1
|
Teixeira W, Mörstedt T, Viitanen A, Kivijärvi H, Gunyhó A, Tiiri M, Kundu S, Sah A, Vadimov V, Möttönen M. Many-excitation removal of a transmon qubit using a single-junction quantum-circuit refrigerator and a two-tone microwave drive. Sci Rep 2024; 14:13755. [PMID: 38877065 PMCID: PMC11178887 DOI: 10.1038/s41598-024-64496-5] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/01/2024] [Accepted: 06/09/2024] [Indexed: 06/16/2024] Open
Abstract
Achieving fast and precise initialization of qubits is a critical requirement for the successful operation of quantum computers. The combination of engineered environments with all-microwave techniques has recently emerged as a promising approach for the reset of superconducting quantum devices. In this work, we experimentally demonstrate the utilization of a single-junction quantum-circuit refrigerator (QCR) for an expeditious removal of several excitations from a transmon qubit. The QCR is indirectly coupled to the transmon through a resonator in the dispersive regime, constituting a carefully engineered environmental spectrum for the transmon. Using single-shot readout, we observe excitation stabilization times down to roughly 500 ns, a 20-fold speedup with QCR and a simultaneous two-tone drive addressing the e-f and f0-g1 transitions of the system. Our results are obtained at a 48-mK fridge temperature and without postselection, fully capturing the advantage of the protocol for the short-time dynamics and the drive-induced detrimental asymptotic behavior in the presence of relatively hot other baths of the transmon. We validate our results with a detailed Liouvillian model truncated up to the three-excitation subspace, from which we estimate the performance of the protocol in optimized scenarios, such as cold transmon baths and fine-tuned driving frequencies. These results pave the way for optimized reset of quantum-electric devices using engineered environments and for dissipation-engineered state preparation.
Collapse
Affiliation(s)
- Wallace Teixeira
- QCD Labs, Department of Applied Physics, QTF Centre of Excellence, Aalto University, P.O. Box 13500, FI-00076, Aalto, Finland.
| | - Timm Mörstedt
- QCD Labs, Department of Applied Physics, QTF Centre of Excellence, Aalto University, P.O. Box 13500, FI-00076, Aalto, Finland
| | - Arto Viitanen
- QCD Labs, Department of Applied Physics, QTF Centre of Excellence, Aalto University, P.O. Box 13500, FI-00076, Aalto, Finland
| | - Heidi Kivijärvi
- QCD Labs, Department of Applied Physics, QTF Centre of Excellence, Aalto University, P.O. Box 13500, FI-00076, Aalto, Finland
| | - András Gunyhó
- QCD Labs, Department of Applied Physics, QTF Centre of Excellence, Aalto University, P.O. Box 13500, FI-00076, Aalto, Finland
| | - Maaria Tiiri
- QCD Labs, Department of Applied Physics, QTF Centre of Excellence, Aalto University, P.O. Box 13500, FI-00076, Aalto, Finland
| | - Suman Kundu
- QCD Labs, Department of Applied Physics, QTF Centre of Excellence, Aalto University, P.O. Box 13500, FI-00076, Aalto, Finland
| | - Aashish Sah
- QCD Labs, Department of Applied Physics, QTF Centre of Excellence, Aalto University, P.O. Box 13500, FI-00076, Aalto, Finland
| | - Vasilii Vadimov
- QCD Labs, Department of Applied Physics, QTF Centre of Excellence, Aalto University, P.O. Box 13500, FI-00076, Aalto, Finland
| | - Mikko Möttönen
- QCD Labs, Department of Applied Physics, QTF Centre of Excellence, Aalto University, P.O. Box 13500, FI-00076, Aalto, Finland
- QTF Center of Excellence, VTT Technical Research Centre of Finland Ltd., P.O. Box 1000, FI-02044, VTT, Finland
| |
Collapse
|
2
|
Venkatraman J, Cortiñas RG, Frattini NE, Xiao X, Devoret MH. A driven Kerr oscillator with two-fold degeneracies for qubit protection. Proc Natl Acad Sci U S A 2024; 121:e2311241121. [PMID: 38838020 PMCID: PMC11181142 DOI: 10.1073/pnas.2311241121] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/03/2023] [Accepted: 03/01/2024] [Indexed: 06/07/2024] Open
Abstract
We present the experimental finding of multiple simultaneous two-fold degeneracies in the spectrum of a Kerr oscillator subjected to a squeezing drive. This squeezing drive resulting from a three-wave mixing process, in combination with the Kerr interaction, creates an effective static two-well potential in the phase space rotating at half the frequency of the sinusoidal drive generating the squeezing. Remarkably, these degeneracies can be turned on-and-off on demand, as well as their number by simply adjusting the frequency of the squeezing drive. We find that when the detuning Δ between the frequency of the oscillator and the second subharmonic of the drive equals an even multiple of the Kerr coefficient K, [Formula: see text], the oscillator displays [Formula: see text] exact, parity-protected, spectral degeneracies, insensitive to the drive amplitude. These degeneracies can be explained by the unusual destructive interference of tunnel paths in the classically forbidden region of the double well static effective potential that models our experiment. Exploiting this interference, we measure a peaked enhancement of the incoherent well-switching lifetime, thus creating a protected cat qubit in the ground state manifold of our oscillator. Our results illustrate the relationship between degeneracies and noise protection in a driven quantum system.
Collapse
Affiliation(s)
- Jayameenakshi Venkatraman
- Department of Applied Physics, Yale University, New Haven, CT06520
- Department of Physics, Yale University, New Haven, CT06520
| | - Rodrigo G. Cortiñas
- Department of Applied Physics, Yale University, New Haven, CT06520
- Department of Physics, Yale University, New Haven, CT06520
| | - Nicholas E. Frattini
- Department of Applied Physics, Yale University, New Haven, CT06520
- Department of Physics, Yale University, New Haven, CT06520
| | - Xu Xiao
- Department of Applied Physics, Yale University, New Haven, CT06520
- Department of Physics, Yale University, New Haven, CT06520
| | - Michel H. Devoret
- Department of Applied Physics, Yale University, New Haven, CT06520
- Department of Physics, Yale University, New Haven, CT06520
| |
Collapse
|
3
|
Connolly T, Kurilovich PD, Diamond S, Nho H, Bøttcher CGL, Glazman LI, Fatemi V, Devoret MH. Coexistence of Nonequilibrium Density and Equilibrium Energy Distribution of Quasiparticles in a Superconducting Qubit. PHYSICAL REVIEW LETTERS 2024; 132:217001. [PMID: 38856268 DOI: 10.1103/physrevlett.132.217001] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/21/2023] [Revised: 08/10/2023] [Accepted: 03/21/2024] [Indexed: 06/11/2024]
Abstract
The density of quasiparticles typically observed in superconducting qubits exceeds the value expected in equilibrium by many orders of magnitude. Can this out-of-equilibrium quasiparticle density still possess an energy distribution in equilibrium with the phonon bath? Here, we answer this question affirmatively by measuring the thermal activation of charge-parity switching in a transmon qubit with a difference in superconducting gap on the two sides of the Josephson junction. We then demonstrate how the gap asymmetry of the device can be exploited to manipulate its parity.
