Quantum error correction below the surface code threshold

Rajeev Acharya, Laleh Aghababaie-Beni, Igor Aleiner, Trond I. Andersen, Markus Ansmann, Frank Arute, Kunal Arya, Abraham Asfaw, Nikita Astrakhantsev, Juan Atalaya, Ryan Babbush, Dave Bacon, Brian Ballard, Joseph C. Bardin, Johannes Bausch, Andreas Bengtsson, Alexander Bilmes, Sam Blackwell, Sergio Boixo, Gina Bortoli, Alexandre Bourassa, Jenna Bovaird, Leon Brill, Michael Broughton, David A. Browne, Brett Buchea, Bob B. Buckley, David A. Buell, Tim Burger, Brian Burkett, Nicholas Bushnell, Anthony Cabrera, Juan Campero, Hung-Shen Chang, Yu Chen, Zijun Chen, Ben Chiaro, Desmond Chik, Charina Chou, Jahan Claes, Agnetta Y. Cleland, Josh Cogan, Roberto Collins, Paul Conner, William Courtney, Alexander L. Crook, Ben Curtin, Sayan Das, Alex Davies, Laura De Lorenzo, Dripto M. Debroy, Sean Demura, Michel Devoret, Agustin Di Paolo, Paul Donohoe, Ilya Drozdov, Andrew Dunsworth, Clint Earle, Thomas Edlich, Alec Eickbusch, Aviv Moshe Elbag, Mahmoud Elzouka, Catherine Erickson, Lara Faoro, Edward Farhi, Vinicius S. Ferreira, Leslie Flores Burgos, Ebrahim Forati, Austin G. Fowler, Brooks Foxen, Suhas Ganjam, Gonzalo Garcia, Robert Gasca, Élie Genois, William Giang, Craig Gidney, Dar Gilboa, Raja Gosula, Alejandro Grajales Dau, Dietrich Graumann, Alex Greene, Jonathan A. Gross, Steve Habegger, John Hall, Michael C. Hamilton, Monica Hansen, Matthew P. Harrigan, Sean D. Harrington, Francisco J. H. Heras, Stephen Heslin, Paula Heu, Oscar Higgott, Gordon Hill, Jeremy Hilton, George Holland, Sabrina Hong, Hsin-Yuan Huang, Ashley Huff, William J. Huggins, Lev B. Ioffe, Sergei V. Isakov, Justin Iveland, Evan Jeffrey, Zhang Jiang, Cody Jones, Stephen Jordan, Chaitali Joshi, Pavol Juhas, Dvir Kafri, Hui Kang, Amir H. Karamlou, Kostyantyn Kechedzhi, Julian Kelly, Trupti Khaire, Tanuj Khattar, Mostafa Khezri, Seon Kim, Paul V. Klimov, Andrey R. Klots, Bryce Kobrin, Pushmeet Kohli, Alexander N. Korotkov, Fedor Kostritsa, Robin Kothari, Borislav Kozlovskii, John Mark Kreikebaum, Vladislav D. Kurilovich, Nathan Lacroix, David Landhuis, Tiano Lange-Dei, Brandon W. Langley, Pavel Laptev, Kim-Ming Lau, Loïck Le Guevel, Justin Ledford, Kenny Lee, Yuri D. Lensky, Shannon Leon, Brian J. Lester, Wing Yan Li, Yin Li, Alexander T. Lill, Wayne Liu, William P. Livingston, Aditya Locharla, Erik Lucero, Daniel Lundahl, Aaron Lunt, Sid Madhuk, Fionn D. Malone, Ashley Maloney, Salvatore Mandrá, Leigh S. Martin, Steven Martin, Orion Martin, Cameron Maxfield, Jarrod R. McClean, Matt McEwen, Seneca Meeks, Anthony Megrant, Xiao Mi, Kevin C. Miao, Amanda Mieszala, Reza Molavi, Sebastian Molina, Shirin Montazeri, Alexis Morvan, Ramis Movassagh, Wojciech Mruczkiewicz, Ofer Naaman, Matthew Neeley, Charles Neill, Ani Nersisyan, Hartmut Neven, Michael Newman, Jiun How Ng, Anthony Nguyen, Murray Nguyen, Chia-Hung Ni, Thomas E. O'Brien, William D. Oliver, Alex Opremcak, Kristoffer Ottosson, Andre Petukhov, Alex Pizzuto, John Platt, Rebecca Potter, Orion Pritchard, Leonid P. Pryadko, Chris Quintana, Ganesh Ramachandran, Matthew J. Reagor, David M. Rhodes, Gabrielle Roberts, Eliott Rosenberg, Emma Rosenfeld, Pedram Roushan, Nicholas C. Rubin, Negar Saei, Daniel Sank, Kannan Sankaragomathi, Kevin J. Satzinger, Henry F. Schurkus, Christopher Schuster, Andrew W. Senior, Michael J. Shearn, Aaron Shorter, Noah Shutty, Vladimir Shvarts, Shraddha Singh, Volodymyr Sivak, Jindra Skruzny, Spencer Small, Vadim Smelyanskiy, W. Clarke Smith, Rolando D. Somma, Sofia Springer, George Sterling, Doug Strain, Jordan Suchard, Aaron Szasz, Alex Sztein, Douglas Thor, Alfredo Torres, M. Mert Torunbalci, Abeer Vaishnav, Justin Vargas, Sergey Vdovichev, Guifre Vidal, Benjamin Villalonga, Catherine Vollgraff Heidweiller, Steven Waltman, Shannon X. Wang, Brayden Ware, Kate Weber, Theodore White, Kristi Wong, Bryan W. K. Woo, Cheng Xing, Z. Jamie Yao, Ping Yeh, Bicheng Ying, Juhwan Yoo, Noureldin Yosri, Grayson Young, Adam Zalcman, Yaxing Zhang, Ningfeng Zhu, Nicholas Zobrist

