Tag Archives: Quantum Computing

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

Overcoming leakage on error-corrected quantum processors

The qubits that make up Google quantum devices are delicate and noisy, so it’s necessary to incorporate error correction procedures that identify and account for qubit errors on the way to building a useful quantum computer. Two of the most prevalent error mechanisms are bit-flip errors (where the energy state of the qubit changes) and phase-flip errors (where the phase of the encoded quantum information changes). Quantum error correction (QEC) promises to address and mitigate these two prominent errors. However, there is an assortment of other error mechanisms that challenges the effectiveness of QEC.

While we want qubits to behave as ideal two-level systems with no loss mechanisms, this is not the case in reality. We use the lowest two energy levels of our qubit (which form the computational basis) to carry out computations. These two levels correspond to the absence (computational ground state) or presence (computational excited state) of an excitation in the qubit, and are labeled |0⟩ (“ket zero”) and |1⟩ (“ket one”), respectively. However, our qubits also host many higher levels called leakage states, which can become occupied. Following the convention of labeling the level by indicating how many excitations are in the qubit, we specify them as |2⟩, |3⟩, |4⟩, and so on.

In “Overcoming leakage in quantum error correction”, published in Nature Physics, we identify when and how our qubits leak energy to higher states, and show that the leaked states can corrupt nearby qubits through our two-qubit gates. We then identify and implement a strategy that can remove leakage and convert it to an error that QEC can efficiently fix. Finally, we show that these operations lead to notably improved performance and stability of the QEC process. This last result is particularly critical, since additional operations take time, usually leading to more errors.

Working with imperfect qubits

Our quantum processors are built from superconducting qubits called transmons. Unlike an ideal qubit, which only has two computational levels — a computational ground state and a computational excited state — transmon qubits have many additional states with higher energy than the computational excited state. These higher leakage states are useful for particular operations that generate entanglement, a necessary resource in quantum algorithms, and also keep transmons from becoming too non-linear and difficult to operate. However, the transmon can also be inadvertently excited into these leakage states through a variety of processes, including imperfections in the control pulses we apply to perform operations or from the small amount of stray heat leftover in our cryogenic refrigerator. These processes are collectively referred to as leakage, which describes the transition of the qubit from computational states to leakage states.

Consider a particular two-qubit operation that is used extensively in our QEC experiments: the CZ gate. This gate operates on two qubits, and when both qubits are in their |1⟩ level, an interaction causes the two individual excitations to briefly “bunch” together in one of the qubits to form |2⟩, while the other qubit becomes |0⟩, before returning to the original configuration where each qubit is in |1⟩. This bunching underlies the entangling power of the CZ gate. However, with a small probability, the gate can encounter an error and the excitations do not return to their original configuration, causing the operation to leave a qubit in |2⟩, a leakage state. When we execute hundreds or more of these CZ gates, this small leakage error probability accumulates.

Transmon qubits support many leakage states (|2⟩, |3⟩, |4⟩, …) beyond the computational basis (|0⟩ and |1⟩). While we typically only use the computational basis to represent quantum information, sometimes the qubit enters these leakage states, and disrupts the normal operation of our qubits.

A single leakage event is especially damaging to normal qubit operation because it induces many individual errors. When one qubit starts in a leaked state, the CZ gate no longer correctly entangles the qubits, preventing the algorithm from executing correctly. Not only that, but CZ gates applied to one qubit in leaked states can cause the other qubit to leak as well, spreading leakage through the device. Our work includes extensive characterization of how leakage is caused and how it interacts with the various operations we use in our quantum processor.

Once the qubit enters a leakage state, it can remain in that state for many operations before relaxing back to the computational states. This means that a single leakage event interferes with many operations on that qubit, creating operational errors that are bunched together in time (time-correlated errors). The ability for leakage to spread between the different qubits in our device through the CZ gates means we also concurrently see bunches of errors on neighboring qubits (space-correlated errors). The fact that leakage induces patterns of space- and time-correlated errors makes it especially hard to diagnose and correct from the perspective of QEC algorithms.

The effect of leakage in QEC

We aim to mitigate qubit errors by implementing surface code QEC, a set of operations applied to a collection of imperfect physical qubits to form a logical qubit, which has properties much closer to an ideal qubit. In a nutshell, we use a set of qubits called data qubits to hold the quantum information, while another set of measure qubits check up on the data qubits, reporting on whether they have suffered any errors, without destroying the delicate quantum state of the data qubits. One of the key underlying assumptions of QEC is that errors occur independently for each operation, but leakage can persist over many operations and cause a correlated pattern of multiple errors. The performance of our QEC strategies is significantly limited when leakage causes this assumption to be violated.

Once leakage manifests in our surface code transmon grid, it persists for a long time relative to a single surface code QEC cycle. To make matters worse, leakage on one qubit can cause its neighbors to leak as well.

Our previous work has shown that we can remove leakage from measure qubits using an operation called multi-level reset (MLR). This is possible because once we perform a measurement on measure qubits, they no longer hold any important quantum information. At this point, we can interact the qubit with a very lossy frequency band, causing whichever state the qubit was in (including leakage states) to decay to the computational ground state |0⟩. If we picture a Jenga tower representing the excitations in the qubit, we tumble the entire stack over. Removing just one brick, however, is much more challenging. Likewise, MLR doesn’t work with data qubits because they always hold important quantum information, so we need a new leakage removal approach that minimally disturbs the computational basis states.

Gently removing leakage

We introduce a new quantum operation called data qubit leakage removal (DQLR), which targets leakage states in a data qubit and converts them into computational states in the data qubit and a neighboring measure qubit. DQLR consists of a two-qubit gate (dubbed Leakage iSWAP — an iSWAP operation with leakage states) inspired by and similar to our CZ gate, followed by a rapid reset of the measure qubit to further remove errors. The Leakage iSWAP gate is very efficient and greatly benefits from our extensive characterization and calibration of CZ gates within the surface code experiment.

Recall that a CZ gate takes two single excitations on two different qubits and briefly brings them to one qubit, before returning them to their respective qubits. A Leakage iSWAP gate operates similarly, but almost in reverse, so that it takes a single qubit with two excitations (otherwise known as |2⟩) and splits them into |1⟩ on two qubits. The Leakage iSWAP gate (and for that matter, the CZ gate) is particularly effective because it does not operate on the qubits if there are fewer than two excitations present. We are precisely removing the |2⟩ Jenga brick without toppling the entire tower.

By carefully measuring the population of leakage states on our transmon grid, we find that DQLR can reduce average leakage state populations over all qubits to about 0.1%, compared to nearly 1% without it. Importantly, we no longer observe a gradual rise in the amount of leakage on the data qubits, which was always present to some extent prior to using DQLR.

This outcome, however, is only half of the puzzle. As mentioned earlier, an operation such as MLR could be used to effectively remove leakage on the data qubits, but it would also completely erase the stored quantum state. We also need to demonstrate that DQLR is compatible with the preservation of a logical quantum state.

The second half of the puzzle comes from executing the QEC experiment with this operation interleaved at the end of each QEC cycle, and observing the logical performance. Here, we use a metric called detection probability to gauge how well we are executing QEC. In the presence of leakage, time- and space-correlated errors will cause a gradual rise in detection probabilities as more and more qubits enter and stay in leakage states. This is most evident when we perform no reset at all, which rapidly leads to a transmon grid plagued by leakage, and it becomes inoperable for the purposes of QEC.

The prior state-of-the-art in our QEC experiments was to use MLR on the measure qubits to remove leakage. While this kept leakage population on the measure qubits (green circles) sufficiently low, data qubit leakage population (green squares) would grow and saturate to a few percent. With DQLR, leakage population on both the measure (blue circles) and data qubits (blue squares) remain acceptably low and stable.

With MLR, the large reduction in leakage population on the measure qubits drastically decreases detection probabilities and mitigates a considerable degree of the gradual rise. This reduction in detection probability happens even though we spend more time dedicated to the MLR gate, when other errors can potentially occur. Put another way, the correlated errors that leakage causes on the grid can be much more damaging than the uncorrelated errors from the qubits waiting idle, and it is well worth it for us to trade the former for the latter.

When only using MLR, we observed a small but persistent residual rise in detection probabilities. We ascribed this residual increase in detection probability to leakage accumulating on the data qubits, and found that it disappeared when we implemented DQLR. And again, the observation that the detection probabilities end up lower compared to only using MLR indicates that our added operation has removed a damaging error mechanism while minimally introducing uncorrelated errors.