Collapse
Affiliation(s)
- Thomas Connolly
- Departments of Applied Physics and Physics, Yale University, New Haven, Connecticut 06520, USA
| | - Pavel D Kurilovich
- Departments of Applied Physics and Physics, Yale University, New Haven, Connecticut 06520, USA
| | - Spencer Diamond
- Departments of Applied Physics and Physics, Yale University, New Haven, Connecticut 06520, USA
| | - Heekun Nho
- Departments of Applied Physics and Physics, Yale University, New Haven, Connecticut 06520, USA
| | - Charlotte G L Bøttcher
- Departments of Applied Physics and Physics, Yale University, New Haven, Connecticut 06520, USA
| | - Leonid I Glazman
- Departments of Applied Physics and Physics, Yale University, New Haven, Connecticut 06520, USA
| | - Valla Fatemi
- Departments of Applied Physics and Physics, Yale University, New Haven, Connecticut 06520, USA
- School of Applied and Engineering Physics, Cornell University, Ithaca, New York 14853, USA
| | - Michel H Devoret
- Departments of Applied Physics and Physics, Yale University, New Haven, Connecticut 06520, USA
| |
Collapse
|
4
|
Kono S, Pan J, Chegnizadeh M, Wang X, Youssefi A, Scigliuzzo M, Kippenberg TJ. Mechanically induced correlated errors on superconducting qubits with relaxation times exceeding 0.4 ms. Nat Commun 2024; 15:3950. [PMID: 38729959 PMCID: PMC11087564 DOI: 10.1038/s41467-024-48230-3] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/06/2023] [Accepted: 04/24/2024] [Indexed: 05/12/2024] Open
Abstract
Superconducting qubits are among the most advanced candidates for achieving fault-tolerant quantum computing. Despite recent significant advancements in the qubit lifetimes, the origin of the loss mechanism for state-of-the-art qubits is still subject to investigation. Furthermore, the successful implementation of quantum error correction requires negligible correlated errors between qubits. Here, we realize long-lived superconducting transmon qubits that exhibit fluctuating lifetimes, averaging 0.2 ms and exceeding 0.4 ms - corresponding to quality factors above 5 million and 10 million, respectively. We then investigate their dominant error mechanism. By introducing novel time-resolved error measurements that are synchronized with the operation of the pulse tube cooler in a dilution refrigerator, we find that mechanical vibrations from the pulse tube induce nonequilibrium dynamics in highly coherent qubits, leading to their correlated bit-flip errors. Our findings not only deepen our understanding of the qubit error mechanisms but also provide valuable insights into potential error-mitigation strategies for achieving fault tolerance by decoupling superconducting qubits from their mechanical environments.
Collapse
Affiliation(s)
- Shingo Kono
- Institute of Physics, Swiss Federal Institute of Technology Lausanne (EPFL), Lausanne, Switzerland.
- Center for Quantum Science and Engineering, EPFL, Lausanne, Switzerland.
| | - Jiahe Pan
- Institute of Physics, Swiss Federal Institute of Technology Lausanne (EPFL), Lausanne, Switzerland
- Center for Quantum Science and Engineering, EPFL, Lausanne, Switzerland
| | - Mahdi Chegnizadeh
- Institute of Physics, Swiss Federal Institute of Technology Lausanne (EPFL), Lausanne, Switzerland
- Center for Quantum Science and Engineering, EPFL, Lausanne, Switzerland
| | - Xuxin Wang
- Institute of Physics, Swiss Federal Institute of Technology Lausanne (EPFL), Lausanne, Switzerland
- Center for Quantum Science and Engineering, EPFL, Lausanne, Switzerland
| | - Amir Youssefi
- Institute of Physics, Swiss Federal Institute of Technology Lausanne (EPFL), Lausanne, Switzerland
- Center for Quantum Science and Engineering, EPFL, Lausanne, Switzerland
| | - Marco Scigliuzzo
- Institute of Physics, Swiss Federal Institute of Technology Lausanne (EPFL), Lausanne, Switzerland
- Center for Quantum Science and Engineering, EPFL, Lausanne, Switzerland
| | - Tobias J Kippenberg
- Institute of Physics, Swiss Federal Institute of Technology Lausanne (EPFL), Lausanne, Switzerland.
- Center for Quantum Science and Engineering, EPFL, Lausanne, Switzerland.
| |
Collapse
|
5
|
Thorbeck T, Xiao Z, Kamal A, Govia LCG. Readout-Induced Suppression and Enhancement of Superconducting Qubit Lifetimes. PHYSICAL REVIEW LETTERS 2024; 132:090602. [PMID: 38489646 DOI: 10.1103/physrevlett.132.090602] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/12/2023] [Accepted: 01/23/2024] [Indexed: 03/17/2024]
Abstract
It has long been known that the lifetimes of superconducting qubits suffer during readout, increasing readout errors. We show that this degradation is due to the anti-Zeno effect, as readout-induced dephasing broadens the qubit so that it overlaps "hot spots" of strong dissipation, likely due to two-level systems in the qubit's bath. Using a flux-tunable qubit to probe the qubit's frequency-dependent loss, we accurately predict the change in lifetime during readout with a new self-consistent master equation that incorporates the modification to qubit relaxation due to measurement-induced dephasing. Moreover, we controllably demonstrate both the Zeno and anti-Zeno effects, which can explain both suppression and the rarer enhancement of qubit lifetimes during readout.
Collapse
Affiliation(s)
- Ted Thorbeck
- IBM Quantum, IBM T. J. Watson Research Center, Yorktown Heights, New York 10598, USA
| | - Zhihao Xiao
- Department of Physics and Applied Physics, University of Massachusetts, Lowell, Massachusetts 01854, USA
| | - Archana Kamal
- Department of Physics and Applied Physics, University of Massachusetts, Lowell, Massachusetts 01854, USA
| | - Luke C G Govia
- IBM Quantum, IBM Almaden Research Center, San Jose, California 95120, USA
| |
Collapse
|
6
|
Willsch D, Rieger D, Winkel P, Willsch M, Dickel C, Krause J, Ando Y, Lescanne R, Leghtas Z, Bronn NT, Deb P, Lanes O, Minev ZK, Dennig B, Geisert S, Günzler S, Ihssen S, Paluch P, Reisinger T, Hanna R, Bae JH, Schüffelgen P, Grützmacher D, Buimaga-Iarinca L, Morari C, Wernsdorfer W, DiVincenzo DP, Michielsen K, Catelani G, Pop IM. Observation of Josephson harmonics in tunnel junctions. NATURE PHYSICS 2024; 20:815-821. [PMID: 38799981 PMCID: PMC11116114 DOI: 10.1038/s41567-024-02400-8] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 10/25/2023] [Accepted: 01/14/2024] [Indexed: 05/29/2024]
Abstract
Approaches to developing large-scale superconducting quantum processors must cope with the numerous microscopic degrees of freedom that are ubiquitous in solid-state devices. State-of-the-art superconducting qubits employ aluminium oxide (AlOx) tunnel Josephson junctions as the sources of nonlinearity necessary to perform quantum operations. Analyses of these junctions typically assume an idealized, purely sinusoidal current-phase relation. However, this relation is expected to hold only in the limit of vanishingly low-transparency channels in the AlOx barrier. Here we show that the standard current-phase relation fails to accurately describe the energy spectra of transmon artificial atoms across various samples and laboratories. Instead, a mesoscopic model of tunnelling through an inhomogeneous AlOx barrier predicts percent-level contributions from higher Josephson harmonics. By including these in the transmon Hamiltonian, we obtain orders of magnitude better agreement between the computed and measured energy spectra. The presence and impact of Josephson harmonics has important implications for developing AlOx-based quantum technologies including quantum computers and parametric amplifiers. As an example, we show that engineered Josephson harmonics can reduce the charge dispersion and associated errors in transmon qubits by an order of magnitude while preserving their anharmonicity.
