrologic models accept as inputs things like precipitation, solar radiation, soil moisture and the like, and produce a forecast for the river discharge (among other things), days into the future. These models are traditionally implemented using a combination of conceptual models approximating different core processes such as snowmelt, surface runoff, evapotranspiration and more.
The core processes of a hydrologic model. Designed by Daniel Klotz, JKU Institute for Machine Learning.
These models also traditionally require a large amount of manual calibration, and tend to underperform in data scarce regions. We are exploring how multi-task learning can be used to address both of these problems — making hydrologic models both more scalable, and more accurate. In research collaboration with JKU Institute For Machine Learning group under Sepp Hochreiter on developing ML-based hydrologic models, Kratzert et al. show how LSTMs perform better than all benchmarked classic hydrologic models.
The distribution of NSE scores on basins across the United States for various models, showing the proposed EA-LSTM consistently outperforming a wide range of commonly used models.
Though this work is still in the basic research stage and not yet operational, we think it is an important first step, and hope it can already be useful for other researchers and hydrologists. It’s an incredible privilege to take part in the large eco-system of researchers, governments, and NGOs working to reduce the harms of flooding. We’re excited about the potential impact this type of research can provide, and look forward to where research in this field will go.

There are many people who contributed to this large effort, and we’d like to highlight some of the key contributors: Aaron Yonas, Adi Mano, Ajai Tirumali, Avinatan Hassidim, Carla Bromberg, Damien Pierce, Gal Elidan, Guy Shalev, John Anderson, Karan Agarwal, Kartik Murthy, Manan Singhi, Mor Schlesinger, Ofir Reich, Oleg Zlydenko, Pete Giencke, Piyush Poddar, Ruha Devanesan, Slava Salasin, Varun Gulshan, Vova Anisimov, Yossi Matias, Yi-fan Chen, Yotam Gigi, Yusef Shafi, Zach Moshe and Zvika Ben-Haim.

Source: Google AI Blog

Learning Better Simulation Methods for Partial Differential Equations

The world’s fastest supercomputers were designed for modeling physical phenomena, yet they still are not fast enough to robustly predict the impacts of climate change, to design controls for airplanes based on airflow or to accurately simulate a fusion reactor. All of these phenomena are modeled by partial differential equations (PDEs), the class of equations that describe everything smooth and continuous in the physical world, and the most common class of simulation problems in science and engineering. To solve these equations, we need faster simulations, but in recent years, Moore’s law has been slowing. At the same time, we’ve seen huge breakthroughs in machine learning (ML) along with faster hardware optimized for it. What does this new paradigm offer for scientific computing?

In “Learning Data Driven Discretizations for Partial Differential Equations”, published in Proceedings of the National Academy of Sciences, we explore a potential path for how ML can offer continued improvements in high-performance computing, both for solving PDEs and, more broadly, for solving hard computational problems in every area of science.

For most real-world problems, closed-form solutions to PDEs don’t exist. Instead, one must find discrete equations (“discretizations”) that a computer can solve to approximate the continuous PDE. Typical approaches to solve PDEs represent equations on a grid, e.g., using finite differences. To achieve convergence, the mesh spacing of the grid needs to be smaller than the smallest feature size of the solutions. This often isn’t feasible because of an unfortunate scaling law: achieving 10x higher resolution requires 10,000x more compute, because the grid must be scaled in four dimensions—three spatial dimensions and time. Instead, in our paper we show that ML can be used to learn better representations for PDEs on coarser grids.
Satellite photo of a hurricane, at both full resolution and simulated resolution in a state of the art weather model. Cumulus clouds (e.g., in the red circle) are responsible for heavy rainfall, but in the weather model the details are entirely blurred out. Instead, models rely on crude approximations for sub-grid physics, a key source of uncertainty in climate models. Image credit: NOAA
The challenge is to retain the accuracy of high-resolution simulations while still using the coarsest grid possible. In our work we’re able to improve upon existing schemes by replacing heuristics based on deep human insight (e.g., “solutions to a PDE should always be smooth away from discontinuities”) with optimized rules based on machine learning. The rules our ML models recover are complex, and we don’t entirely understand them, but they incorporate sophisticated physical principles like the idea of “upwinding”—to accurately model what’s coming towards you in a fluid flow, you should look upstream in the direction the wind is coming from. An example of our results on a simple model of fluid dynamics are shown below:
Simulations of Burgers’ equation, a model for shock waves in fluids, solved with either a standard finite volume method (left) or our neural network based method (right). The orange squares represent simulations with each method on low resolution grids. These points are fed back into the model at each time step, which then predicts how they should change. Blue lines show the exact simulations used for training. The neural network solution is much better, even on a 4x coarser grid, as indicated by the orange squares smoothly tracing the blue line.
Our research also illustrates a broader lesson about how to effectively combine machine learning and physics. Rather than attempting to learn physics from scratch, we combined neural networks with components from traditional simulation methods, including the known form of the equations we’re solving and finite volume methods. This means that laws such as conservation of momentum are exactly satisfied, by construction, and allows our machine learning models to focus on what they do best, learning optimal rules for interpolation in complex, high-dimensional spaces.