Leakage manifests during surface code operation as increased errors (shown as error detection probabilities) over the number of cycles. With DQLR, we no longer see a notable rise in detection probability over more surface code cycles.

Prospects for QEC scale-up

Given these promising results, we are eager to implement DQLR in future QEC experiments, where we expect error mechanisms outside of leakage to be greatly improved, and sensitivity to leakage to be enhanced as we work with larger and larger transmon grids. In particular, our simulations indicate that scale-up of our surface code will almost certainly require a large reduction in leakage generation rates, or an active leakage removal technique over all qubits, such as DQLR.

Having laid the groundwork by understanding where leakage is generated, capturing the dynamics of leakage after it presents itself in a transmon grid, and showing that we have an effective mitigation strategy in DQLR, we believe that leakage and its associated errors no longer pose an existential threat to the prospects of executing a surface code QEC protocol on a large grid of transmon qubits. With one fewer challenge standing in the way of demonstrating working QEC, the pathway to a useful quantum computer has never been more promising.


This work would not have been possible without the contributions of the entire Google Quantum AI Team.

Source: Google AI Blog

Measurement-induced entanglement phase transitions in a quantum circuit

Quantum mechanics allows many phenomena that are classically impossible: a quantum particle can exist in a superposition of two states simultaneously or be entangled with another particle, such that anything you do to one seems to instantaneously also affect the other, regardless of the space between them. But perhaps no aspect of quantum theory is as striking as the act of measurement. In classical mechanics, a measurement need not affect the system being studied. But a measurement on a quantum system can profoundly influence its behavior. For example, when a quantum bit of information, called a qubit, that is in a superposition of both “0” and “1” is measured, its state will suddenly collapse to one of the two classically allowed states: it will be either “0” or “1,” but not both. This transition from the quantum to classical worlds seems to be facilitated by the act of measurement. How exactly it occurs is one of the fundamental unanswered questions in physics.

In a large system comprising many qubits, the effect of measurements can cause new phases of quantum information to emerge. Similar to how changing parameters such as temperature and pressure can cause a phase transition in water from liquid to solid, tuning the strength of measurements can induce a phase transition in the entanglement of qubits.

Today in “Measurement-induced entanglement and teleportation on a noisy quantum processor”, published in Nature, we describe experimental observations of measurement-induced effects in a system of 70 qubits on our Sycamore quantum processor. This is, by far, the largest system in which such a phase transition has been observed. Additionally, we detected "quantum teleportation" — when a quantum state is transferred from one set of qubits to another, detectable even if the details of that state are unknown — which emerged from measurements of a random circuit. We achieved this breakthrough by implementing a few clever “tricks” to more readily see the signatures of measurement-induced effects in the system.

Background: Measurement-induced entanglement

Consider a system of qubits that start out independent and unentangled with one another. If they interact with one another , they will become entangled. You can imagine this as a web, where the strands represent the entanglement between qubits. As time progresses, this web grows larger and more intricate, connecting increasingly disparate points together.

A full measurement of the system completely destroys this web, since every entangled superposition of qubits collapses when it’s measured. But what happens when we make a measurement on only a few of the qubits? Or if we wait a long time between measurements? During the intervening time, entanglement continues to grow. The web’s strands may not extend as vastly as before, but there are still patterns in the web.

There is a balancing point between the strength of interactions and measurements, which compete to affect the intricacy of the web. When interactions are strong and measurements are weak, entanglement remains robust and the web’s strands extend farther, but when measurements begin to dominate, the entanglement web is destroyed. We call the crossover between these two extremes the measurement-induced phase transition.

In our quantum processor, we observe this measurement-induced phase transition by varying the relative strengths between interactions and measurement. We induce interactions by performing entangling operations on pairs of qubits. But to actually see this web of entanglement in an experiment is notoriously challenging. First, we can never actually look at the strands connecting the qubits — we can only infer their existence by seeing statistical correlations between the measurement outcomes of the qubits. So, we need to repeat the same experiment many times to infer the pattern of the web. But there’s another complication: the web pattern is different for each possible measurement outcome. Simply averaging all of the experiments together without regard for their measurement outcomes would wash out the webs’ patterns. To address this, some previous experiments used “post-selection,” where only data with a particular measurement outcome is used and the rest is thrown away. This, however, causes an exponentially decaying bottleneck in the amount of “usable” data you can acquire. In addition, there are also practical challenges related to the difficulty of mid-circuit measurements with superconducting qubits and the presence of noise in the system.

How we did it

To address these challenges, we introduced three novel tricks to the experiment that enabled us to observe measurement-induced dynamics in a system of up to 70 qubits.

Trick 1: Space and time are interchangeable

As counterintuitive as it may seem, interchanging the roles of space and time dramatically reduces the technical challenges of the experiment. Before this “space-time duality” transformation, we would have had to interleave measurements with other entangling operations, frequently checking the state of selected qubits. Instead, after the transformation, we can postpone all measurements until after all other operations, which greatly simplifies the experiment. As implemented here, this transformation turns the original 1-spatial-dimensional circuit we were interested in studying into a 2-dimensional one. Additionally, since all measurements are now at the end of the circuit, the relative strength of measurements and entangling interactions is tuned by varying the number of entangling operations performed in the circuit.

Exchanging space and time. To avoid the complication of interleaving measurements into our experiment (shown as gauges in the left panel), we utilize a space-time duality mapping to exchange the roles of space and time. This mapping transforms the 1D circuit (left) into a 2D circuit (right), where the circuit depth (T) now tunes the effective measurement rate.

Trick 2: Overcoming the post-selection bottleneck

Since each combination of measurement outcomes on all of the qubits results in a unique web pattern of entanglement, researchers often use post-selection to examine the details of a particular web. However, because this method is very inefficient, we developed a new “decoding” protocol that compares each instance of the real “web” of entanglement to the same instance in a classical simulation. This avoids post-selection and is sensitive to features that are common to all of the webs. This common feature manifests itself into a combined classical–quantum “order parameter”, akin to the cross-entropy benchmark used in the random circuit sampling used in our beyond-classical demonstration.

This order parameter is calculated by selecting one of the qubits in the system as the “probe” qubit, measuring it, and then using the measurement record of the nearby qubits to classically “decode” what the state of the probe qubit should be. By cross-correlating the measured state of the probe with this “decoded” prediction, we can obtain the entanglement between the probe qubit and the rest of the (unmeasured) qubits. This serves as an order parameter, which is a proxy for determining the entanglement characteristics of the entire web.

In the decoding procedure we choose a “probe” qubit (pink) and classically compute its expected value, conditional on the measurement record of the surrounding qubits (yellow). The order parameter is then calculated by the cross correlation between the measured probe bit and the classically computed value.

Trick 3: Using noise to our advantage

A key feature of the so-called “disentangling phase” — where measurements dominate and entanglement is less widespread — is its insensitivity to noise. We can therefore look at how the probe qubit is affected by noise in the system and use that to differentiate between the two phases. In the disentangling phase, the probe will be sensitive only to local noise that occurs within a particular area near the probe. On the other hand, in the entangling phase, any noise in the system can affect the probe qubit. In this way, we are turning something that is normally seen as a nuisance in experiments into a unique probe of the system.

What we saw

We first studied how the order parameter was affected by noise in each of the two phases. Since each of the qubits is noisy, adding more qubits to the system adds more noise. Remarkably, we indeed found that in the disentangling phase the order parameter is unaffected by adding more qubits to the system. This is because, in this phase, the strands of the web are very short, so the probe qubit is only sensitive to the noise of its nearest qubits. In contrast, we found that in the entangling phase, where the strands of the entanglement web stretch longer, the order parameter is very sensitive to the size of the system, or equivalently, the amount of noise in the system. The transition between these two sharply contrasting behaviors indicates a transition in the entanglement character of the system as the “strength” of measurement is increased.

Order parameter vs. gate density (number of entangling operations) for different numbers of qubits. When the number of entangling operations is low, measurements play a larger role in limiting the entanglement across the system. When the number of entangling operations is high, entanglement is widespread, which results in the dependence of the order parameter on system size (inset).

In our experiment, we also demonstrated a novel form of quantum teleportation that arises in the entangling phase. Typically, a specific set of operations are necessary to implement quantum teleportation, but here, the teleportation emerges from the randomness of the non-unitary dynamics. When all qubits, except the probe and another system of far away qubits, are measured, the remaining two systems are strongly entangled with each other. Without measurement, these two systems of qubits would be too far away from each other to know about the existence of each other. With measurements, however, entanglement can be generated faster than the limits typically imposed by locality and causality. This “measurement-induced entanglement” between the qubits (that must also be aided with a classical communications channel) is what allows for quantum teleportation to occur.