Collapse
Affiliation(s)
- Dennis Willsch
- Jülich Supercomputing Centre, Forschungszentrum Jülich, Jülich, Germany
| | - Dennis Rieger
- IQMT, Karlsruhe Institute of Technology, Eggenstein-Leopoldshafen, Germany
- PHI, Karlsruhe Institute of Technology, Karlsruhe, Germany
| | - Patrick Winkel
- IQMT, Karlsruhe Institute of Technology, Eggenstein-Leopoldshafen, Germany
- PHI, Karlsruhe Institute of Technology, Karlsruhe, Germany
- Departments of Applied Physics and Physics, Yale University, New Haven, CT USA
- Yale Quantum Institute, Yale University, New Haven, CT USA
| | - Madita Willsch
- Jülich Supercomputing Centre, Forschungszentrum Jülich, Jülich, Germany
- AIDAS, Jülich, Germany
| | | | - Jonas Krause
- Physics Institute II, University of Cologne, Köln, Germany
| | - Yoichi Ando
- Physics Institute II, University of Cologne, Köln, Germany
| | - Raphaël Lescanne
- LPENS, Mines Paris-PSL, ENS-PSL, Inria, Université PSL, CNRS, Paris, France
- Alice & Bob, Paris, France
| | - Zaki Leghtas
- LPENS, Mines Paris-PSL, ENS-PSL, Inria, Université PSL, CNRS, Paris, France
| | - Nicholas T. Bronn
- IBM Quantum, IBM T. J. Watson Research Center, Yorktown Heights, NY USA
| | - Pratiti Deb
- IBM Quantum, IBM T. J. Watson Research Center, Yorktown Heights, NY USA
| | - Olivia Lanes
- IBM Quantum, IBM T. J. Watson Research Center, Yorktown Heights, NY USA
| | - Zlatko K. Minev
- IBM Quantum, IBM T. J. Watson Research Center, Yorktown Heights, NY USA
| | - Benedikt Dennig
- IQMT, Karlsruhe Institute of Technology, Eggenstein-Leopoldshafen, Germany
- PHI, Karlsruhe Institute of Technology, Karlsruhe, Germany
| | - Simon Geisert
- IQMT, Karlsruhe Institute of Technology, Eggenstein-Leopoldshafen, Germany
| | - Simon Günzler
- IQMT, Karlsruhe Institute of Technology, Eggenstein-Leopoldshafen, Germany
| | - Sören Ihssen
- IQMT, Karlsruhe Institute of Technology, Eggenstein-Leopoldshafen, Germany
| | - Patrick Paluch
- IQMT, Karlsruhe Institute of Technology, Eggenstein-Leopoldshafen, Germany
- PHI, Karlsruhe Institute of Technology, Karlsruhe, Germany
| | - Thomas Reisinger
- IQMT, Karlsruhe Institute of Technology, Eggenstein-Leopoldshafen, Germany
| | - Roudy Hanna
- PGI-9, Forschungszentrum Jülich and JARA Jülich-Aachen Research Alliance, Jülich, Germany
- RWTH Aachen University, Aachen, Germany
| | - Jin Hee Bae
- PGI-9, Forschungszentrum Jülich and JARA Jülich-Aachen Research Alliance, Jülich, Germany
| | - Peter Schüffelgen
- PGI-9, Forschungszentrum Jülich and JARA Jülich-Aachen Research Alliance, Jülich, Germany
| | - Detlev Grützmacher
- PGI-9, Forschungszentrum Jülich and JARA Jülich-Aachen Research Alliance, Jülich, Germany
- RWTH Aachen University, Aachen, Germany
| | | | | | - Wolfgang Wernsdorfer
- IQMT, Karlsruhe Institute of Technology, Eggenstein-Leopoldshafen, Germany
- PHI, Karlsruhe Institute of Technology, Karlsruhe, Germany
| | - David P. DiVincenzo
- RWTH Aachen University, Aachen, Germany
- PGI-2, Forschungszentrum Jülich, Jülich, Germany
| | - Kristel Michielsen
- Jülich Supercomputing Centre, Forschungszentrum Jülich, Jülich, Germany
- AIDAS, Jülich, Germany
- RWTH Aachen University, Aachen, Germany
| | - Gianluigi Catelani
- PGI-11, Forschungszentrum Jülich, Jülich, Germany
- Quantum Research Center, Technology Innovation Institute, Abu Dhabi, UAE
| | - Ioan M. Pop
- IQMT, Karlsruhe Institute of Technology, Eggenstein-Leopoldshafen, Germany
- PHI, Karlsruhe Institute of Technology, Karlsruhe, Germany
| |
Collapse
|
7
|
Lu Y, Maiti A, Garmon JWO, Ganjam S, Zhang Y, Claes J, Frunzio L, Girvin SM, Schoelkopf RJ. High-fidelity parametric beamsplitting with a parity-protected converter. Nat Commun 2023; 14:5767. [PMID: 37723141 PMCID: PMC10507116 DOI: 10.1038/s41467-023-41104-0] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/07/2023] [Accepted: 08/23/2023] [Indexed: 09/20/2023] Open
Abstract
Fast, high-fidelity operations between microwave resonators are an important tool for bosonic quantum computation and simulation with superconducting circuits. An attractive approach for implementing these operations is to couple these resonators via a nonlinear converter and actuate parametric processes with RF drives. It can be challenging to make these processes simultaneously fast and high fidelity, since this requires introducing strong drives without activating parasitic processes or introducing additional decoherence channels. We show that in addition to a careful management of drive frequencies and the spectrum of environmental noise, leveraging the inbuilt symmetries of the converter Hamiltonian can suppress unwanted nonlinear interactions, preventing converter-induced decoherence. We demonstrate these principles using a differentially-driven DC-SQUID as our converter, coupled to two high-Q microwave cavities. Using this architecture, we engineer a highly-coherent beamsplitter and fast (~100 ns) swaps between the cavities, limited primarily by their intrinsic single-photon loss. We characterize this beamsplitter in the cavities' joint single-photon subspace, and show that we can detect and post-select photon loss events to achieve a beamsplitter gate fidelity exceeding 99.98%, which to our knowledge far surpasses the current state of the art.
Collapse
Affiliation(s)
- Yao Lu
- Departments of Applied Physics and Physics, Yale University, New Haven, 06511, CT, USA.
- Yale Quantum Institute, Yale University, New Haven, 06511, CT, USA.
| | - Aniket Maiti
- Departments of Applied Physics and Physics, Yale University, New Haven, 06511, CT, USA.