Quantum error correction provides a path to reach practical quantum computing by combining multiple physical qubits into a logical qubit, where the logical error rate is suppressed exponentially as more qubits are added. However, this exponential suppression only occurs if the physical error rate is below a critical threshold. In this work, we present two surface code memories operating below this threshold: a distance-7 code and a distance-5 code integrated with a real-time decoder. The logical error rate of our larger quantum memory is suppressed by a factor of $Λ$ = 2.14 $\pm$ 0.02 when increasing the code distance by two, culminating in a 101-qubit distance-7 code with 0.143% $\pm$ 0.003% error per cycle of error correction. This logical memory is also beyond break-even, exceeding its best physical qubit's lifetime by a factor of 2.4 $\pm$ 0.3. We maintain below-threshold performance when decoding in real time, achieving an average decoder latency of 63 $μ$s at distance-5 up to a million cycles, with a cycle time of 1.1 $μ$s. To probe the limits of our error-correction performance, we run repetition codes up to distance-29 and find that logical performance is limited by rare correlated error events occurring approximately once every hour, or 3 $\times$ 10$^9$ cycles. Our results present device performance that, if scaled, could realize the operational requirements of large scale fault-tolerant quantum algorithms.

Reduced phase error through optimized control of a superconducting qubit

Erik Lucero, Julian Kelly, Radoslaw C. Bialczak, Mike Lenander, Matteo Mariantoni, Matthew Neeley, A. D. O'Connell, Daniel Sank, H. Wang, Martin Weides, James Wenner, Tsuyoshi Yamamoto, A. N. Cleland, John Martinis

Minimizing phase and other errors in experimental quantum gates allows higher fidelity quantum processing. To quantify and correct for phase errors in particular, we have developed a new experimental metrology --- amplified phase error (APE) pulses --- that amplifies and helps identify phase errors in general multi-level qubit architectures. In order to correct for both phase and amplitude errors specific to virtual transitions and leakage outside of the qubit manifold, we implement "half derivative" an experimental simplification of derivative reduction by adiabatic gate (DRAG) control theory. The phase errors are lowered by about a factor of five using this method to $\sim 1.6^{\circ}$ per gate, and can be tuned to zero. Leakage outside the qubit manifold, to the qubit $|2\rangle$ state, is also reduced to $\sim 10^{-4}$ for $20\%$ faster gates.

Formation of robust bound states of interacting microwave photons

Alexis Morvan, Trond I. Andersen, Xiao Mi, Charles Neill, Andre Petukhov, Kostyantyn Kechedzhi, Dmitry Abanin, Rajeev Acharya, Frank Arute, Kunal Arya, Abraham Asfaw, Juan Atalaya, Ryan Babbush, Dave Bacon, Joseph C. Bardin, Joao Basso, Andreas Bengtsson, Gina Bortoli, Alexandre Bourassa, Jenna Bovaird, Leon Brill, Michael Broughton, Bob B. Buckley, David A. Buell, Tim Burger, Brian Burkett, Nicholas Bushnell, Zijun Chen, Ben Chiaro, Roberto Collins, Paul Conner, William Courtney, Alexander L. Crook, Ben Curtin, Dripto M. Debroy, Alexander Del Toro Barba, Sean Demura, Andrew Dunsworth, Daniel Eppens, Catherine Erickson, Lara Faoro, Edward Farhi, Reza Fatemi, Leslie Flores Burgos, Ebrahim Forati, Austin G. Fowler, Brooks Foxen, William Giang, Craig Gidney, Dar Gilboa, Marissa Giustina, Alejandro Grajales Dau, Jonathan A. Gross, Steve Habegger, Michael C. Hamilton, Matthew P. Harrigan, Sean D. Harrington, Jeremy Hilton, Markus Hoffmann, Sabrina Hong, Trent Huang, Ashley Huff, William J. Huggins, Sergei V. Isakov, Justin Iveland, Evan Jeffrey, Zhang Jiang, Cody Jones, Pavol Juhas, Dvir Kafri, Tanuj Khattar, Mostafa Khezri, Marika Kieferova, Seon Kim, Alexei Kitaev, Paul V. Klimov, Andrey R. Klots, Alexander N. Korotkov, Fedor Kostritsa, John Mark Kreikebaum, David Landhuis, Pavel Laptev, Kim-Ming Lau, Lily Laws, Joonho Lee, Kenny Lee, Brian J. Lester, Alexander Lill, Wayne Liu, Aditya Locharla, Erik Lucero, Fionn D. Malone, Orion Martin, Jarrod R. McClean, Matt McEwen, Bernardo Meurer Costa, Kevin C. Miao, Masoud Mohseni, Shirin Montazeri, Emily Mount, Wojciech Mruczkiewicz, Ofer Naaman, Matthew Neeley, Ani Nersisyan, Michael Newman, Anthony Nguyen, Murray Nguyen, Murphy Yuezhen Niu, Thomas E. O'Brien, Ricardo Olenewa, Alex Opremcak, Rebecca Potter, Chris Quintana, Nicholas C. Rubin, Negar Saei, Daniel Sank, Kannan Sankaragomathi, Kevin J. Satzinger, Henry F. Schurkus, Christopher Schuster, Michael J. Shearn, Aaron Shorter, Vladimir Shvarts, Jindra Skruzny, W. Clarke Smith, George Sterling, Doug Strain, Yuan Su, Marco Szalay, Alfredo Torres, Guifre Vidal, Benjamin Villalonga, Catherine Vollgraff Heidweiller, Theodore White, Cheng Xing, Z. Jamie Yao, Ping Yeh, Juhwan Yoo, Adam Zalcman, Yaxing Zhang, Ningfeng Zhu, Hartmut Neven, Sergio Boixo, Anthony Megrant, Julian Kelly, Yu Chen, Vadim Smelyanskiy, Igor Aleiner, Lev B. Ioffe, Pedram Roushan