Proxy entropy vs. gate density for two far separated subsystems (pink and black qubits) when all other qubits are measured. There is a finite-size crossing at ~0.9. Above this gate density, the probe qubit is entangled with qubits on the opposite side of the system and is a signature of the teleporting phase.


Our experiments demonstrate the effect of measurements on a quantum circuit. We show that by tuning the strength of measurements, we can induce transitions to new phases of quantum entanglement within the system and even generate an emergent form of quantum teleportation. This work could potentially have relevance to quantum computing schemes, where entanglement and measurements both play a role.


This work was done while Jesse Hoke was interning at Google from Stanford University. We would like to thank Katie McCormick, our Quantum Science Communicator, for helping to write this blog post.

Source: Google AI Blog

Developing industrial use cases for physical simulation on future error-corrected quantum computers

If you’ve paid attention to the quantum computing space, you’ve heard the claim that in the future, quantum computers will solve certain problems exponentially more efficiently than classical computers can. They have the potential to transform many industries, from pharmaceuticals to energy.

For the most part, these claims have rested on arguments about the asymptotic scaling of algorithms as the problem size approaches infinity, but this tells us very little about the practical performance of quantum computers for finite-sized problems. We want to be more concrete: Exactly which problems are quantum computers more suited to tackle than their classical counterparts, and exactly what quantum algorithms could we run to solve these problems? Once we’ve designed an algorithm, we can go beyond analysis based on asymptotic scaling — we can determine the actual resources required to compile and run the algorithm on a quantum computer, and how that compares to a classical computation.

Over the last few years, Google Quantum AI has collaborated with industry and academic partners to assess the prospects for quantum simulation to revolutionize specific technologies and perform concrete analyses of the resource requirements. In 2022, we developed quantum algorithms to analyze the chemistry of an important enzyme family called cytochrome P450. Then, in our paper released this fall, we demonstrated how to use a quantum computer to study sustainable alternatives to cobalt for use in lithium ion batteries. And most recently, as we report in a preprint titled “Quantum computation of stopping power for inertial fusion target design,” we’ve found a new application in modeling the properties of materials in inertial confinement fusion experiments, such as those at the National Ignition Facility (NIF) at Lawrence Livermore National Laboratory, which recently made headlines for a breakthrough in nuclear fusion.

Below, we describe these three industrially relevant applications for simulations with quantum computers. While running the algorithms will require an error-corrected quantum computer, which is still years away, working on this now will ensure that we are ready with efficient quantum algorithms when such a quantum computer is built. Already, our work has reduced the cost of compiling and running the algorithms significantly, as we have reported in the past. Our work is essential for demonstrating the potential of quantum computing, but it also provides our hardware team with target specifications for the number of qubits and time needed to run useful quantum algorithms in the future.

Application 1: The CYP450 mechanism

The pharmaceutical industry is often touted as a field ripe for discovery using quantum computers. But concrete examples of such potential applications are few and far between. Working with collaborators at the pharmaceutical company Boehringer Ingelheim, our partners at the startup QSimulate, and academic colleagues at Columbia University, we explored one example in the 2022 PNAS article, “Reliably assessing the electronic structure of cytochrome P450 on today’s classical computers and tomorrow’s quantum computers”.

Cytochrome P450 is an enzyme family naturally found in humans that helps us metabolize drugs. It excels at its job: more than 70% of all drug metabolism is performed by enzymes of the P450 family. The enzymes work by oxidizing the drug — a process that depends on complex correlations between electrons. The details of the interactions are too complicated for scientists to know a priori how effective the enzyme will be on a particular drug.

In the paper, we showed how a quantum computer could approach this problem. The CYP450 metabolic process is a complex chain of reactions with many intermediate changes in the electronic structure of the enzymes throughout. We first use state-of-the-art classical methods to determine the resources required to simulate this problem on a classical computer. Then we imagine implementing a phase-estimation algorithm — which is needed to compute the ground-state energies of the relevant electronic configurations throughout the reaction chain — on a surface-code error-corrected quantum computer.

With a quantum computer, we could follow the chain of changing electronic structure with greater accuracy and fewer resources. In fact, we find that the higher accuracy offered by a quantum computer is needed to correctly resolve the chemistry in this system, so not only will a quantum computer be better, it will be necessary. And as the system size gets bigger, i.e., the more quantum energy levels we include in the simulation, the more the quantum computer wins over the classical computer. Ultimately, we show that a few million physical qubits would be required to reach quantum advantage for this problem.

Left: Example of an electron orbital (red and blue) of a CYP enzyme. More than 60 such orbitals are required to model the CYP system. Right: Comparison of actual runtime (CPU) of various classical techniques (blue) to hypothetical runtime (QPU) of a quantum algorithm (green). The lower slope of the quantum algorithm demonstrates the favorable asymptotic scaling over classical methods. Already at about 20-30 orbitals, we see a crossover to the regime where a quantum algorithm would be more efficient than classical methods.

Application 2: Lithium-ion batteries

Lithium-ion batteries rely on the electrochemical potential difference between two lithium containing materials. One material used today for the cathodes of Li-ion batteries is LiCoO2. Unfortunately, it has drawbacks from a manufacturing perspective. Cobalt mining is expensive, destructive to the environment, and often utilizes unsafe or abusive labor practices. Consequently, many in the field are interested in alternatives to cobalt for lithium-ion cathodes.

In the 1990’s, researchers discovered that nickel could replace cobalt to form LiNiO2 (called “lithium nickel oxide” or “LNO”) for cathodes. While pure LNO was found to be unstable in production, many cathode materials used in the automotive industry today use a high fraction of nickel and hence, resemble LNO. Despite its applications to industry, however, not all of the chemical properties of LNO are understood — even the properties of its ground state remains a subject of debate.

In our recent paper, “Fault tolerant quantum simulation of materials using Bloch orbitals,” we worked with the chemical company, BASF, the molecular modeling startup, QSimulate, and collaborators at Macquarie University in Australia to develop techniques to perform quantum simulations on systems with periodic, regularly spaced atomic structure, such as LNO. We then applied these techniques to design algorithms to study the relative energies of a few different candidate structures of LNO. With classical computers, high accuracy simulations of the quantum wavefunction are considered too expensive to perform. In our work, we found that a quantum computer would need tens of millions of physical qubits to calculate the energies of each of the four candidate ground-state LNO structures. This is out of reach of the first error-corrected quantum computers, but we expect this number to come down with future algorithmic improvements.

Four candidate structures of LNO. In the paper, we consider the resources required to compare the energies of these structures in order to find the ground state of LNO.

Application 3: Fusion reactor dynamics

In our third and most recent example, we collaborated with theorists at Sandia National Laboratories and our Macquarie University collaborators to put our hypothetical quantum computer to the task of simulating dynamics of charged particles in the extreme conditions typical of inertial confinement fusion (ICF) experiments, like those at the National Ignition Facility. In those experiments, high-intensity lasers are focused into a metallic cavity (hohlraum) that holds a target capsule consisting of an ablator surrounding deuterium–tritium fuel. When the lasers heat the inside of the hohlraum, its walls radiate x-rays that compress the capsule, heating the deuterium and tritium inside to 10s of millions of Kelvin. This allows the nucleons in the fuel to overcome their mutual electrostatic repulsion and start fusing into helium nuclei, also called alpha particles.

Simulations of these experiments are computationally demanding and rely on models of material properties that are themselves uncertain. Even testing these models, using methods similar to those in quantum chemistry, is extremely computationally expensive. In some cases, such test calculations have consumed >100 million CPU hours. One of the most expensive and least accurate aspects of the simulation is the dynamics of the plasma prior to the sustained fusion stage (>10s of millions of Kelvin), when parts of the capsule and fuel are a more balmy 100k Kelvin. In this “warm dense matter” regime, quantum correlations play a larger role in the behavior of the system than in the “hot dense matter” regime when sustained fusion takes place.

In our new preprint, “Quantum computation of stopping power for inertial fusion target design”, we present a quantum algorithm to compute the so-called “stopping power” of the warm dense matter in a nuclear fusion experiment. The stopping power is the rate at which a high energy alpha particle slows down due to Coulomb interactions with the surrounding plasma. Understanding the stopping power of the system is vital for optimizing the efficiency of the reactor. As the alpha particle is slowed by the plasma around it, it transfers its energy to the plasma, heating it up. This self-heating process is the mechanism by which fusion reactions sustain the burning plasma. Detailed modeling of this process will help inform future reactor designs.