- Yale Quantum Institute, Yale University, New Haven, 06511, CT, USA.
| | - John W O Garmon
- Departments of Applied Physics and Physics, Yale University, New Haven, 06511, CT, USA
- Yale Quantum Institute, Yale University, New Haven, 06511, CT, USA
| | - Suhas Ganjam
- Departments of Applied Physics and Physics, Yale University, New Haven, 06511, CT, USA
- Yale Quantum Institute, Yale University, New Haven, 06511, CT, USA
| | - Yaxing Zhang
- Departments of Applied Physics and Physics, Yale University, New Haven, 06511, CT, USA
- Yale Quantum Institute, Yale University, New Haven, 06511, CT, USA
| | - Jahan Claes
- Departments of Applied Physics and Physics, Yale University, New Haven, 06511, CT, USA
- Yale Quantum Institute, Yale University, New Haven, 06511, CT, USA
| | - Luigi Frunzio
- Departments of Applied Physics and Physics, Yale University, New Haven, 06511, CT, USA
- Yale Quantum Institute, Yale University, New Haven, 06511, CT, USA
| | - Steven M Girvin
- Departments of Applied Physics and Physics, Yale University, New Haven, 06511, CT, USA
- Yale Quantum Institute, Yale University, New Haven, 06511, CT, USA
| | - Robert J Schoelkopf
- Departments of Applied Physics and Physics, Yale University, New Haven, 06511, CT, USA.
- Yale Quantum Institute, Yale University, New Haven, 06511, CT, USA.
| |
Collapse
|
8
|
Hassani F, Peruzzo M, Kapoor LN, Trioni A, Zemlicka M, Fink JM. Inductively shunted transmons exhibit noise insensitive plasmon states and a fluxon decay exceeding 3 hours. Nat Commun 2023; 14:3968. [PMID: 37407570 DOI: 10.1038/s41467-023-39656-2] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/19/2022] [Accepted: 06/22/2023] [Indexed: 07/07/2023] Open
Abstract
Currently available quantum processors are dominated by noise, which severely limits their applicability and motivates the search for new physical qubit encodings. In this work, we introduce the inductively shunted transmon, a weakly flux-tunable superconducting qubit that offers charge offset protection for all levels and a 20-fold reduction in flux dispersion compared to the state-of-the-art resulting in a constant coherence over a full flux quantum. The parabolic confinement provided by the inductive shunt as well as the linearity of the geometric superinductor facilitates a high-power readout that resolves quantum jumps with a fidelity and QND-ness of >90% and without the need for a Josephson parametric amplifier. Moreover, the device reveals quantum tunneling physics between the two prepared fluxon ground states with a measured average decay time of up to 3.5 h. In the future, fast time-domain control of the transition matrix elements could offer a new path forward to also achieve full qubit control in the decay-protected fluxon basis.
Collapse
Affiliation(s)
- F Hassani
- Institute of Science and Technology Austria, 3400, Klosterneuburg, Austria.
| | - M Peruzzo
- Institute of Science and Technology Austria, 3400, Klosterneuburg, Austria
| | - L N Kapoor
- Institute of Science and Technology Austria, 3400, Klosterneuburg, Austria
| | - A Trioni
- Institute of Science and Technology Austria, 3400, Klosterneuburg, Austria
| | - M Zemlicka
- Institute of Science and Technology Austria, 3400, Klosterneuburg, Austria
| | - J M Fink
- Institute of Science and Technology Austria, 3400, Klosterneuburg, Austria.
| |
Collapse
|
9
|
Marques JF, Ali H, Varbanov BM, Finkel M, Veen HM, van der Meer SLM, Valles-Sanclemente S, Muthusubramanian N, Beekman M, Haider N, Terhal BM, DiCarlo L. All-Microwave Leakage Reduction Units for Quantum Error Correction with Superconducting Transmon Qubits. PHYSICAL REVIEW LETTERS 2023; 130:250602. [PMID: 37418741 DOI: 10.1103/physrevlett.130.250602] [Citation(s) in RCA: 2] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/22/2023] [Accepted: 05/24/2023] [Indexed: 07/09/2023]
Abstract
Minimizing leakage from computational states is a challenge when using many-level systems like superconducting quantum circuits as qubits. We realize and extend the quantum-hardware-efficient, all-microwave leakage reduction unit (LRU) for transmons in a circuit QED architecture proposed by Battistel et al. This LRU effectively reduces leakage in the second- and third-excited transmon states with up to 99% efficacy in 220 ns, with minimum impact on the qubit subspace. As a first application in the context of quantum error correction, we show how multiple simultaneous LRUs can reduce the error detection rate and suppress leakage buildup within 1% in data and ancilla qubits over 50 cycles of a weight-2 stabilizer measurement.
Collapse
Affiliation(s)
- J F Marques
- QuTech, Delft University of Technology, P.O. Box 5046, 2600 GA Delft, Netherlands
- Kavli Institute of Nanoscience, Delft University of Technology, P.O. Box 5046, 2600 GA Delft, Netherlands
| | - H Ali
- QuTech, Delft University of Technology, P.O. Box 5046, 2600 GA Delft, Netherlands
- Kavli Institute of Nanoscience, Delft University of Technology, P.O. Box 5046, 2600 GA Delft, Netherlands
| | - B M Varbanov
- QuTech, Delft University of Technology, P.O. Box 5046, 2600 GA Delft, Netherlands
| | - M Finkel
- QuTech, Delft University of Technology, P.O. Box 5046, 2600 GA Delft, Netherlands
- Kavli Institute of Nanoscience, Delft University of Technology, P.O. Box 5046, 2600 GA Delft, Netherlands
| | - H M Veen
- QuTech, Delft University of Technology, P.O. Box 5046, 2600 GA Delft, Netherlands
- Kavli Institute of Nanoscience, Delft University of Technology, P.O. Box 5046, 2600 GA Delft, Netherlands
| | - S L M van der Meer
- QuTech, Delft University of Technology, P.O. Box 5046, 2600 GA Delft, Netherlands
- Kavli Institute of Nanoscience, Delft University of Technology, P.O. Box 5046, 2600 GA Delft, Netherlands
| | - S Valles-Sanclemente
- QuTech, Delft University of Technology, P.O. Box 5046, 2600 GA Delft, Netherlands
- Kavli Institute of Nanoscience, Delft University of Technology, P.O. Box 5046, 2600 GA Delft, Netherlands
| | - N Muthusubramanian
- QuTech, Delft University of Technology, P.O. Box 5046, 2600 GA Delft, Netherlands
- Kavli Institute of Nanoscience, Delft University of Technology, P.O. Box 5046, 2600 GA Delft, Netherlands
| | - M Beekman
- QuTech, Delft University of Technology, P.O. Box 5046, 2600 GA Delft, Netherlands
- Netherlands Organisation for Applied Scientific Research (TNO), P.O. Box 96864, 2509 JG The Hague, Netherlands
| | - N Haider
- QuTech, Delft University of Technology, P.O. Box 5046, 2600 GA Delft, Netherlands
- Netherlands Organisation for Applied Scientific Research (TNO), P.O. Box 96864, 2509 JG The Hague, Netherlands
| | - B M Terhal
- QuTech, Delft University of Technology, P.O. Box 5046, 2600 GA Delft, Netherlands
- EEMCS Department, Delft University of Technology, P.O. Box 5046, 2600 GA Delft, Netherlands
| | - L DiCarlo
- QuTech, Delft University of Technology, P.O. Box 5046, 2600 GA Delft, Netherlands
- Kavli Institute of Nanoscience, Delft University of Technology, P.O. Box 5046, 2600 GA Delft, Netherlands
| |
Collapse
|
10
|
Sundaresan N, Yoder TJ, Kim Y, Li M, Chen EH, Harper G, Thorbeck T, Cross AW, Córcoles AD, Takita M. Demonstrating multi-round subsystem quantum error correction using matching and maximum likelihood decoders. Nat Commun 2023; 14:2852. [PMID: 37202409 DOI: 10.1038/s41467-023-38247-5] [Citation(s) in RCA: 4] [Impact Index Per Article: 4.0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/15/2022] [Accepted: 04/19/2023] [Indexed: 05/20/2023] Open
Abstract
Quantum error correction offers a promising path for performing high fidelity quantum computations. Although fully fault-tolerant executions of algorithms remain unrealized, recent improvements in control electronics and quantum hardware enable increasingly advanced demonstrations of the necessary operations for error correction. Here, we perform quantum error correction on superconducting qubits connected in a heavy-hexagon lattice. We encode a logical qubit with distance three and perform several rounds of fault-tolerant syndrome measurements that allow for the correction of any single fault in the circuitry. Using real-time feedback, we reset syndrome and flag qubits conditionally after each syndrome extraction cycle. We report decoder dependent logical error, with average logical error per syndrome measurement in Z(X)-basis of ~0.040 (~0.088) and ~0.037 (~0.087) for matching and maximum likelihood decoders, respectively, on leakage post-selected data.