Next Steps
We are focused on scaling up the techniques outlined in our paper to solve larger scale simulation problems with real-world impacts, such as weather and climate prediction. We’re excited about the broad potential of blending machine learning into the complex algorithms of scientific computing.

Thanks to co-authors Yohai Bar-Sinari, Jason Hickey and Michael Brenner; and Google collaborators Peyman Milanfar, Pascal Getreur, Ignacio Garcia Dorado, Dmitrii Kochkov, Jiawei Zhuang and Anton Geraschenko.

Source: Google AI Blog

Introducing TensorNetwork, an Open Source Library for Efficient Tensor Calculations

Originally posted on the Google AI Blog.

Many of the world's toughest scientific challenges, like developing high-temperature superconductors and understanding the true nature of space and time, involve dealing with the complexity of quantum systems. What makes these challenges difficult is that the number of quantum states in these systems is exponentially large, making brute-force computation infeasible. To deal with this, data structures called tensor networks are used. Tensor networks let one focus on the quantum states that are most relevant for real-world problems—the states of low energy, say—while ignoring other states that aren't relevant. Tensor networks are also increasingly finding applications in machine learning (ML). However, there remain difficulties that prohibit them from widespread use in the ML community: 1) a production-level tensor network library for accelerated hardware has not been available to run tensor network algorithms at scale, and 2) most of the tensor network literature is geared toward physics applications and creates the false impression that expertise in quantum mechanics is required to understand the algorithms.

In order to address these issues, we are releasing TensorNetwork, a brand new open source library to improve the efficiency of tensor calculations, developed in collaboration with the Perimeter Institute for Theoretical Physics and X. TensorNetwork uses TensorFlow as a backend and is optimized for GPU processing, which can enable speedups of up to 100x when compared to work on a CPU. We introduce TensorNetwork in a series of papers, the first of which presents the new library and its API, and provides an overview of tensor networks for a non-physics audience. In our second paper we focus on a particular use case in physics, demonstrating the speedup that one gets using GPUs.

How are Tensor Networks Useful?

Tensors are multidimensional arrays, categorized in a hierarchy according to their order: e.g., an ordinary number is a tensor of order zero (also known as a scalar), a vector is an order-one tensor, a matrix is an order-two tensorDiagrammatic notation for tensors. and so on. While low-order tensors can easily be represented by an explicit array of numbers or with a mathematical symbol such as Tijnklm (where the number of indices represents the order of the tensor), that notation becomes very cumbersome once we start talking about high-order tensors. At that point it's useful to start using diagrammatic notation, where one simply draws a circle (or some other shape) with a number of lines, or legs, coming out of it—the number of legs being the same as the order of the tensor. In this notation, a scalar is just a circle, a vector has a single leg, a matrix has two legs, etc. Each leg of the tensor also has a dimension, which is the size of that leg. For example, a vector representing an object's velocity through space would be a three-dimensional, order-one tensor.
Diagrammatic notation for tensors.
The benefit of representing tensors in this way is to succinctly encode mathematical operations, e.g., multiplying a matrix by a vector to produce another vector, or multiplying two vectors to make a scalar. These are all examples of a more general concept called tensor contraction.
Diagrammatic notation for tensor contraction. Vector and matrix multiplication, as well as the matrix trace (i.e., the sum of the diagonal elements of a matrix), are all examples.
These are also simple examples of tensor networks, which are graphical ways of encoding the pattern of tensor contractions of several constituent tensors to form a new one. Each constituent tensor has an order determined by its own number of legs. Legs that are connected, forming an edge in the diagram, represent contraction, while the number of remaining dangling legs determines the order of the resultant tensor.
Left: The trace of the product of four matrices, tr(ABCD), which is a scalar. You can see that it has no dangling legs. Right: Three order-three tensors being contracted with three legs dangling, resulting in a new order-three tensor.
While these examples are very simple, the tensor networks of interest often represent hundreds of tensors contracted in a variety of ways. Describing such a thing would be very obscure using traditional notation, which is why the diagrammatic notation was invented by Roger Penrose in 1971.