We estimate that the quantum algorithm needed to calculate the stopping power would require resources somewhere between the P450 application and the battery application. But since this is the first case study on first-principles dynamics (or any application at finite temperature), such estimates are just a starting point and we again expect to find algorithmic improvements to bring this cost down in the future. Despite this uncertainty, it is still certainly better than the classical alternative, for which the only tractable approaches for these simulations are mean-field methods. While these methods incur unknown systematic errors when describing the physics of these systems, they are currently the only meaningful means of performing such simulations.

Left: A projectile (red) passing through a medium (blue) with initial velocity vproj. Right: To calculate the stopping power, we monitor the energy transfer between the projectile and the medium (blue solid line) and determine its average slope (red dashed line).

Discussion and conclusion

The examples described above are just three of a large and growing body of concrete applications for a future error-corrected quantum computer in simulating physical systems. This line of research helps us understand the classes of problems that will most benefit from the power of quantum computing. In particular, the last example is distinct from the other two in that it is simulating a dynamical system. In contrast to the other problems, which focus on finding the lowest energy, static ground state of a quantum system, quantum dynamics is concerned with how a quantum system changes over time. Since quantum computers are inherently dynamic — the qubit states evolve and change as each operation is performed — they are particularly well suited to solving these kinds of problems. Together with collaborators at Columbia, Harvard, Sandia National Laboratories and Macquarie University in Australia we recently published a paper in Nature Communications demonstrating that quantum algorithms for simulating electron dynamics can be more efficient even than approximate, “mean-field” classical calculations, while simultaneously offering much higher accuracy.

Developing and improving algorithms today prepares us to take full advantage of them when an error-corrected quantum computer is eventually realized. Just as in the classical computing case, we expect improvements at every level of the quantum computing stack to further lower the resource requirements. But this first step helps separate hyperbole from genuine applications amenable to quantum computational speedups.


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

Source: Google AI Blog

How to compare a noisy quantum processor to a classical computer

A full-scale error-corrected quantum computer will be able to solve some problems that are impossible for classical computers, but building such a device is a huge endeavor. We are proud of the milestones that we have achieved toward a fully error-corrected quantum computer, but that large-scale computer is still some number of years away. Meanwhile, we are using our current noisy quantum processors as flexible platforms for quantum experiments.

In contrast to an error-corrected quantum computer, experiments in noisy quantum processors are currently limited to a few thousand quantum operations or gates, before noise degrades the quantum state. In 2019 we implemented a specific computational task called random circuit sampling on our quantum processor and showed for the first time that it outperformed state-of-the-art classical supercomputing.

Although they have not yet reached beyond-classical capabilities, we have also used our processors to observe novel physical phenomena, such as time crystals and Majorana edge modes, and have made new experimental discoveries, such as robust bound states of interacting photons and the noise-resilience of Majorana edge modes of Floquet evolutions.

We expect that even in this intermediate, noisy regime, we will find applications for the quantum processors in which useful quantum experiments can be performed much faster than can be calculated on classical supercomputers — we call these "computational applications" of the quantum processors. No one has yet demonstrated such a beyond-classical computational application. So as we aim to achieve this milestone, the question is: What is the best way to compare a quantum experiment run on such a quantum processor to the computational cost of a classical application?

We already know how to compare an error-corrected quantum algorithm to a classical algorithm. In that case, the field of computational complexity tells us that we can compare their respective computational costs — that is, the number of operations required to accomplish the task. But with our current experimental quantum processors, the situation is not so well defined.

In “Effective quantum volume, fidelity and computational cost of noisy quantum processing experiments”, we provide a framework for measuring the computational cost of a quantum experiment, introducing the experiment’s “effective quantum volume”, which is the number of quantum operations or gates that contribute to a measurement outcome. We apply this framework to evaluate the computational cost of three recent experiments: our random circuit sampling experiment, our experiment measuring quantities known as “out of time order correlators” (OTOCs), and a recent experiment on a Floquet evolution related to the Ising model. We are particularly excited about OTOCs because they provide a direct way to experimentally measure the effective quantum volume of a circuit (a sequence of quantum gates or operations), which is itself a computationally difficult task for a classical computer to estimate precisely. OTOCs are also important in nuclear magnetic resonance and electron spin resonance spectroscopy. Therefore, we believe that OTOC experiments are a promising candidate for a first-ever computational application of quantum processors.

Plot of computational cost and impact of some recent quantum experiments. While some (e.g., QC-QMC 2022) have had high impact and others (e.g., RCS 2023) have had high computational cost, none have yet been both useful and hard enough to be considered a “computational application.” We hypothesize that our future OTOC experiment could be the first to pass this threshold. Other experiments plotted are referenced in the text.

Random circuit sampling: Evaluating the computational cost of a noisy circuit

When it comes to running a quantum circuit on a noisy quantum processor, there are two competing considerations. On one hand, we aim to do something that is difficult to achieve classically. The computational cost — the number of operations required to accomplish the task on a classical computer — depends on the quantum circuit’s effective quantum volume: the larger the volume, the higher the computational cost, and the more a quantum processor can outperform a classical one.

But on the other hand, on a noisy processor, each quantum gate can introduce an error to the calculation. The more operations, the higher the error, and the lower the fidelity of the quantum circuit in measuring a quantity of interest. Under this consideration, we might prefer simpler circuits with a smaller effective volume, but these are easily simulated by classical computers. The balance of these competing considerations, which we want to maximize, is called the "computational resource", shown below.

Graph of the tradeoff between quantum volume and noise in a quantum circuit, captured in a quantity called the “computational resource.” For a noisy quantum circuit, this will initially increase with the computational cost, but eventually, noise will overrun the circuit and cause it to decrease.

We can see how these competing considerations play out in a simple “hello world” program for quantum processors, known as random circuit sampling (RCS), which was the first demonstration of a quantum processor outperforming a classical computer. Any error in any gate is likely to make this experiment fail. Inevitably, this is a hard experiment to achieve with significant fidelity, and thus it also serves as a benchmark of system fidelity. But it also corresponds to the highest known computational cost achievable by a quantum processor. We recently reported the most powerful RCS experiment performed to date, with a low measured experimental fidelity of 1.7x10-3, and a high theoretical computational cost of ~1023. These quantum circuits had 700 two-qubit gates. We estimate that this experiment would take ~47 years to simulate in the world's largest supercomputer. While this checks one of the two boxes needed for a computational application — it outperforms a classical supercomputer — it is not a particularly useful application per se.

OTOCs and Floquet evolution: The effective quantum volume of a local observable

There are many open questions in quantum many-body physics that are classically intractable, so running some of these experiments on our quantum processor has great potential. We typically think of these experiments a bit differently than we do the RCS experiment. Rather than measuring the quantum state of all qubits at the end of the experiment, we are usually concerned with more specific, local physical observables. Because not every operation in the circuit necessarily impacts the observable, a local observable’s effective quantum volume might be smaller than that of the full circuit needed to run the experiment.

We can understand this by applying the concept of a light cone from relativity, which determines which events in space-time can be causally connected: some events cannot possibly influence one another because information takes time to propagate between them. We say that two such events are outside their respective light cones. In a quantum experiment, we replace the light cone with something called a “butterfly cone,” where the growth of the cone is determined by the butterfly speed — the speed with which information spreads throughout the system. (This speed is characterized by measuring OTOCs, discussed later.) The effective quantum volume of a local observable is essentially the volume of the butterfly cone, including only the quantum operations that are causally connected to the observable. So, the faster information spreads in a system, the larger the effective volume and therefore the harder it is to simulate classically.

A depiction of the effective volume Veff of the gates contributing to the local observable B. A related quantity called the effective area Aeff is represented by the cross-section of the plane and the cone. The perimeter of the base corresponds to the front of information travel that moves with the butterfly velocity vB.

We apply this framework to a recent experiment implementing a so-called Floquet Ising model, a physical model related to the time crystal and Majorana experiments. From the data of this experiment, one can directly estimate an effective fidelity of 0.37 for the largest circuits. With the measured gate error rate of ~1%, this gives an estimated effective volume of ~100. This is much smaller than the light cone, which included two thousand gates on 127 qubits. So, the butterfly velocity of this experiment is quite small. Indeed, we argue that the effective volume covers only ~28 qubits, not 127, using numerical simulations that obtain a larger precision than the experiment. This small effective volume has also been corroborated with the OTOC technique. Although this was a deep circuit, the estimated computational cost is 5x1011, almost one trillion times less than the recent RCS experiment. Correspondingly, this experiment can be simulated in less than a second per data point on a single A100 GPU. So, while this is certainly a useful application, it does not fulfill the second requirement of a computational application: substantially outperforming a classical simulation.