Collapse
Affiliation(s)
- Neereja Sundaresan
- IBM Quantum, IBM T.J. Watson Research Center, Yorktown Heights, NY, 10598, USA.
| | - Theodore J Yoder
- IBM Quantum, IBM T.J. Watson Research Center, Yorktown Heights, NY, 10598, USA.
| | - Youngseok Kim
- IBM Quantum, IBM T.J. Watson Research Center, Yorktown Heights, NY, 10598, USA
| | - Muyuan Li
- IBM Quantum, IBM T.J. Watson Research Center, Yorktown Heights, NY, 10598, USA
| | - Edward H Chen
- IBM Quantum, IBM Almaden Research Center, San Jose, CA, 95120, USA
| | - Grace Harper
- IBM Quantum, IBM T.J. Watson Research Center, Yorktown Heights, NY, 10598, USA
| | - Ted Thorbeck
- IBM Quantum, IBM T.J. Watson Research Center, Yorktown Heights, NY, 10598, USA
| | - Andrew W Cross
- IBM Quantum, IBM T.J. Watson Research Center, Yorktown Heights, NY, 10598, USA
| | - Antonio D Córcoles
- IBM Quantum, IBM T.J. Watson Research Center, Yorktown Heights, NY, 10598, USA
| | - Maika Takita
- IBM Quantum, IBM T.J. Watson Research Center, Yorktown Heights, NY, 10598, USA
| |
Collapse
|
11
|
Zhang X, Kim E, Mark DK, Choi S, Painter O. A superconducting quantum simulator based on a photonic-bandgap metamaterial. Science 2023; 379:278-283. [PMID: 36656924 DOI: 10.1126/science.ade7651] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/20/2023]
Abstract
Synthesizing many-body quantum systems with various ranges of interactions facilitates the study of quantum chaotic dynamics. Such extended interaction range can be enabled by using nonlocal degrees of freedom such as photonic modes in an otherwise locally connected structure. Here, we present a superconducting quantum simulator in which qubits are connected through an extensible photonic-bandgap metamaterial, thus realizing a one-dimensional Bose-Hubbard model with tunable hopping range and on-site interaction. Using individual site control and readout, we characterize the statistics of measurement outcomes from many-body quench dynamics, which enables in situ Hamiltonian learning. Further, the outcome statistics reveal the effect of increased hopping range, showing the predicted crossover from integrability to ergodicity. Our work enables the study of emergent randomness from chaotic many-body evolution and, more broadly, expands the accessible Hamiltonians for quantum simulation using superconducting circuits.
Collapse
Affiliation(s)
- Xueyue Zhang
- Thomas J. Watson, Sr., Laboratory of Applied Physics and Kavli Nanoscience Institute, California Institute of Technology, Pasadena, CA 91125, USA.,Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, CA 91125, USA
| | - Eunjong Kim
- Thomas J. Watson, Sr., Laboratory of Applied Physics and Kavli Nanoscience Institute, California Institute of Technology, Pasadena, CA 91125, USA.,Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, CA 91125, USA
| | - Daniel K Mark
- Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
| | - Soonwon Choi
- Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
| | - Oskar Painter
- Thomas J. Watson, Sr., Laboratory of Applied Physics and Kavli Nanoscience Institute, California Institute of Technology, Pasadena, CA 91125, USA.,Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, CA 91125, USA.,AWS Center for Quantum Computing, Pasadena, CA 91125, USA
| |
Collapse
|
12
|
Pereira L, García-Ripoll JJ, Ramos T. Complete Physical Characterization of Quantum Nondemolition Measurements via Tomography. PHYSICAL REVIEW LETTERS 2022; 129:010402. [PMID: 35841584 DOI: 10.1103/physrevlett.129.010402] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/29/2021] [Accepted: 06/03/2022] [Indexed: 06/15/2023]
Abstract
We introduce a self-consistent tomography for arbitrary quantum nondemolition (QND) detectors. Based on this, we build a complete physical characterization of the detector, including the measurement processes and a quantification of the fidelity, ideality, and backaction of the measurement. This framework is a diagnostic tool for the dynamics of QND detectors, allowing us to identify errors, and to improve their calibration and design. We illustrate this on a realistic Jaynes-Cummings simulation of a superconducting qubit readout. We characterize nondispersive errors, quantify the backaction introduced by the readout cavity, and calibrate the optimal measurement point.
Collapse
Affiliation(s)
- L Pereira
- Instituto de Física Fundamental IFF-CSIC, Calle Serrano 113b, Madrid 28006, Spain
| | - J J García-Ripoll
- Instituto de Física Fundamental IFF-CSIC, Calle Serrano 113b, Madrid 28006, Spain
| | - T Ramos
- Instituto de Física Fundamental IFF-CSIC, Calle Serrano 113b, Madrid 28006, Spain
| |
Collapse
|
13
|
Krinner S, Lacroix N, Remm A, Di Paolo A, Genois E, Leroux C, Hellings C, Lazar S, Swiadek F, Herrmann J, Norris GJ, Andersen CK, Müller M, Blais A, Eichler C, Wallraff A. Realizing repeated quantum error correction in a distance-three surface code. Nature 2022; 605:669-674. [PMID: 35614249 DOI: 10.1038/s41586-022-04566-8] [Citation(s) in RCA: 39] [Impact Index Per Article: 19.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/15/2021] [Accepted: 02/09/2022] [Indexed: 11/09/2022]
Abstract
Quantum computers hold the promise of solving computational problems that are intractable using conventional methods1. For fault-tolerant operation, quantum computers must correct errors occurring owing to unavoidable decoherence and limited control accuracy2. Here we demonstrate quantum error correction using the surface code, which is known for its exceptionally high tolerance to errors3-6. Using 17 physical qubits in a superconducting circuit, we encode quantum information in a distance-three logical qubit, building on recent distance-two error-detection experiments7-9. In an error-correction cycle taking only 1.1 μs, we demonstrate the preservation of four cardinal states of the logical qubit. Repeatedly executing the cycle, we measure and decode both bit-flip and phase-flip error syndromes using a minimum-weight perfect-matching algorithm in an error-model-free approach and apply corrections in post-processing. We find a low logical error probability of 3% per cycle when rejecting experimental runs in which leakage is detected. The measured characteristics of our device agree well with a numerical model. Our demonstration of repeated, fast and high-performance quantum error-correction cycles, together with recent advances in ion traps10, support our understanding that fault-tolerant quantum computation will be practically realizable.