Controlled catch and release of microwave photon states

Yi Yin, Yu Chen, Daniel Sank, P. J. J. O'Malley, T. C. White, R. Barends, J. Kelly, Erik Lucero, Matteo Mariantoni, A. Megrant, C. Neill, A. Vainsencher, J. Wenner, Alexander N. Korotkov, A. N. Cleland, John M. Martinis

The quantum behavior of superconducting qubits coupled to resonators is very similar to that of atoms in optical cavities [1, 2], in which the resonant cavity confines photons and promotes strong light-matter interactions. The cavity end-mirrors determine the performance of the coupled system, with higher mirror reflectivity yielding better quantum coherence, but higher mirror transparency giving improved measurement and control, forcing a compromise. An alternative is to control the mirror transparency, enabling switching between long photon lifetime during quantum interactions and large signal strength when performing measurements. Here we demonstrate the superconducting analogue, using a quantum system comprising a resonator and a qubit, with variable coupling to a measurement transmission line. The coupling can be adjusted through zero to a photon emission rate 1,000 times the intrinsic photon decay rate. We use this system to control photons in coherent states as well as in non-classical Fock states, and dynamically shape the waveform of released photons. This has direct applications to circuit quantum electrodynamics [3], and may enable high-fidelity quantum state transfer between distant qubits, for which precisely-controlled waveform shaping is a critical and non-trivial requirement [4, 5].

Measurement-Induced State Transitions in a Superconducting Qubit: Within the Rotating Wave Approximation

Mostafa Khezri, Alex Opremcak, Zijun Chen, Kevin C. Miao, Matt McEwen, Andreas Bengtsson, Theodore White, Ofer Naaman, Daniel Sank, Alexander N. Korotkov, Yu Chen, Vadim Smelyanskiy

Superconducting qubits typically use a dispersive readout scheme, where a resonator is coupled to a qubit such that its frequency is qubit-state dependent. Measurement is performed by driving the resonator, where the transmitted resonator field yields information about the resonator frequency and thus the qubit state. Ideally, we could use arbitrarily strong resonator drives to achieve a target signal-to-noise ratio in the shortest possible time. However, experiments have shown that when the average resonator photon number exceeds a certain threshold, the qubit is excited out of its computational subspace in a process we refer to as a measurement-induced state transition (MIST). These transitions degrade readout fidelity, and constitute leakage which precludes further operation of the qubit in, for example, error correction. Here we study these transitions experimentally with a transmon qubit by measuring their dependence on qubit frequency, average resonator photon number, and qubit state, in the regime where the resonator frequency is lower than the qubit frequency. We observe signatures of resonant transitions between levels in the coupled qubit-resonator system that exhibit noisy behavior when measured repeatedly in time. We provide a semi-classical model of these transitions based on the rotating wave approximation and use it to predict the onset of state transitions in our experiments. Our results suggest the transmon is excited to levels near the top of its cosine potential following a state transition, where the charge dispersion of higher transmon levels explains the observed noisy behavior of state transitions. Moreover, we show that occupation in these higher energy levels poses a major challenge for fast qubit reset.

Compressed sensing quantum process tomography for superconducting quantum gates

Andrey V. Rodionov, Andrzej Veitia, R. Barends, J. Kelly, Daniel Sank, J. Wenner, John M. Martinis, Robert L. Kosut, Alexander N. Korotkov

We apply the method of compressed sensing (CS) quantum process tomography (QPT) to characterize quantum gates based on superconducting Xmon and phase qubits. Using experimental data for a two-qubit controlled-Z gate, we obtain an estimate for the process matrix $χ$ with reasonably high fidelity compared to full QPT, but using a significantly reduced set of initial states and measurement configurations. We show that the CS method still works when the amount of used data is so small that the standard QPT would have an underdetermined system of equations. We also apply the CS method to the analysis of the three-qubit Toffoli gate with numerically added noise, and similarly show that the method works well for a substantially reduced set of data. For the CS calculations we use two different bases in which the process matrix $χ$ is approximately sparse, and show that the resulting estimates of the process matrices match each ther with reasonably high fidelity. For both two-qubit and three-qubit gates, we characterize the quantum process by not only its process matrix and fidelity, but also by the corresponding standard deviation, defined via variation of the state fidelity for different initial states.

Hartree-Fock on a superconducting qubit quantum computer

Frank Arute, Kunal Arya, Ryan Babbush, Dave Bacon, Joseph C. Bardin, Rami Barends, Sergio Boixo, Michael Broughton, Bob B. Buckley, David A. Buell, Brian Burkett, Nicholas Bushnell, Yu Chen, Zijun Chen, Benjamin Chiaro, Roberto Collins, William Courtney, Sean Demura, Andrew Dunsworth, Daniel Eppens, Edward Farhi, Austin Fowler, Brooks Foxen, Craig Gidney, Marissa Giustina, Rob Graff, Steve Habegger, Matthew P. Harrigan, Alan Ho, Sabrina Hong, Trent Huang, William J. Huggins, Lev Ioffe, Sergei V. Isakov, Evan Jeffrey, Zhang Jiang, Cody Jones, Dvir Kafri, Kostyantyn Kechedzhi, Julian Kelly, Seon Kim, Paul V. Klimov, Alexander Korotkov, Fedor Kostritsa, David Landhuis, Pavel Laptev, Mike Lindmark, Erik Lucero, Orion Martin, John M. Martinis, Jarrod R. McClean, Matt McEwen, Anthony Megrant, Xiao Mi, Masoud Mohseni, Wojciech Mruczkiewicz, Josh Mutus, Ofer Naaman, Matthew Neeley, Charles Neill, Hartmut Neven, Murphy Yuezhen Niu, Thomas E. O'Brien, Eric Ostby, Andre Petukhov, Harald Putterman, Chris Quintana, Pedram Roushan, Nicholas C. Rubin, Daniel Sank, Kevin J. Satzinger, Vadim Smelyanskiy, Doug Strain, Kevin J. Sung, Marco Szalay, Tyler Y. Takeshita, Amit Vainsencher, Theodore White, Nathan Wiebe, Z. Jamie Yao, Ping Yeh, Adam Zalcman