Information scrambling experiments with OTOCs are a promising avenue for a computational application. OTOCs can tell us important physical information about a system, such as the butterfly velocity, which is critical for precisely measuring the effective quantum volume of a circuit. OTOC experiments with fast entangling gates offer a potential path for a first beyond-classical demonstration of a computational application with a quantum processor. Indeed, in our experiment from 2021 we achieved an effective fidelity of Feff ~ 0.06 with an experimental signal-to-noise ratio of ~1, corresponding to an effective volume of ~250 gates and a computational cost of 2x1012.

While these early OTOC experiments are not sufficiently complex to outperform classical simulations, there is a deep physical reason why OTOC experiments are good candidates for the first demonstration of a computational application. Most of the interesting quantum phenomena accessible to near-term quantum processors that are hard to simulate classically correspond to a quantum circuit exploring many, many quantum energy levels. Such evolutions are typically chaotic and standard time-order correlators (TOC) decay very quickly to a purely random average in this regime. There is no experimental signal left. This does not happen for OTOC measurements, which allows us to grow complexity at will, only limited by the error per gate. We anticipate that a reduction of the error rate by half would double the computational cost, pushing this experiment to the beyond-classical regime.


Using the effective quantum volume framework we have developed, we have determined the computational cost of our RCS and OTOC experiments, as well as a recent Floquet evolution experiment. While none of these meet the requirements yet for a computational application, we expect that with improved error rates, an OTOC experiment will be the first beyond-classical, useful application of a quantum processor.

Source: Google AI Blog

The world’s first braiding of non-Abelian anyons

Imagine you’re shown two identical objects and then asked to close your eyes. When you open your eyes, you see the same two objects in the same position. How can you determine if they have been swapped back and forth? Intuition and the laws of quantum mechanics agree: If the objects are truly identical, there is no way to tell.

While this sounds like common sense, it only applies to our familiar three-dimensional world. Researchers have predicted that for a special type of particle, called an anyon, that is restricted to move only in a two-dimensional (2D) plane, quantum mechanics allows for something quite different. Anyons are indistinguishable from one another and some, non-Abelian anyons, have a special property that causes observable differences in the shared quantum state under exchange, making it possible to tell when they have been exchanged, despite being fully indistinguishable from one another. While researchers have managed to detect their relatives, Abelian anyons, whose change under exchange is more subtle and impossible to directly detect, realizing “non-Abelian exchange behavior” has proven more difficult due to challenges with both control and detection.

In “Non-Abelian braiding of graph vertices in a superconducting processor”, published in Nature, we report the observation of this non-Abelian exchange behavior for the first time. Non-Abelian anyons could open a new avenue for quantum computation, in which quantum operations are achieved by swapping particles around one another like strings are swapped around one another to create braids. Realizing this new exchange behavior on our superconducting quantum processor could be an alternate route to so-called topological quantum computation, which benefits from being robust against environmental noise.

Exchange statistics and non-Abelian anyons

In order to understand how this strange non-Abelian behavior can occur, it’s helpful to consider an analogy with the braiding of two strings. Take two identical strings and lay them parallel next to one another. Swap their ends to form a double-helix shape. The strings are identical, but because they wrap around one another when the ends are exchanged, it is very clear when the two ends are swapped.

The exchange of non-Abelian anyons can be visualized in a similar way, where the strings are made from extending the particles’ positions into the time dimension to form “world-lines.” Imagine plotting two particles’ locations vs. time. If the particles stay put, the plot would simply be two parallel lines, representing their constant locations. But if we exchange the locations of the particles, the world lines wrap around one another. Exchange them a second time, and you’ve made a knot.

While a bit difficult to visualize, knots in four dimensions (three spatial plus one time dimension) can always easily be undone. They are trivial — like a shoelace, simply pull one end and it unravels. But when the particles are restricted to two spatial dimensions, the knots are in three total dimensions and — as we know from our everyday 3D lives — cannot always be easily untied. The braiding of the non-Abelian anyons’ world lines can be used as quantum computing operations to transform the state of the particles.

A key aspect of non-Abelian anyons is “degeneracy”: the full state of several separated anyons is not completely specified by local information, allowing the same anyon configuration to represent superpositions of several quantum states. Winding non-Abelian anyons about each other can change the encoded state.

How to make a non-Abelian anyon

So how do we realize non-Abelian braiding with one of Google’s quantum processors? We start with the familiar surface code, which we recently used to achieve a milestone in quantum error correction, where qubits are arranged on the vertices of a checkerboard pattern. Each color square of the checkerboard represents one of two possible joint measurements that can be made of the qubits on the four corners of the square. These so-called “stabilizer measurements” can return a value of either + or – 1. The latter is referred to as a plaquette violation, and can be created and moved diagonally — just like bishops in chess — by applying single-qubit X- and Z-gates. Recently, we showed that these bishop-like plaquette violations are Abelian anyons. In contrast to non-Abelian anyons, the state of Abelian anyons changes only subtly when they are swapped — so subtly that it is impossible to directly detect. While Abelian anyons are interesting, they do not hold the same promise for topological quantum computing that non-Abelian anyons do.

To produce non-Abelian anyons, we need to control the degeneracy (i.e., the number of wavefunctions that causes all stabilizer measurements to be +1). Since a stabilizer measurement returns two possible values, each stabilizer cuts the degeneracy of the system in half, and with sufficiently many stabilizers, only one wave function satisfies the criterion. Hence, a simple way to increase the degeneracy is to merge two stabilizers together. In the process of doing so, we remove one edge in the stabilizer grid, giving rise to two points where only three edges intersect. These points, referred to as “degree-3 vertices” (D3Vs), are predicted to be non-Abelian anyons.

In order to braid the D3Vs, we have to move them, meaning that we have to stretch and squash the stabilizers into new shapes. We accomplish this by implementing two-qubit gates between the anyons and their neighbors (middle and right panels shown below).

Non-Abelian anyons in stabilizer codes. a: Example of a knot made by braiding two anyons’ world lines. b: Single-qubit gates can be used to create and move stabilizers with a value of –1 (red squares). Like bishops in chess, these can only move diagonally and are therefore constrained to one sublattice in the regular surface code. This constraint is broken when D3Vs (yellow triangles) are introduced. c: Process to form and move D3Vs (predicted to be non-Abelian anyons). We start with the surface code, where each square corresponds to a joint measurement of the four qubits on its corners (left panel). We remove an edge separating two neighboring squares, such that there is now a single joint measurement of all six qubits (middle panel). This creates two D3Vs, which are non-Abelian anyons. We move the D3Vs by applying two-qubit gates between neighboring sites (right panel).

Now that we have a way to create and move the non-Abelian anyons, we need to verify their anyonic behavior. For this we examine three characteristics that would be expected of non-Abelian anyons:

  1. The “fusion rules” — What happens when non-Abelian anyons collide with each other?
  2. Exchange statistics — What happens when they are braided around one another?
  3. Topological quantum computing primitives — Can we encode qubits in the non-Abelian anyons and use braiding to perform two-qubit entangling operations?

The fusion rules of non-Abelian anyons

We investigate fusion rules by studying how a pair of D3Vs interact with the bishop-like plaquette violations introduced above. In particular, we create a pair of these and bring one of them around a D3V by applying single-qubit gates.

While the rules of bishops in chess dictate that the plaquette violations can never meet, the dislocation in the checkerboard lattice allows them to break this rule, meet its partner and annihilate with it. The plaquette violations have now disappeared! But bring the non-Abelian anyons back in contact with one another, and the anyons suddenly morph into the missing plaquette violations. As weird as this behavior seems, it is a manifestation of exactly the fusion rules that we expect these entities to obey. This establishes confidence that the D3Vs are, indeed, non-Abelian anyons.

Demonstration of anyonic fusion rules (starting with panel I, in the lower left). We form and separate two D3Vs (yellow triangles), then form two adjacent plaquette violations (red squares) and pass one between the D3Vs. The D3Vs deformation of the “chessboard” changes the bishop rules of the plaquette violations. While they used to lie on adjacent squares, they are now able to move along the same diagonals and collide (as shown by the red lines). When they do collide, they annihilate one another. The D3Vs are brought back together and surprisingly morph into the missing adjacent red plaquette violations.