Collapse
Affiliation(s)
| | | | - Ants Remm
- Department of Physics, ETH Zurich, Zurich, Switzerland
| | - Agustin Di Paolo
- Institut Quantique, Université de Sherbrooke, Sherbrooke, Québec, Canada.,Département de Physique, Université de Sherbrooke, Sherbrooke, Québec, Canada
| | - Elie Genois
- Institut Quantique, Université de Sherbrooke, Sherbrooke, Québec, Canada.,Département de Physique, Université de Sherbrooke, Sherbrooke, Québec, Canada
| | - Catherine Leroux
- Institut Quantique, Université de Sherbrooke, Sherbrooke, Québec, Canada.,Département de Physique, Université de Sherbrooke, Sherbrooke, Québec, Canada
| | | | | | | | | | | | - Christian Kraglund Andersen
- Department of Physics, ETH Zurich, Zurich, Switzerland.,QuTech and Kavli Institute for Nanoscience, Delft University of Technology, Delft, The Netherlands
| | - Markus Müller
- Institute for Quantum Information, RWTH Aachen University, Aachen, Germany.,Peter Grünberg Institute, Theoretical Nanoelectronics, Forschungszentrum Jülich, Jülich, Germany
| | - Alexandre Blais
- Institut Quantique, Université de Sherbrooke, Sherbrooke, Québec, Canada.,Département de Physique, Université de Sherbrooke, Sherbrooke, Québec, Canada.,Canadian Institute for Advanced Research, Toronto, Ontario, Canada
| | | | - Andreas Wallraff
- Department of Physics, ETH Zurich, Zurich, Switzerland.,Quantum Center, ETH Zurich, Zurich, Switzerland
| |
Collapse
|
14
|
Yin Z, Li C, Allcock J, Zheng Y, Gu X, Dai M, Zhang S, An S. Shortcuts to adiabaticity for open systems in circuit quantum electrodynamics. Nat Commun 2022; 13:188. [PMID: 35013301 PMCID: PMC8748912 DOI: 10.1038/s41467-021-27900-6] [Citation(s) in RCA: 2] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/23/2021] [Accepted: 12/16/2021] [Indexed: 11/09/2022] Open
Abstract
Shortcuts to adiabaticity are powerful quantum control methods, allowing quick evolution into target states of otherwise slow adiabatic dynamics. Such methods have widespread applications in quantum technologies, and various shortcuts to adiabaticity protocols have been demonstrated in closed systems. However, realizing shortcuts to adiabaticity for open quantum systems has presented a challenge due to the complex controls in existing proposals. Here, we present the experimental demonstration of shortcuts to adiabaticity for open quantum systems, using a superconducting circuit quantum electrodynamics system. By applying a counterdiabatic driving pulse, we reduce the adiabatic evolution time of a single lossy mode from 800 ns to 100 ns. In addition, we propose and implement an optimal control protocol to achieve fast and qubit-unconditional equilibrium of multiple lossy modes. Our results pave the way for precise time-domain control of open quantum systems and have potential applications in designing fast open-system protocols of physical and interdisciplinary interest, such as accelerating bioengineering and chemical reaction dynamics.
Collapse
Affiliation(s)
- Zelong Yin
- Tencent Quantum Laboratory, Tencent, 518057, Shenzhen, Guangdong, China
| | - Chunzhen Li
- Tencent Quantum Laboratory, Tencent, 518057, Shenzhen, Guangdong, China
| | - Jonathan Allcock
- Tencent Quantum Laboratory, Tencent, 518057, Shenzhen, Guangdong, China
| | - Yicong Zheng
- Tencent Quantum Laboratory, Tencent, 518057, Shenzhen, Guangdong, China
| | - Xiu Gu
- Tencent Quantum Laboratory, Tencent, 518057, Shenzhen, Guangdong, China
| | - Maochun Dai
- Tencent Quantum Laboratory, Tencent, 518057, Shenzhen, Guangdong, China
| | - Shengyu Zhang
- Tencent Quantum Laboratory, Tencent, 518057, Shenzhen, Guangdong, China
| | - Shuoming An
- Tencent Quantum Laboratory, Tencent, 518057, Shenzhen, Guangdong, China.
| |
Collapse
|
15
|
Zhou Y, Zhang Z, Yin Z, Huai S, Gu X, Xu X, Allcock J, Liu F, Xi G, Yu Q, Zhang H, Zhang M, Li H, Song X, Wang Z, Zheng D, An S, Zheng Y, Zhang S. Rapid and unconditional parametric reset protocol for tunable superconducting qubits. Nat Commun 2021; 12:5924. [PMID: 34635663 PMCID: PMC8505451 DOI: 10.1038/s41467-021-26205-y] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/04/2021] [Accepted: 09/13/2021] [Indexed: 11/09/2022] Open
Abstract
Qubit initialization is a critical task in quantum computation and communication. Extensive efforts have been made to achieve this with high speed, efficiency and scalability. However, previous approaches have either been measurement-based and required fast feedback, suffered from crosstalk or required sophisticated calibration. Here, we report a fast and high-fidelity reset scheme, avoiding the issues above without any additional chip architecture. By modulating the flux through a transmon qubit, we realize a swap between the qubit and its readout resonator that suppresses the excited state population to 0.08% ± 0.08% within 34 ns (284 ns if photon depletion of the resonator is required). Furthermore, our approach (i) can achieve effective second excited state depletion, (ii) has negligible effects on neighboring qubits, and (iii) offers a way to entangle the qubit with an itinerant single photon, useful in quantum communication applications.
Collapse
Affiliation(s)
- Yu Zhou
- Tencent Quantum Laboratory, Tencent, Shenzhen, Guangdong, 518057, China
| | - Zhenxing Zhang
- Tencent Quantum Laboratory, Tencent, Shenzhen, Guangdong, 518057, China
| | - Zelong Yin
- Tencent Quantum Laboratory, Tencent, Shenzhen, Guangdong, 518057, China
| | - Sainan Huai
- Tencent Quantum Laboratory, Tencent, Shenzhen, Guangdong, 518057, China
| | - Xiu Gu
- Tencent Quantum Laboratory, Tencent, Shenzhen, Guangdong, 518057, China
| | - Xiong Xu
- Tencent Quantum Laboratory, Tencent, Shenzhen, Guangdong, 518057, China
| | - Jonathan Allcock
- Tencent Quantum Laboratory, Tencent, Shenzhen, Guangdong, 518057, China
| | - Fuming Liu
- Tencent Quantum Laboratory, Tencent, Shenzhen, Guangdong, 518057, China
| | - Guanglei Xi
- Tencent Quantum Laboratory, Tencent, Shenzhen, Guangdong, 518057, China
| | - Qiaonian Yu
- Tencent Quantum Laboratory, Tencent, Shenzhen, Guangdong, 518057, China
| | - Hualiang Zhang
- Tencent Quantum Laboratory, Tencent, Shenzhen, Guangdong, 518057, China
| | - Mengyu Zhang
- Tencent Quantum Laboratory, Tencent, Shenzhen, Guangdong, 518057, China
| | - Hekang Li
- Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing, 100190, China
- School of Physical Sciences, University of Chinese Academy of Sciences, Beijing, 100049, China
| | - Xiaohui Song
- Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing, 100190, China
- School of Physical Sciences, University of Chinese Academy of Sciences, Beijing, 100049, China
| | - Zhan Wang
- Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing, 100190, China
- School of Physical Sciences, University of Chinese Academy of Sciences, Beijing, 100049, China
| | - Dongning Zheng
- Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing, 100190, China
- School of Physical Sciences, University of Chinese Academy of Sciences, Beijing, 100049, China
| | - Shuoming An
- Tencent Quantum Laboratory, Tencent, Shenzhen, Guangdong, 518057, China.