As the search continues for useful applications of noisy intermediate scale quantum devices, variational simulations of fermionic systems remain one of the most promising directions. Here, we perform a series of quantum simulations of chemistry the largest of which involved a dozen qubits, 78 two-qubit gates, and 114 one-qubit gates. We model the binding energy of ${\rm H}_6$, ${\rm H}_8$, ${\rm H}_{10}$ and ${\rm H}_{12}$ chains as well as the isomerization of diazene. We also demonstrate error-mitigation strategies based on $N$-representability which dramatically improve the effective fidelity of our experiments. Our parameterized ansatz circuits realize the Givens rotation approach to non-interacting fermion evolution, which we variationally optimize to prepare the Hartree-Fock wavefunction. This ubiquitous algorithmic primitive corresponds to a rotation of the orbital basis and is required by many proposals for correlated simulations of molecules and Hubbard models. Because non-interacting fermion evolutions are classically tractable to simulate, yet still generate highly entangled states over the computational basis, we use these experiments to benchmark the performance of our hardware while establishing a foundation for scaling up more complex correlated quantum simulations of chemistry.

Quantum Approximate Optimization of Non-Planar Graph Problems on a Planar Superconducting Processor

Matthew P. Harrigan, Kevin J. Sung, Matthew Neeley, Kevin J. Satzinger, Frank Arute, Kunal Arya, Juan Atalaya, Joseph C. Bardin, Rami Barends, Sergio Boixo, Michael Broughton, Bob B. Buckley, David A. Buell, Brian Burkett, Nicholas Bushnell, Yu Chen, Zijun Chen, Ben Chiaro, Roberto Collins, William Courtney, Sean Demura, Andrew Dunsworth, Daniel Eppens, Austin Fowler, Brooks Foxen, Craig Gidney, Marissa Giustina, Rob Graff, Steve Habegger, Alan Ho, Sabrina Hong, Trent Huang, L. B. Ioffe, Sergei V. Isakov, Evan Jeffrey, Zhang Jiang, Cody Jones, Dvir Kafri, Kostyantyn Kechedzhi, Julian Kelly, Seon Kim, Paul V. Klimov, Alexander N. Korotkov, Fedor Kostritsa, David Landhuis, Pavel Laptev, Mike Lindmark, Martin Leib, Orion Martin, John M. Martinis, Jarrod R. McClean, Matt McEwen, Anthony Megrant, Xiao Mi, Masoud Mohseni, Wojciech Mruczkiewicz, Josh Mutus, Ofer Naaman, Charles Neill, Florian Neukart, Murphy Yuezhen Niu, Thomas E. O'Brien, Bryan O'Gorman, Eric Ostby, Andre Petukhov, Harald Putterman, Chris Quintana, Pedram Roushan, Nicholas C. Rubin, Daniel Sank, Andrea Skolik, Vadim Smelyanskiy, Doug Strain, Michael Streif, Marco Szalay, Amit Vainsencher, Theodore White, Z. Jamie Yao, Ping Yeh, Adam Zalcman, Leo Zhou, Hartmut Neven, Dave Bacon, Erik Lucero, Edward Farhi, Ryan Babbush

We demonstrate the application of the Google Sycamore superconducting qubit quantum processor to combinatorial optimization problems with the quantum approximate optimization algorithm (QAOA). Like past QAOA experiments, we study performance for problems defined on the (planar) connectivity graph of our hardware; however, we also apply the QAOA to the Sherrington-Kirkpatrick model and MaxCut, both high dimensional graph problems for which the QAOA requires significant compilation. Experimental scans of the QAOA energy landscape show good agreement with theory across even the largest instances studied (23 qubits) and we are able to perform variational optimization successfully. For problems defined on our hardware graph we obtain an approximation ratio that is independent of problem size and observe, for the first time, that performance increases with circuit depth. For problems requiring compilation, performance decreases with problem size but still provides an advantage over random guessing for circuits involving several thousand gates. This behavior highlights the challenge of using near-term quantum computers to optimize problems on graphs differing from hardware connectivity. As these graphs are more representative of real world instances, our results advocate for more emphasis on such problems in the developing tradition of using the QAOA as a holistic, device-level benchmark of quantum processors.