Observation of non-Abelian exchange statistics

After establishing the fusion rules, we want to see the real smoking gun of non-Abelian anyons: non-Abelian exchange statistics. We create two pairs of non-Abelian anyons, then braid them by wrapping one from each pair around each other (shown below). When we fuse the two pairs back together, two pairs of plaquette violations appear. The simple act of braiding the anyons around one another changed the observables of our system. In other words, if you closed your eyes while the non-Abelian anyons were being exchanged, you would still be able to tell that they had been exchanged once you opened your eyes. This is the hallmark of non-Abelian statistics.

Braiding non-Abelian anyons. We make two pairs of D3Vs (panel II), then bring one from each pair around each other (III-XI). When fusing the two pairs together again in panel XII, two pairs of plaquette violations appear! Braiding the non-Abelian anyons changed the observables of the system from panel I to panel XII; a direct manifestation of non-Abelian exchange statistics.

Topological quantum computing

Finally, after establishing their fusion rules and exchange statistics, we demonstrate how we can use these particles in quantum computations. The non-Abelian anyons can be used to encode information, represented by logical qubits, which should be distinguished from the actual physical qubits used in the experiment. The number of logical qubits encoded in N D3Vs can be shown to be N/2–1, so we use N=8 D3Vs to encode three logical qubits, and perform braiding to entangle them. By studying the resulting state, we find that the braiding has indeed led to the formation of the desired, well-known quantum entangled state called the Greenberger-Horne-Zeilinger (GHZ) state.

Using non-Abelian anyons as logical qubits. a, We braid the non-Abelian anyons to entangle three qubits encoded in eight D3Vs. b, Quantum state tomography allows for reconstructing the density matrix, which can be represented in a 3D bar plot and is found to be consistent with the desired highly entangled GHZ-state.


Our experiments show the first observation of non-Abelian exchange statistics, and that braiding of the D3Vs can be used to perform quantum computations. With future additions, including error correction during the braiding procedure, this could be a major step towards topological quantum computation, a long-sought method to endow qubits with intrinsic resilience against fluctuations and noise that would otherwise cause errors in computations.


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

Source: Google AI Blog

Suppressing quantum errors by scaling a surface code logical qubit

Many years from today, scientists will be able to use fault-tolerant quantum computers for large-scale computations with applications across science and industry. These quantum computers will be much bigger than today, consisting of millions of coherent quantum bits, or qubits. But there’s a catch — these basic building blocks must be good enough or the systems will be overrun with errors.

Currently, the error rates of the qubits on our 3rd generation Sycamore processor are typically between 1 in 10,000 to 1 in 100. Through our work and that of others, we understand that developing large-scale quantum computers will require far lower error rates. We will need rates in the range of 1 in 109 to 1 in 106 to run quantum circuits that can solve industrially relevant problems.

So how do we get there, knowing that squeezing three to six orders of magnitude of better performance from our current physical qubits is unlikely? Our team has created a roadmap that has directed our research for the last several years, improving the performance of our quantum computers in gradual steps toward a fault-tolerant quantum computer.

Roadmap for building a useful error-corrected quantum computer with key milestones. We are currently building one logical qubit that we will scale in the future.

Today, in “Suppressing Quantum Errors by Scaling a Surface Code Logical Qubit”, published in Nature, we are announcing that we have reached the second milestone on our roadmap. Our experimental results demonstrate a prototype of the basic unit of an error-corrected quantum computer known as a logical qubit, with performance nearing the regime that enables scalable fault-tolerant quantum computing.

From physical qubits to logical qubits

Quantum error correction (QEC) represents a significant shift from today’s quantum computing, where each physical qubit on the processor acts as a unit of computation. It provides the recipe to reach low errors by trading many good qubits for an excellent one: information is encoded across several physical qubits to construct a single logical qubit that is more resilient and capable of running large-scale quantum algorithms. Under the right conditions, the more physical qubits used to build a logical qubit, the better that logical qubit becomes.

However, this will not work if the added errors from each additional physical qubit outweigh the benefits of QEC. Until now, the high physical error rates have always won out.

To that end, we use a particular error-correcting code called a surface code and show for the first time that increasing the size of the code decreases the error rate of the logical qubit. A first-ever for any quantum computing platform, this was achieved by painstakingly mitigating many error sources as we scaled from 17 to 49 physical qubits. This work is evidence that with enough care, we can produce the logical qubits necessary for a large-scale error-corrected quantum computer.

Quantum error correction with surface codes

How does an error-correcting code protect information? Take a simple example from classical communication: Bob wants to send Alice a single bit that reads “1” across a noisy communication channel. Recognizing that the message is lost if the bit flips to “0”, Bob instead sends three bits: “111”. If one erroneously flips, Alice could take a majority vote (a simple error-correcting code) of all the received bits and still understand the intended message. Repeating the information more than three times — increasing the “size” of the code — would enable the code to tolerate more individual errors.

Many physical qubits on a quantum processor acting as one logical qubit in an error-correcting code called a surface code.

A surface code takes this principle and imagines a practical quantum implementation. It has to satisfy two additional constraints. First, the surface code must be able to correct not just bit flips, taking a qubit from |0 to |1, but also phase flips. This error is unique to quantum states and transforms a qubit in a superposition state, for example from “|0 + |1” to “|0 - |1”. Second, checking the qubits’ states would destroy their superpositions, so one needs a way of detecting errors without measuring the states directly.

To address these constraints, we arrange two types of qubits on a checkerboard. “Data” qubits on the vertices make up the logical qubit, while “measure” qubits at the center of each square are used for so-called “stabilizer measurements.” These measurements tell us whether the qubits are all the same, as desired, or different, signaling that an error occurred, without actually revealing the value of the individual data qubits.

We tile two types of stabilizer measurements in a checkerboard pattern to protect the logical data from bit- and phase-flips. If some of the stabilizer measurements register an error, then correlations in the stabilizer measurements are used to identify which error(s) occurred and where.

Surface-code QEC. Data qubits (yellow) are at the vertices of a checkerboard. Measure qubits at the center of each square are used for stabilizer measurements (blue squares). Dark blue squares check for bit-flip errors, while light blue squares check for phase-flip errors. Left: A phase-flip error. The two nearest light blue stabilizer measurements register the error (light red). Right: A bit-flip error. The two nearest dark blue stabilizer measurements register the error (dark red).

Just as Bob’s message to Alice in the example above became more robust against errors with increasing code size, a larger surface code better protects the logical information it contains. The surface code can withstand a number of bit- and phase-flip errors each equal to less than half the distance, where the distance is the number of data qubits that span the surface code in either dimension.

But here’s the problem: every individual physical qubit is prone to errors, so the more qubits in a code, the more opportunity for errors. We want the higher protection offered by QEC to outweigh the increased opportunities for errors as we increase the number of qubits. For this to happen, the physical qubits must have errors below the so-called “fault-tolerant threshold.” For the surface code, this threshold is quite low. So low that it hasn’t been experimentally feasible until recently. We are now on the precipice of reaching this coveted regime.

Making and controlling high-quality physical qubits

Entering the regime where QEC improves with scale required improving every aspect of our quantum computers, from nanofabrication of the physical qubits to the optimized control of the full quantum system. These experiments ran on a state-of-the-art 3rd generation Sycamore processor architecture optimized for QEC using the surface code with improvements across the board:

  • Increased qubit relaxation and dephasing lifetimes through an improved fabrication process and environmental noise reduction near the quantum processor.
  • Lowered cross-talk between all physical qubits during parallel operation by optimizing quantum processor circuit design and nanofabrication.
  • Reduced drift and improved qubit control fidelity through upgraded custom electronics.
  • Implemented faster and higher-fidelity readout and reset operations compared with previous generations of the Sycamore processor.
  • Reduced calibration errors by extensively modeling the full quantum system and employing better system-optimization algorithms.
  • Developed context-aware and fully parallel calibrations to minimize drift and optimize control parameters for QEC circuits.
  • Enhanced dynamical decoupling protocols to protect physical qubits from noise and cross-talk during idling operations.

Running surface code circuits

With these upgrades in place, we ran experiments to compare the ratio (𝚲3,5) between the logical error rate of a distance-3 surface code (ε3) with 17 qubits to that of a distance-5 surface code (ε5) with 49 qubits — 𝚲3,5 = ε3 / ε5.