Tensor Networks in Practice

Consider a collection of black-and-white images, each of which can be thought of as a list of N pixel values. A single pixel of a single image can be one-hot-encoded into a two-dimensional vector, and by combining these pixel encodings together we can make a 2N-dimensional one-hot encoding of the entire image. We can reshape that high-dimensional vector into an order-N tensor, and then add up all of the tensors in our collection of images to get a total tensor Ti1,i2,...,iN encapsulating the collection.
This sounds like a very wasteful thing to do: encoding images with about 50 pixels in this way would already take petabytes of memory. That's where tensor networks come in. Rather than storing or manipulating the tensor T directly, we instead represent T as the contraction of many smaller constituent tensors in the shape of a tensor network. That turns out to be much more efficient. For instance, the popular matrix product state (MPS) network would write T in terms of N much smaller tensors, so that the total number of parameters is only linear in N, rather than exponential.
The high-order tensor T is represented in terms of many low-order tensors in a matrix product state tensor network.
It's not obvious that large tensor networks can be efficiently created or manipulated while consistently avoiding the need for a huge amount of memory. But it turns out that this is possible in many cases, which is why tensor networks have been used extensively in quantum physics and, now, in machine learning. Stoudenmire and Schwab used the encoding just described to make an image classification model, demonstrating a new use for tensor networks. The TensorNetwork library is designed to facilitate exactly that kind of work, and our first paper describes how the library functions for general tensor network manipulations.

Performance in Physics Use-Cases

TensorNetwork is a general-purpose library for tensor network algorithms, and so it should prove useful for physicists as well. Approximating quantum states is a typical use-case for tensor networks in physics, and is well-suited to illustrate the capabilities of the TensorNetwork library. In our second paper, we describe a tree tensor network (TTN) algorithm for approximating the ground state of either a periodic quantum spin chain (1D) or a lattice model on a thin torus (2D), and implement the algorithm using TensorNetwork. We compare the use of CPUs with GPUs and observe significant computational speed-ups, up to a factor of 100, when using a GPU and the TensorNetwork library.
Computational time as a function of the bond dimension, χ. The bond dimension determines the size of the constituent tensors of the tensor network. A larger bond dimension means the tensor network is more powerful, but requires more computational resources to manipulate.

Conclusion and Future Work

These are the first in a series of planned papers to illustrate the power of TensorNetwork in real-world applications. In our next paper we will use TensorNetwork to classify images in the MNIST and Fashion-MNIST datasets. Future plans include time series analysis on the ML side, and quantum circuit simulation on the physics side. With the open source community, we are also always adding new features to TensorNetwork itself. We hope that TensorNetwork will become a valuable tool for physicists and machine learning practitioners.


The TensorNetwork library was developed by Chase Roberts, Adam Zalcman, and Bruce Fontaine of Google AI; Ashley Milsted, Martin Ganahl, and Guifre Vidal of the Perimeter Institute; and Jack Hidary and Stefan Leichenauer of X. We'd also like to thank Stavros Efthymiou at X for valuable contributions.

by Chase Roberts, Research Engineer, Google AI and Stefan Leichenauer, Research Scientist, X