Dynamics of magnetization at infinite temperature in a Heisenberg spin chain

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

Non-Abelian braiding of graph vertices in a superconducting processor

Trond I. Andersen, Yuri D. Lensky, Kostyantyn Kechedzhi, Ilya Drozdov, Andreas Bengtsson, Sabrina Hong, Alexis Morvan, Xiao Mi, Alex Opremcak, Rajeev Acharya, Richard Allen, Markus Ansmann, Frank Arute, Kunal Arya, Abraham Asfaw, Juan Atalaya, Ryan Babbush, Dave Bacon, Joseph C. Bardin, Gina Bortoli, Alexandre Bourassa, Jenna Bovaird, Leon Brill, Michael Broughton, Bob B. Buckley, David A. Buell, Tim Burger, Brian Burkett, Nicholas Bushnell, Zijun Chen, Ben Chiaro, Desmond Chik, Charina Chou, Josh Cogan, Roberto Collins, Paul Conner, William Courtney, Alexander L. Crook, Ben Curtin, Dripto M. Debroy, Alexander Del Toro Barba, Sean Demura, Andrew Dunsworth, Daniel Eppens, Catherine Erickson, Lara Faoro, Edward Farhi, Reza Fatemi, Vinicius S. Ferreira, Leslie Flores Burgos, Ebrahim Forati, Austin G. Fowler, Brooks Foxen, William Giang, Craig Gidney, Dar Gilboa, Marissa Giustina, Raja Gosula, Alejandro Grajales Dau, Jonathan A. Gross, Steve Habegger, Michael C. Hamilton, Monica Hansen, Matthew P. Harrigan, Sean D. Harrington, Paula Heu, Jeremy Hilton, Markus R. Hoffmann, Trent Huang, Ashley Huff, William J. Huggins, Lev B. Ioffe, Sergei V. Isakov, Justin Iveland, Evan Jeffrey, Zhang Jiang, Cody Jones, Pavol Juhas, Dvir Kafri, Tanuj Khattar, Mostafa Khezri, Mária Kieferová, Seon Kim, Alexei Kitaev, Paul V. Klimov, Andrey R. Klots, Alexander N. Korotkov, Fedor Kostritsa, John Mark Kreikebaum, David Landhuis, Pavel Laptev, Kim-Ming Lau, Lily Laws, Joonho Lee, Kenny Lee, Brian J. Lester, Alexander Lill, Wayne Liu, Aditya Locharla, Erik Lucero, Fionn D. Malone, Orion Martin, Jarrod R. McClean, Trevor McCourt, Matt McEwen, Kevin C. Miao, Amanda Mieszala, Masoud Mohseni, Shirin Montazeri, Emily Mount, Ramis Movassagh, Wojciech Mruczkiewicz, Ofer Naaman, Matthew Neeley, Charles Neill, Ani Nersisyan, Michael Newman, Jiun How Ng, Anthony Nguyen, Murray Nguyen, Murphy Yuezhen Niu, Thomas E. O'Brien, Seun Omonije, Andre Petukhov, Rebecca Potter, Leonid P. Pryadko, Chris Quintana, Charles Rocque, Nicholas C. Rubin, Negar Saei, Daniel Sank, Kannan Sankaragomathi, Kevin J. Satzinger, Henry F. Schurkus, Christopher Schuster, Michael J. Shearn, Aaron Shorter, Noah Shutty, Vladimir Shvarts, Jindra Skruzny, W. Clarke Smith, Rolando Somma, George Sterling, Doug Strain, Marco Szalay, Alfredo Torres, Guifre Vidal, Benjamin Villalonga, Catherine Vollgraff Heidweiller, Theodore White, Bryan W. K. Woo, Cheng Xing, Z. Jamie Yao, Ping Yeh, Juhwan Yoo, Grayson Young, Adam Zalcman, Yaxing Zhang, Ningfeng Zhu, Nicholas Zobrist, Hartmut Neven, Sergio Boixo, Anthony Megrant, Julian Kelly, Yu Chen, Vadim Smelyanskiy, Eun-Ah Kim, Igor Aleiner, Pedram Roushan