Comparison of logical fidelity (defined as 1-ε) between distance-3 (d=3) and distance-5 (d=5) surface codes. The distance-5 code contains four possible distance-3 arrangements, with one example shown in the red outline (left). As improvements were made, the d=5 fidelity increased faster than that of the d=3, eventually overtaking the distance-3 code, as shown in the top-right data points (right), whose average lies slightly to the left of the ε3 = ε5 line.

The results of these experiments are shown above on the right. Continued improvements over several months allowed us to reduce the logical errors of both grids, leading to the distance-5 grid (ε5 = 2.914%) outperforming the distance-3 grids (ε3 = 3.028%) by 4% (𝚲3,5 = 1.04) with 5𝛔 confidence. While this might seem like a small improvement, it’s important to emphasize that the result represents a first for the field since Peter Shor’s 1995 QEC proposal. A larger code outperforming a smaller one is a key signature of QEC, and all quantum computing architectures will need to pass this hurdle to realize a path to the low errors that are necessary for quantum applications.

The path forward

These results indicate that we are entering a new era of practical QEC. The Google Quantum AI team has spent the last few years thinking about how we define success in this new era, and how we measure progress along the way.

The ultimate goal is to demonstrate a pathway to achieving the low errors needed for using quantum computers in meaningful applications. To this end, our target remains achieving logical error rates of 1 in 106 or lower per cycle of QEC. In the figure below on the left, we outline the path that we anticipate to reach this target. As we continue improving our physical qubits (and hence the performance of our logical qubits), we expect to gradually increase 𝚲 from close to 1 in this work to larger numbers. The figure below shows that a value of 𝚲 = 4 and a code distance of 17 (577 physical qubits with good enough quality) will yield a logical error rate below our target of 1 in 106.

While this result is still a few years out, we have an experimental technique to probe error rates this low with today’s hardware, albeit in limited circumstances. While two-dimensional surface codes allow us to correct both bit- and phase-flip errors, we can also construct one-dimensional repetition codes that are only able to solve one type of error with relaxed requirements. On the right below, we show that a distance-25 repetition code can reach error rates per cycle close to 1 in 106. At such low errors, we see new kinds of error mechanisms that are not yet observable with our surface codes. By controlling for these error mechanisms, we can improve repetition codes to error rates near 1 in 107.

Left: Expected progression as we improve performance (quantified by 𝚲) and scale (quantified by code distance) for surface codes. Right: Experimentally measured logical error rates per cycle versus the distance of one-dimensional repetition codes and two-dimensional surface codes.

Reaching this milestone reflects three years of focused work by the entire Google Quantum AI team following our demonstration of a quantum computer outperforming a classical computer. In our march toward building fault-tolerant quantum computers, we will continue to use the target error rates in the figure above to measure our progress. With further improvements toward our next milestone, we anticipate entering the fault-tolerant regime, where we can exponentially suppress logical errors and unlock the first useful error-corrected quantum applications. In the meantime, we continue to explore various ways of solving problems using quantum computers in topics ranging from condensed matter physics to chemistry, machine learning, and materials science.

Source: Google AI Blog

Google Research, 2022 & beyond: Natural sciences

(This is Part 7 in our series of posts covering different topical areas of research at Google. You can find other posts in the series here.)

It's an incredibly exciting time to be a scientist. With the amazing advances in machine learning (ML) and quantum computing, we now have powerful new tools that enable us to act on our curiosity, collaborate in new ways, and radically accelerate progress toward breakthrough scientific discoveries.

Since joining Google Research eight years ago, I’ve had the privilege of being part of a community of talented researchers fascinated by applying cutting-edge computing to push the boundaries of what is possible in applied science. Our teams are exploring topics across the physical and natural sciences. So, for this year’s blog post I want to focus on high-impact advances we’ve made recently in the fields of biology and physics, from helping to organize the world’s protein and genomics information to benefit people's lives to improving our understanding of the nature of the universe with quantum computers. We are inspired by the great potential of this work.

Using machine learning to unlock mysteries in biology

Many of our researchers are fascinated by the extraordinary complexity of biology, from the mysteries of the brain, to the potential of proteins, and to the genome, which encodes the very language of life. We’ve been working alongside scientists from other leading organizations around the world to tackle important challenges in the fields of connectomics, protein function prediction, and genomics, and to make our innovations accessible and useful to the greater scientific community.


One exciting application of our Google-developed ML methods was to explore how information travels through the neuronal pathways in the brains of zebrafish, which provides insight into how the fish engage in social behavior like swarming. In collaboration with researchers from the Max Planck Institute for Biological Intelligence, we were able to computationally reconstruct a portion of zebrafish brains imaged with 3D electron microscopy — an exciting advance in the use of imaging and computational pipelines to map out the neuronal circuitry in small brains, and another step forward in our long-standing contributions to the field of connectomics.

Reconstruction of the neural circuitry of a larval zebrafish brain, courtesy of the Max Planck Institute for Biological Intelligence.

The technical advances necessary for this work will have applications even beyond neuroscience. For example, to address the difficulty of working with such large connectomics datasets, we developed and released TensorStore, an open-source C++ and Python software library designed for storage and manipulation of n-dimensional data. We look forward to seeing the ways it is used in other fields for the storage of large datasets.

We're also using ML to shed light on how human brains perform remarkable feats like language by comparing human language processing and autoregressive deep language models (DLMs). For this study, a collaboration with colleagues at Princeton University and New York University Grossman School of Medicine, participants listened to a 30-minute podcast while their brain activity was recorded using electrocorticography. The recordings suggested that the human brain and DLMs share computational principles for processing language, including continuous next-word prediction, reliance on contextual embeddings, and calculation of post-onset surprise based on word match (we can measure how surprised the human brain is by the word, and correlate that surprise signal with how well the word is predicted by the DLM). These results provide new insights into language processing in the human brain, and suggest that DLMs can be used to reveal valuable insights about the neural basis of language.


ML has also allowed us to make significant advances in understanding biological sequences. In 2022, we leveraged recent advances in deep learning to accurately predict protein function from raw amino acid sequences. We also worked in close collaboration with the European Molecular Biology Laboratory's European Bioinformatics Institute (EMBL-EBI) to carefully assess model performance and add hundreds of millions of functional annotations to the public protein databases UniProt, Pfam/InterPro, and MGnify. Human annotation of protein databases can be a laborious and slow process and our ML methods enabled a giant leap forward — for example, increasing the number of Pfam annotations by a larger number than all other efforts during the past decade combined. The millions of scientists worldwide who access these databases each year can now use our annotations for their research.

Google Research contributions to Pfam exceed in size all expansion efforts made to the database over the last decade.

Although the first draft of the human genome was released in 2003, it was incomplete and had many gaps due to technical limitations in the sequencing technologies. In 2022 we celebrated the remarkable achievements of the Telomere-2-Telomere (T2T) Consortium in resolving these previously unavailable regions — including five full chromosome arms and nearly 200 million base pairs of novel DNA sequences — which are interesting and important for questions of human biology, evolution, and disease. Our open source genomics variant caller, DeepVariant, was one of the tools used by the T2T Consortium to prepare their release of a complete 3.055 billion base pair sequence of a human genome. The T2T Consortium is also using our newer open source method DeepConsensus, which provides on-device error correction for Pacific Biosciences long-read sequencing instruments, in their latest research toward comprehensive pan-genome resources that can represent the breadth of human genetic diversity.

Using quantum computing for new physics discoveries

When it comes to making scientific discoveries, quantum computing is still in its infancy, but has a lot of potential. We’re exploring ways of advancing the capabilities of quantum computing so that it can become a tool for scientific discovery and breakthroughs. In collaboration with physicists from around the world, we are also starting to use our existing quantum computers to create interesting new experiments in physics.

As an example of such experiments, consider the problem where a sensor measures something, and a computer then processes the data from the sensor. Traditionally, this means the sensor’s data is processed as classical information on our computers. Instead, one idea in quantum computing is to directly process quantum data from sensors. Feeding data from quantum sensors directly to quantum algorithms without going through classical measurements may provide a large advantage. In a recent Science paper written in collaboration with researchers from multiple universities, we show that quantum computing can extract information from exponentially fewer experiments than classical computing, as long as the quantum computer is coupled directly to the quantum sensors and is running a learning algorithm. This “quantum machine learning” can yield an exponential advantage in dataset size, even with today’s noisy intermediate-scale quantum computers. Because experimental data is often the limiting factor in scientific discovery, quantum ML has the potential to unlock the vast power of quantum computers for scientists. Even better, the insights from this work are also applicable to learning on the output of quantum computations, such as the output of quantum simulations that may otherwise be difficult to extract.