| | - Yarui Zheng
- Tencent Quantum Laboratory, Tencent, Shenzhen, Guangdong, 518057, China
| | - Shengyu Zhang
- Tencent Quantum Laboratory, Tencent, Shenzhen, Guangdong, 518057, China
| |
Collapse
|
16
|
McEwen M, Kafri D, Chen Z, Atalaya J, Satzinger KJ, Quintana C, Klimov PV, Sank D, Gidney C, Fowler AG, Arute F, Arya K, Buckley B, Burkett B, Bushnell N, Chiaro B, Collins R, Demura S, Dunsworth A, Erickson C, Foxen B, Giustina M, Huang T, Hong S, Jeffrey E, Kim S, Kechedzhi K, Kostritsa F, Laptev P, Megrant A, Mi X, Mutus J, Naaman O, Neeley M, Neill C, Niu M, Paler A, Redd N, Roushan P, White TC, Yao J, Yeh P, Zalcman A, Chen Y, Smelyanskiy VN, Martinis JM, Neven H, Kelly J, Korotkov AN, Petukhov AG, Barends R. Removing leakage-induced correlated errors in superconducting quantum error correction. Nat Commun 2021; 12:1761. [PMID: 33741936 PMCID: PMC7979694 DOI: 10.1038/s41467-021-21982-y] [Citation(s) in RCA: 17] [Impact Index Per Article: 5.7] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/29/2020] [Accepted: 02/23/2021] [Indexed: 11/30/2022] Open
Abstract
Quantum computing can become scalable through error correction, but logical error rates only decrease with system size when physical errors are sufficiently uncorrelated. During computation, unused high energy levels of the qubits can become excited, creating leakage states that are long-lived and mobile. Particularly for superconducting transmon qubits, this leakage opens a path to errors that are correlated in space and time. Here, we report a reset protocol that returns a qubit to the ground state from all relevant higher level states. We test its performance with the bit-flip stabilizer code, a simplified version of the surface code for quantum error correction. We investigate the accumulation and dynamics of leakage during error correction. Using this protocol, we find lower rates of logical errors and an improved scaling and stability of error suppression with increasing qubit number. This demonstration provides a key step on the path towards scalable quantum computing.
Collapse
Affiliation(s)
- M McEwen
- Department of Physics, University of California, Santa Barbara, CA, USA
- Google, Santa Barbara, CA, USA
| | | | - Z Chen
- Google, Santa Barbara, CA, USA
| | | | | | | | | | - D Sank
- Google, Santa Barbara, CA, USA
| | | | | | - F Arute
- Google, Santa Barbara, CA, USA
| | - K Arya
- Google, Santa Barbara, CA, USA
| | | | | | | | | | | | | | | | | | - B Foxen
- Google, Santa Barbara, CA, USA
| | | | - T Huang
- Google, Santa Barbara, CA, USA
| | - S Hong
- Google, Santa Barbara, CA, USA
| | | | - S Kim
- Google, Santa Barbara, CA, USA
| | | | | | | | | | - X Mi
- Google, Santa Barbara, CA, USA
| | - J Mutus
- Google, Santa Barbara, CA, USA
| | | | | | - C Neill
- Google, Santa Barbara, CA, USA
| | | | - A Paler
- Johannes Kepler University, Linz, Austria
- University of Texas at Dallas, Richardson, TX, USA
| | - N Redd
- Google, Santa Barbara, CA, USA
| | | | | | - J Yao
- Google, Santa Barbara, CA, USA
| | - P Yeh
- Google, Santa Barbara, CA, USA
| | | | - Yu Chen
- Google, Santa Barbara, CA, USA
| | | | - John M Martinis
- Department of Physics, University of California, Santa Barbara, CA, USA
| | - H Neven
- Google, Santa Barbara, CA, USA
| | - J Kelly
- Google, Santa Barbara, CA, USA
| | - A N Korotkov
- Google, Santa Barbara, CA, USA
- Department of Electrical and Computer Engineering, University of California, Riverside, CA, USA
| | | | | |
Collapse
|
17
|
Rosenthal EI, Schneider CMF, Malnou M, Zhao Z, Leditzky F, Chapman BJ, Wustmann W, Ma X, Palken DA, Zanner MF, Vale LR, Hilton GC, Gao J, Smith G, Kirchmair G, Lehnert KW. Efficient and Low-Backaction Quantum Measurement Using a Chip-Scale Detector. PHYSICAL REVIEW LETTERS 2021; 126:090503. [PMID: 33750151 DOI: 10.1103/physrevlett.126.090503] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/08/2020] [Accepted: 01/21/2021] [Indexed: 06/12/2023]
Abstract
Superconducting qubits are a leading platform for scalable quantum computing and quantum error correction. One feature of this platform is the ability to perform projective measurements orders of magnitude more quickly than qubit decoherence times. Such measurements are enabled by the use of quantum-limited parametric amplifiers in conjunction with ferrite circulators-magnetic devices which provide isolation from noise and decoherence due to amplifier backaction. Because these nonreciprocal elements have limited performance and are not easily integrated on chip, it has been a long-standing goal to replace them with a scalable alternative. Here, we demonstrate a solution to this problem by using a superconducting switch to control the coupling between a qubit and amplifier. Doing so, we measure a transmon qubit using a single, chip-scale device to provide both parametric amplification and isolation from the bulk of amplifier backaction. This measurement is also fast, high fidelity, and has 70% efficiency, comparable to the best that has been reported in any superconducting qubit measurement. As such, this work constitutes a high-quality platform for the scalable measurement of superconducting qubits.