Constructive interference at the edge of quantum ergodic dynamics

Dmitry A. Abanin, Rajeev Acharya, Laleh Aghababaie-Beni, Georg Aigeldinger, Ashok Ajoy, Ross Alcaraz, Igor Aleiner, Trond I. Andersen, Markus Ansmann, Frank Arute, Kunal Arya, Abraham Asfaw, Nikita Astrakhantsev, Juan Atalaya, Ryan Babbush, Dave Bacon, Brian Ballard, Joseph C. Bardin, Christian Bengs, Andreas Bengtsson, Alexander Bilmes, Sergio Boixo, Gina Bortoli, Alexandre Bourassa, Jenna Bovaird, Dylan Bowers, Leon Brill, Michael Broughton, David A. Browne, Brett Buchea, Bob B. Buckley, David A. Buell, Tim Burger, Brian Burkett, Nicholas Bushnell, Anthony Cabrera, Juan Campero, Hung-Shen Chang, Yu Chen, Zijun Chen, Ben Chiaro, Liang-Ying Chih, Desmond Chik, Charina Chou, Jahan Claes, Agnetta Y. Cleland, Josh Cogan, Saul Cohen, Roberto Collins, Paul Conner, William Courtney, Alexander L. Crook, Ben Curtin, Sayan Das, Laura De Lorenzo, Dripto M. Debroy, Sean Demura, Michel Devoret, Agustin Di Paolo, Paul Donohoe, Ilya Drozdov, Andrew Dunsworth, Clint Earle, Alec Eickbusch, Aviv Moshe Elbag, Mahmoud Elzouka, Catherine Erickson, Lara Faoro, Edward Farhi, Vinicius S. Ferreira, Leslie Flores Burgos, Ebrahim Forati, Austin G. Fowler, Brooks Foxen, Suhas Ganjam, Gonzalo Garcia, Robert Gasca, Elie Genois, William Giang, Craig Gidney, Dar Gilboa, Raja Gosula, Alejandro Grajales Dau, Dietrich Graumann, Alex Greene, Jonathan A. Gross, Hanfeng Gu, Steve Habegger, John Hall, Ikko Hamamura, Michael C. Hamilton, Monica Hansen, Matthew P. Harrigan, Sean D. Harrington, Stephen Heslin, Paula Heu, Oscar Higgott, Gordon Hill, Jeremy Hilton, Sabrina Hong, Hsin-Yuan Huang, Ashley Huff, William J. Huggins, Lev B. Ioffe, Sergei V. Isakov, Justin Iveland, Evan Jeffrey, Zhang Jiang, Xiaoxuan Jin, Cody Jones, Stephen Jordan, Chaitali Joshi, Pavol Juhas, Andreas Kabel, Dvir Kafri, Hui Kang, Amir H. Karamlou, Kostyantyn Kechedzhi, Julian Kelly, Trupti Khaire, Tanuj Khattar, Mostafa Khezri, Seon Kim, Robbie King, Paul V. Klimov, Andrey R. Klots, Bryce Kobrin, Alexander N. Korotkov, Fedor Kostritsa, Robin Kothari, John Mark Kreikebaum, Vladislav D. Kurilovich, Elica Kyoseva, David Landhuis, Tiano Lange-Dei, Brandon W. Langley, Pavel Laptev, Kim-Ming Lau, Loick Le Guevel, Justin Ledford, Joonho Lee, Kenny Lee, Yuri D. Lensky, Shannon Leon, Brian J. Lester, Wing Yan Li, Alexander T. Lill, Wayne Liu, William P. Livingston, Aditya Locharla, Erik Lucero, Daniel Lundahl, Aaron Lunt, Sid Madhuk, Fionn D. Malone, Ashley Maloney, Salvatore Mandra, James M. Manyika, Leigh S. Martin, Orion Martin, Steven Martin, Yossi Matias, Cameron Maxfield, Jarrod R. McClean, Matt McEwen, Seneca Meeks, Anthony Megrant, Xiao Mi, Kevin C. Miao, Amanda Mieszala, Reza Molavi, Sebastian Molina, Shirin Montazeri, Alexis Morvan, Ramis Movassagh, Wojciech Mruczkiewicz, Ofer Naaman, Matthew Neeley, Charles Neill, Ani Nersisyan, Hartmut Neven, Michael Newman, Jiun How Ng, Anthony Nguyen, Murray Nguyen, Chia-Hung Ni, Murphy Yuezhen Niu, Logan Oas, Thomas E. O'Brien, William D. Oliver, Alex Opremcak, Kristoffer Ottosson, Andre Petukhov, Alex Pizzuto, John Platt, Rebecca Potter, Orion Pritchard, Leonid P. Pryadko, Chris Quintana, Ganesh Ramachandran, Chandrasekhar Ramanathan, Matthew J. Reagor, John Redding, David M. Rhodes, Gabrielle Roberts, Eliott Rosenberg, Emma Rosenfeld, Pedram Roushan, Nicholas C. Rubin, Negar Saei, Daniel Sank, Kannan Sankaragomathi, Kevin J. Satzinger, Alexander Schmidhuber, Henry F. Schurkus, Christopher Schuster, Thomas Schuster, Michael J. Shearn, Aaron Shorter, Noah Shutty, Vladimir Shvarts, Volodymyr Sivak, Jindra Skruzny, Spencer Small, Vadim Smelyanskiy, W. Clarke Smith, Rolando D. Somma, Sofia Springer, George Sterling, Doug Strain, Jordan Suchard, Philippe Suchsland, Aaron Szasz, Alex Sztein, Douglas Thor, Eifu Tomita, Alfredo Torres, M. Mert Torunbalci, Abeer Vaishnav, Justin Vargas, Sergey Vdovichev, Guifre Vidal, Benjamin Villalonga, Catherine Vollgraff Heidweiller, Steven Waltman, Shannon X. Wang, Brayden Ware, Kate Weber, Travis Weidel, Tom Westerhout, Theodore White, Kristi Wong, Bryan W. K. Woo, Cheng Xing, Z. Jamie Yao, Ping Yeh, Bicheng Ying, Juhwan Yoo, Noureldin Yosri, Grayson Young, Adam Zalcman, Chongwei Zhang, Yaxing Zhang, Ningfeng Zhu, Nicholas Zobrist

Resolving catastrophic error bursts from cosmic rays in large arrays of superconducting qubits

Matt McEwen, Lara Faoro, Kunal Arya, Andrew Dunsworth, Trent Huang, Seon Kim, Brian Burkett, Austin Fowler, Frank Arute, Joseph C. Bardin, Andreas Bengtsson, Alexander Bilmes, Bob B. Buckley, Nicholas Bushnell, Zijun Chen, Roberto Collins, Sean Demura, Alan R. Derk, Catherine Erickson, Marissa Giustina, Sean D. Harrington, Sabrina Hong, Evan Jeffrey, Julian Kelly, Paul V. Klimov, Fedor Kostritsa, Pavel Laptev, Aditya Locharla, Xiao Mi, Kevin C. Miao, Shirin Montazeri, Josh Mutus, Ofer Naaman, Matthew Neeley, Charles Neill, Alex Opremcak, Chris Quintana, Nicholas Redd, Pedram Roushan, Daniel Sank, Kevin J. Satzinger, Vladimir Shvarts, Theodore White, Z. Jamie Yao, Ping Yeh, Juhwan Yoo, Yu Chen, Vadim Smelyanskiy, John M. Martinis, Hartmut Neven, Anthony Megrant, Lev Ioffe, Rami Barends

Scalable quantum computing can become a reality with error correction, provided coherent qubits can be constructed in large arrays. The key premise is that physical errors can remain both small and sufficiently uncorrelated as devices scale, so that logical error rates can be exponentially suppressed. However, energetic impacts from cosmic rays and latent radioactivity violate both of these assumptions. An impinging particle ionizes the substrate, radiating high energy phonons that induce a burst of quasiparticles, destroying qubit coherence throughout the device. High-energy radiation has been identified as a source of error in pilot superconducting quantum devices, but lacking a measurement technique able to resolve a single event in detail, the effect on large scale algorithms and error correction in particular remains an open question. Elucidating the physics involved requires operating large numbers of qubits at the same rapid timescales as in error correction, exposing the event's evolution in time and spread in space. Here, we directly observe high-energy rays impacting a large-scale quantum processor. We introduce a rapid space and time-multiplexed measurement method and identify large bursts of quasiparticles that simultaneously and severely limit the energy coherence of all qubits, causing chip-wide failure. We track the events from their initial localised impact to high error rates across the chip. Our results provide direct insights into the scale and dynamics of these damaging error bursts in large-scale devices, and highlight the necessity of mitigation to enable quantum computing to scale.