Even without quantum ML, a powerful application of quantum computers is to experimentally explore quantum systems that would be otherwise impossible to observe or simulate. In 2022, the Quantum AI team used this approach to observe the first experimental evidence of multiple microwave photons in a bound state using superconducting qubits. Photons typically do not interact with one another, and require an additional element of non-linearity to cause them to interact. The results of our quantum computer simulations of these interactions surprised us — we thought the existence of these bound states relied on fragile conditions, but instead we found that they were robust even to relatively strong perturbations that we applied.

Occupation probability versus discrete time step for n-photon bound states. We observe that the majority of the photons (darker colors) remain bound together.

Given the initial successes we have had in applying quantum computing to make physics breakthroughs, we are hopeful about the possibility of this technology to enable future groundbreaking discoveries that could have as significant a societal impact as the creation of transistors or GPS. The future of quantum computing as a scientific tool is exciting!


I would like to thank everyone who worked hard on the advances described in this post, including the Google Applied Sciences, Quantum AI, Genomics and Brain teams and their collaborators across Google Research and externally. Finally, I would like to thank the many Googlers who provided feedback in the writing of this post, including Lizzie Dorfman, Erica Brand, Elise Kleeman, Abe Asfaw, Viren Jain, Lucy Colwell, Andrew Carroll, Ariel Goldstein and Charina Chou.


Google Research, 2022 & beyond

This was the seventh blog post in the “Google Research, 2022 & Beyond” series. Other posts in this series are listed in the table below:

Source: Google AI Blog

Amplification at the Quantum limit

The Google Quantum AI team is building quantum computers with superconducting microwave circuits, but much like a classical computer the superconducting processor at the heart of these computers is only part of the story. An entire technology stack of peripheral hardware is required to make the quantum computer work properly. In many cases these parts must be custom designed, requiring extensive research and development to reach the highest levels of performance.

In this post, we highlight one aspect of this supplemental hardware: our superconducting microwave amplifiers. In “Readout of a Quantum Processor with High Dynamic Range Josephson Parametric Amplifiers”, published in Applied Physics Letters, we describe how we increased the maximum output power of our superconducting microwave amplifiers by a factor of over 100x. We discuss how this work can pave the way for the operation of larger quantum processor chips with improved performance.

Why microwave amplifiers?

One of the challenges of operating a superconducting quantum processor is measuring the state of a qubit without disturbing its operation. Fundamentally, this comes down to a microwave engineering problem, where we need to be able to measure the energy inside the qubit resonator without exposing it to noisy or lossy wiring. This can be accomplished by adding an additional microwave resonator to the system that is coupled to the qubit, but far from the qubit’s resonance frequency. The resonator acts as a filter that isolates the qubit from the control lines but also picks up a state-dependent frequency shift from the qubit. Just like in the binary phase shift keying (BPSK) encoding technique, the digital state of the qubit (0 or 1) is translated into a phase for a probe tone (microwave signal) reflecting off of this auxiliary resonator. Measuring the phase of this probe tone allows us to infer the state of the qubit without directly interfacing with the qubit itself.

While this sounds simple, the qubit actually imposes a severe cap on how much power can be used for this probe tone. In normal operation, a qubit should be in the 0 state or the 1 state or some superposition of the two. A measurement pulse should collapse the qubit into one of these two states, but using too much power can push it into a higher excited state and corrupt the computation. A safe measurement power is typically around -125 dBm, which amounts to only a handful of microwave photons interacting with the processor during the measurement. Typically, small signals are measured using microwave amplifiers, which increase the signal level, but also add their own noise. How much noise is acceptable? If the measurement process takes too long, the qubit state can change due to energy loss in the circuit. This means that these very small signals must be measured in just a few hundred nanoseconds with very high (>99%) fidelity. We therefore cannot afford to average the signal over a longer time to reduce the noise. Unfortunately, even the best semiconductor low-noise amplifiers are still almost a factor of 10 too noisy.

The solution is to design our own custom amplifiers based on the same circuit elements as the qubits themselves. These amplifiers typically consist of Josephson junctions to provide a tunable inductance wired into a superconducting resonant circuit. By constructing a resonant circuit out of these elements, you can create a parametric amplifier where amplification is achieved by modulating the tunable inductance at twice the frequency you want to amplify. Additionally, because all of the wiring is made of lossless superconductors, these devices operate near the quantum limit of added noise, where the only noise in the signal is coming from amplification of the zero point quantum voltage fluctuations.

The one downside to these devices is that the Josephson junctions constrain the power of the signals we can measure. If the signal is too large, the drive current can approach the junction critical current and degrade the amplifier performance. Even if this limit was sufficient to measure a single qubit, our goal was to increase efficiency by measuring up to six qubits at a time using the same amplifier. Some groups get around this limit by making traveling wave amplifiers, where the signals are distributed across thousands of junctions. This increases the saturation power, but the amplifiers get very complicated to produce and take up a lot of space on the chip. Our goal was to create an amplifier that could handle as much power as a traveling wave amplifier but with the same simple and compact design we were used to.


The critical current of each Josephson junction limits our amplifier’s power handling. However, increasing this critical current also changes the inductance and, thus, the operating frequency of the amplifier. To avoid these constraints, we replaced a standard 2-junction DC SQUID with a nonlinear tunable inductor made up of two RF-SQUID arrays in parallel, which we call a snake inductor. Each RF-SQUID consists of a Josephson junction and geometric inductances L1 and L2, and each array contains 20 RF-SQUIDs. In this case, each junction of a standard DC SQUID is replaced by one of these RF-SQUID arrays. While the critical current of each RF-SQUID is much higher, we chain them together to keep the inductance and operating frequency the same. While this is a relatively modest increase in device complexity, it enables us to increase the power handling of each amplifier by roughly a factor of 100x. It is also fully compatible with existing designs that use impedance matching circuits to provide large measurement bandwidth.

Circuit diagram of our superconducting microwave amplifier. A split bias coil allows both DC and RF modulation of the snake inductor, while a shunt capacitor sets the frequency range. The flow of current is illustrated in the animation where an applied current (blue) on the bias line causes a circulating current (red) in the snake. A tapered impedance transformer lowers the loaded Q of the device. Since the Q is defined as frequency divided by bandwidth, lowering the Q with a constant frequency increases the bandwidth of the amplifier. Example circuit parameters used for a real device are Cs=6.0 pF, L1=2.6 pH, L2=8.0 pH, Lb=30 pH, M=50 pH, Z0 = 50 Ohms, and Zfinal = 18 ohms. The device operation is illustrated with a small signal (magenta) reflecting off the input of the amplifier. When the large pump tone (blue) is applied to the bias port, it generates amplified versions of the signal (gold) and a secondary tone known as an idler (also gold).
Microscope image of the nonlinear resonator showing the resonant circuit that consists of a large parallel plate capacitor, nonlinear snake inductor, and a current bias transformer to tune the inductance.

We measure this performance improvement by measuring the saturation power of the amplifier, or the point at which the gain is compressed by 1 dB. We also measure this power value vs. frequency to see how it scales with amplifier gain and distance from the center of the amplifier bandwidth. Since the amplifier gain is symmetric about its center frequency we measure this in terms of absolute detuning, which is just the absolute value of the difference between the center frequency of the amplifier and the probe tone frequency.

Input and output saturation power (1-dB gain compression point), calibrated using a superconducting quantum processor vs. absolute detuning from the amplifier center frequency.

Conclusion and future directions

The new microwave amplifiers represent a big step forward for our qubit measurement system. They will allow us to measure more qubits using a single device, and enable techniques that require higher power for each measurement tone. However, there are still quite a few areas we would like to explore. For example, we are currently investigating the application of snake inductors in amplifiers with advanced impedance matching techniques, directional amplifiers, and non-reciprocal devices like microwave circulators.


We would like to thank the Quantum AI team for the infrastructure and support that enabled the creation and measurement of our microwave amplifier devices. Thanks to our cohort of talented Google Research Interns that contributed to the future work mentioned above: Andrea Iorio for developing algorithms that automatically tune amplifiers and provide a snapshot of the local parameter space, Ryan Kaufman for measuring a new class of amplifiers using multi-pole impedance matching networks, and Randy Kwende for designing and testing a range of parametric devices based on snake inductors. With their contributions, we are gaining a better understanding of our amplifiers and designing the next generation of parametrically-driven devices.

Source: Google AI Blog