Exponential suppression of bit or phase flip errors with repetitive error correction

Zijun Chen, Kevin J. Satzinger, Juan Atalaya, Alexander N. Korotkov, Andrew Dunsworth, Daniel Sank, Chris Quintana, Matt McEwen, Rami Barends, Paul V. Klimov, Sabrina Hong, Cody Jones, Andre Petukhov, Dvir Kafri, Sean Demura, Brian Burkett, Craig Gidney, Austin G. Fowler, Harald Putterman, Igor Aleiner, Frank Arute, Kunal Arya, Ryan Babbush, Joseph C. Bardin, Andreas Bengtsson, Alexandre Bourassa, Michael Broughton, Bob B. Buckley, David A. Buell, Nicholas Bushnell, Benjamin Chiaro, Roberto Collins, William Courtney, Alan R. Derk, Daniel Eppens, Catherine Erickson, Edward Farhi, Brooks Foxen, Marissa Giustina, Jonathan A. Gross, Matthew P. Harrigan, Sean D. Harrington, Jeremy Hilton, Alan Ho, Trent Huang, William J. Huggins, L. B. Ioffe, Sergei V. Isakov, Evan Jeffrey, Zhang Jiang, Kostyantyn Kechedzhi, Seon Kim, Fedor Kostritsa, David Landhuis, Pavel Laptev, Erik Lucero, Orion Martin, Jarrod R. McClean, Trevor McCourt, Xiao Mi, Kevin C. Miao, Masoud Mohseni, Wojciech Mruczkiewicz, Josh Mutus, Ofer Naaman, Matthew Neeley, Charles Neill, Michael Newman, Murphy Yuezhen Niu, Thomas E. O'Brien, Alex Opremcak, Eric Ostby, Bálint Pató, Nicholas Redd, Pedram Roushan, Nicholas C. Rubin, Vladimir Shvarts, Doug Strain, Marco Szalay, Matthew D. Trevithick, Benjamin Villalonga, Theodore White, Z. Jamie Yao, Ping Yeh, Adam Zalcman, Hartmut Neven, Sergio Boixo, Vadim Smelyanskiy, Yu Chen, Anthony Megrant, Julian Kelly

Realizing the potential of quantum computing will require achieving sufficiently low logical error rates. Many applications call for error rates in the $10^{-15}$ regime, but state-of-the-art quantum platforms typically have physical error rates near $10^{-3}$. Quantum error correction (QEC) promises to bridge this divide by distributing quantum logical information across many physical qubits so that errors can be detected and corrected. Logical errors are then exponentially suppressed as the number of physical qubits grows, provided that the physical error rates are below a certain threshold. QEC also requires that the errors are local and that performance is maintained over many rounds of error correction, two major outstanding experimental challenges. Here, we implement 1D repetition codes embedded in a 2D grid of superconducting qubits which demonstrate exponential suppression of bit or phase-flip errors, reducing logical error per round by more than $100\times$ when increasing the number of qubits from 5 to 21. Crucially, this error suppression is stable over 50 rounds of error correction. We also introduce a method for analyzing error correlations with high precision, and characterize the locality of errors in a device performing QEC for the first time. Finally, we perform error detection using a small 2D surface code logical qubit on the same device, and show that the results from both 1D and 2D codes agree with numerical simulations using a simple depolarizing error model. These findings demonstrate that superconducting qubits are on a viable path towards fault tolerant quantum computing.

A 28nm Bulk-CMOS 4-to-8GHz <2mW Cryogenic Pulse Modulator for Scalable Quantum Computing

Joseph C Bardin, Evan Jeffrey, Erik Lucero, Trent Huang, Ofer Naaman, Rami Barends, Ted White, Marissa Giustina, Daniel Sank, Pedram Roushan, Kunal Arya, Benjamin Chiaro, Julian Kelly, Jimmy Chen, Brian Burkett, Yu Chen, Andrew Dunsworth, Austin Fowler, Brooks Foxen, Craig Gidney, Rob Graff, Paul Klimov, Josh Mutus, Matthew McEwen, Anthony Megrant, Matthew Neeley, Charles Neill, Chris Quintana, Amit Vainsencher, Hartmut Neven, John Martinis

Future quantum computing systems will require cryogenic integrated circuits to control and measure millions of qubits. In this paper, we report the design and characterization of a prototype cryogenic CMOS integrated circuit that has been optimized for the control of transmon qubits. The circuit has been integrated into a quantum measurement setup and its performance has been validated through multiple quantum control experiments.

Fluctuations of Energy-Relaxation Times in Superconducting Qubits

P. V. Klimov, J. Kelly, Z. Chen, M. Neeley, A. Megrant, B. Burkett, R. Barends, K. Arya, B. Chiaro, Yu Chen, A. Dunsworth, A. Fowler, B. Foxen, C. Gidney, M. Giustina, R. Graff, T. Huang, E. Jeffrey, Erik Lucero, J. Y. Mutus, O. Naaman, C. Neill, C. Quintana, P. Roushan, Daniel Sank, A. Vainsencher, J. Wenner, T. C. White, S. Boixo, R. Babbush, V. N. Smelyanskiy, H. Neven, John M. Martinis

Superconducting qubits are an attractive platform for quantum computing since they have demonstrated high-fidelity quantum gates and extensibility to modest system sizes. Nonetheless, an outstanding challenge is stabilizing their energy-relaxation times, which can fluctuate unpredictably in frequency and time. Here, we use qubits as spectral and temporal probes of individual two-level-system defects to provide direct evidence that they are responsible for the largest fluctuations. This research lays the foundation for stabilizing qubit performance through calibration, design, and fabrication.

Strong environmental coupling in a Josephson parametric amplifier

Josh Mutus, Ted White, Rami Barends, Yu Chen, Zijun Chen, Ben Chiaro, Andrew Dunsworth, Evan Jeffrey, Julian Kelly, Anthony Megrant, Charles Neill, Peter O'Malley, Pedram Roushan, Daniel Sank, Amit Vainsencher, James Wenner, Kyle Sundqvist, Andrew Cleland, John Martinis

We present a lumped-element Josephson parametric amplifier designed to operate with strong coupling to the environment. In this regime, we observe broadband frequency dependent amplification with multi-peaked gain profiles. We account for this behaviour using the "pumpistor" model which allows for frequency dependent variation of the external impedance. Using this understanding, we demonstrate control over gain profiles through changes in the environment impedance at a given frequency. With strong coupling to a suitable external impedance we observe a significant increase in dynamic range, and large amplification bandwidth up to 700 MHz giving near quantum-limited performance.

Sputtered TiN films for superconducting coplanar waveguide resonators

Shinobu Ohya, Ben Chiaro, Anthony Megrant, Charles Neill, Rami Barends, Yu Chen, Julian Kelly, David Low, Josh Mutus, Peter O'Malley, Pedram Roushan, Daniel Sank, Amit Vainsencher, James Wenner, Theodore C. White, Yi Yin, B. D. Schultz, Chris J Palmstrøm, Benjamin A. Mazin, Andrew N. Cleland, John M. Martinis

We present a systematic study of the properties of TiN films by varying the deposition conditions in an ultra-high-vacuum reactive magnetron sputtering chamber. By increasing the deposition pressure from 2 to 9 mTorr while keeping a nearly stoichiometric composition of Ti(1-x)N(x) (x=0.5), the film resistivity increases, the dominant crystal orientation changes from (100) to (111), grain boundaries become clearer, and the strong compressive strain changes to weak tensile strain. The TiN films absorb a high concentration of contaminants including hydrogen, carbon, and oxygen when they are exposed to air after deposition. With the target-substrate distance set to 88 mm the contaminant levels increase from ~0.1% to ~10% as the pressure is increased from 2 to 9 mTorr. The contaminant concentrations also correlate with in-plane distance from the center of the substrate and increase by roughly two orders of magnitude as the target-substrate distance is increased from 88 mm to 266 mm. These contaminants are found to strongly influence the properties of TiN films. For instance, the resistivity of stoichiometric films increases by around a factor of 5 as the oxygen content increases from 0.1% to 11%. These results suggest that the sputtered TiN particle energy is critical in determining the TiN film properties, and that it is important to control this energy to obtain high-quality TiN films. Superconducting coplanar waveguide resonators made from a series of nearly stoichiometric films grown at pressures from 2 mTorr to 7 mTorr show an increase in intrinsic quality factor from ~10^4 to ~10^6 as the magnitude of the compressive strain decreases from nearly 3800 MPa to approximately 150 MPa and the oxygen content increases from 0.1% to 8%. The films with a higher oxygen content exhibit lower loss, but the nonuniformity of the oxygen incorporation hinders the use of sputtered TiN in larger circuits.

Supplementary information for "Quantum supremacy using a programmable superconducting processor"

Frank Arute, Kunal Arya, Ryan Babbush, Dave Bacon, Joseph C. Bardin, Rami Barends, Rupak Biswas, Sergio Boixo, Fernando G. S. L. Brandao, David A. Buell, Brian Burkett, Yu Chen, Zijun Chen, Ben Chiaro, Roberto Collins, William Courtney, Andrew Dunsworth, Edward Farhi, Brooks Foxen, Austin Fowler, Craig Gidney, Marissa Giustina, Rob Graff, Keith Guerin, Steve Habegger, Matthew P. Harrigan, Michael J. Hartmann, Alan Ho, Markus R. Hoffmann, Trent Huang, Travis S. Humble, Sergei V. Isakov, Evan Jeffrey, Zhang Jiang, Dvir Kafri, Kostyantyn Kechedzhi, Julian Kelly, Paul V. Klimov, Sergey Knysh, Alexander N. Korotkov, Fedor Kostritsa, David Landhuis, Mike Lindmark, Erik Lucero, Dmitry Lyakh, Salvatore Mandra, Jarrod R. McClean, Matt McEwen, Anthony Megrant, Xiao Mi, Kristel Michielsen, Masoud Mohseni, Josh Mutus, Ofer Naaman, Matthew Neeley, Charles Neill, Murphy Yuezhen Niu, Eric Ostby, Andre Petukhov, John C. Platt, Chris Quintana, Eleanor G. Rieffel, Pedram Roushan, Nicholas C. Rubin, Daniel Sank, Kevin J. Satzinger, Vadim Smelyanskiy, Kevin J. Sung, Matthew D. Trevithick, Amit Vainsencher, Benjamin Villalonga, Theodore White, Z. Jamie Yao, Ping Yeh, Adam Zalcman, Hartmut Neven, John M. Martinis

This is an updated version of supplementary information to accompany "Quantum supremacy using a programmable superconducting processor", an article published in the October 24, 2019 issue of Nature. The main article is freely available at https://www.nature.com/articles/s41586-019-1666-5. Summary of changes since arXiv:1910.11333v1 (submitted 23 Oct 2019): added URL for qFlex source code; added Erratum section; added Figure S41 comparing statistical and total uncertainty for log and linear XEB; new References [1,65]; miscellaneous updates for clarity and style consistency; miscellaneous typographical and formatting corrections.