Collapse
Affiliation(s)
- Eric I Rosenthal
- JILA, University of Colorado, Boulder, Colorado 80309, USA
- Department of Physics, University of Colorado, Boulder, Colorado 80309, USA
- National Institute of Standards and Technology, Boulder, Colorado 80305, USA
| | - Christian M F Schneider
- Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences, A-6020 Innsbruck, Austria
- Institute for Experimental Physics, University of Innsbruck, A-6020 Innsbruck, Austria
| | - Maxime Malnou
- Department of Physics, University of Colorado, Boulder, Colorado 80309, USA
- National Institute of Standards and Technology, Boulder, Colorado 80305, USA
| | - Ziyi Zhao
- JILA, University of Colorado, Boulder, Colorado 80309, USA
- Department of Physics, University of Colorado, Boulder, Colorado 80309, USA
- National Institute of Standards and Technology, Boulder, Colorado 80305, USA
| | - Felix Leditzky
- JILA, University of Colorado, Boulder, Colorado 80309, USA
- National Institute of Standards and Technology, Boulder, Colorado 80305, USA
- Department of Mathematics & Illinois Quantum Information Science and Technology Center, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA
- Center for Theory of Quantum Matter, University of Colorado, Boulder, Colorado 80309, USA
| | - Benjamin J Chapman
- Department of Applied Physics, Yale University, New Haven, Connecticut 06511, USA
| | - Waltraut Wustmann
- The Laboratory for Physical Sciences, College Park, Maryland 20740, USA
| | - Xizheng Ma
- JILA, University of Colorado, Boulder, Colorado 80309, USA
- Department of Physics, University of Colorado, Boulder, Colorado 80309, USA
- National Institute of Standards and Technology, Boulder, Colorado 80305, USA
| | - Daniel A Palken
- JILA, University of Colorado, Boulder, Colorado 80309, USA
- Department of Physics, University of Colorado, Boulder, Colorado 80309, USA
- National Institute of Standards and Technology, Boulder, Colorado 80305, USA
| | - Maximilian F Zanner
- Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences, A-6020 Innsbruck, Austria
- Institute for Experimental Physics, University of Innsbruck, A-6020 Innsbruck, Austria
| | - Leila R Vale
- National Institute of Standards and Technology, Boulder, Colorado 80305, USA
| | - Gene C Hilton
- National Institute of Standards and Technology, Boulder, Colorado 80305, USA
| | - Jiansong Gao
- Department of Physics, University of Colorado, Boulder, Colorado 80309, USA
- National Institute of Standards and Technology, Boulder, Colorado 80305, USA
| | - Graeme Smith
- JILA, University of Colorado, Boulder, Colorado 80309, USA
- Department of Physics, University of Colorado, Boulder, Colorado 80309, USA
- National Institute of Standards and Technology, Boulder, Colorado 80305, USA
- Center for Theory of Quantum Matter, University of Colorado, Boulder, Colorado 80309, USA
| | - Gerhard Kirchmair
- Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences, A-6020 Innsbruck, Austria
- Institute for Experimental Physics, University of Innsbruck, A-6020 Innsbruck, Austria
| | - K W Lehnert
- JILA, University of Colorado, Boulder, Colorado 80309, USA
- Department of Physics, University of Colorado, Boulder, Colorado 80309, USA
- National Institute of Standards and Technology, Boulder, Colorado 80305, USA
| |
Collapse
|
18
|
Wang C, Chen MC, Lu CY, Pan JW. Optimal readout of superconducting qubits exploiting high-level states. FUNDAMENTAL RESEARCH 2021. [DOI: 10.1016/j.fmre.2020.12.008] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/22/2022] Open
|
19
|
Stabilization and operation of a Kerr-cat qubit. Nature 2020; 584:205-209. [PMID: 32788737 DOI: 10.1038/s41586-020-2587-z] [Citation(s) in RCA: 51] [Impact Index Per Article: 12.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/28/2019] [Accepted: 05/20/2020] [Indexed: 11/08/2022]
Abstract
Quantum superpositions of macroscopically distinct classical states-so-called Schrödinger cat states-are a resource for quantum metrology, quantum communication and quantum computation. In particular, the superpositions of two opposite-phase coherent states in an oscillator encode a qubit protected against phase-flip errors1,2. However, several challenges have to be overcome for this concept to become a practical way to encode and manipulate error-protected quantum information. The protection must be maintained by stabilizing these highly excited states and, at the same time, the system has to be compatible with fast gates on the encoded qubit and a quantum non-demolition readout of the encoded information. Here we experimentally demonstrate a method for the generation and stabilization of Schrödinger cat states based on the interplay between Kerr nonlinearity and single-mode squeezing1,3 in a superconducting microwave resonator4. We show an increase in the transverse relaxation time of the stabilized, error-protected qubit of more than one order of magnitude compared with the single-photon Fock-state encoding. We perform all single-qubit gate operations on timescales more than sixty times faster than the shortest coherence time and demonstrate single-shot readout of the protected qubit under stabilization. Our results showcase the combination of fast quantum control and robustness against errors, which is intrinsic to stabilized macroscopic states, as well as the potential of of these states as resources in quantum information processing5-8.
Collapse
|
20
|
Howe L, Castellanos-Beltran MA, Sirois AJ, Olaya D, Biesecker J, Dresselhaus PD, Benz SP, Hopkins PF. Digital Control of a Superconducting Qubit Using a Josephson Pulse Generator at 3 K. PRX QUANTUM : A PHYSICAL REVIEW JOURNAL 2020; 3:10.1103/prxquantum.3.010350. [PMID: 36726390 PMCID: PMC9888300 DOI: 10.1103/prxquantum.3.010350] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/15/2023]
Abstract
Scaling of quantum computers to fault-tolerant levels relies critically on the integration of energy-efficient, stable, and reproducible qubit control and readout electronics. In comparison to traditional semiconductor-control electronics (TSCE) located at room temperature, the signals generated by rf sources based on Josephson-junctions (JJs) benefit from small device sizes, low power dissipation, intrinsic calibration, superior reproducibility, and insensitivity to ambient fluctuations. Previous experiments to colocate qubits and JJ-based control electronics have resulted in quasiparticle poisoning of the qubit, degrading the coherence and lifetime of the qubit. In this paper, we digitally control a 0.01-K transmon qubit with pulses from a Josephson pulse generator (JPG) located at the 3-K stage of a dilution refrigerator. We directly compare the qubit lifetime T 1, the coherence time T 2 * , and the thermal occupation P th when the qubit is controlled by the JPG circuit versus the TSCE setup. We find agreement to within the daily fluctuations of ±0.5 μs and ±2 μs for T 1 and T 2 * , respectively, and agreement to within the 1% error for P th. Additionally, we perform randomized benchmarking to measure an average JPG gate error of 2.1 × 10-2. In combination with a small device size (< 25 mm2) and low on-chip power dissipation (≪100 μW), these results are an important step toward demonstrating the viability of using JJ-based control electronics located at temperature stages higher than the mixing-chamber stage in highly scaled superconducting quantum information systems.
Collapse
Affiliation(s)
- L. Howe
- National Institute of Standards and Technology, Boulder, Colorado 80305, USA
| | | | - A. J. Sirois
- National Institute of Standards and Technology, Boulder, Colorado 80305, USA
| | - D. Olaya
- National Institute of Standards and Technology, Boulder, Colorado 80305, USA
- University of Colorado, Boulder, Colorado 80309, USA
| | - J. Biesecker
- National Institute of Standards and Technology, Boulder, Colorado 80305, USA
| | - P. D. Dresselhaus
- National Institute of Standards and Technology, Boulder, Colorado 80305, USA
| | - S. P. Benz
- National Institute of Standards and Technology, Boulder, Colorado 80305, USA
| | - P. F. Hopkins
- National Institute of Standards and Technology, Boulder, Colorado 80305, USA
| |
Collapse
|
21
|
Touzard S, Kou A, Frattini NE, Sivak VV, Puri S, Grimm A, Frunzio L, Shankar S, Devoret MH. Gated Conditional Displacement Readout of Superconducting Qubits. PHYSICAL REVIEW LETTERS 2019; 122:080502. [PMID: 30932609 DOI: 10.1103/physrevlett.122.080502] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/19/2018] [Indexed: 06/09/2023]
Abstract
We have realized a new interaction between superconducting qubits and a readout cavity that results in the displacement of a coherent state in the cavity, conditioned on the state of the qubit. This conditional state, when it reaches the cavity-following, phase-sensitive amplifier, matches its measured observable, namely, the in phase quadrature. In a setup where several qubits are coupled to the same readout resonator, we show it is possible to measure the state of a target qubit with minimal dephasing of the other qubits. Our results suggest novel directions for faster readout of superconducting qubits and implementations of bosonic quantum error-correcting codes.
Collapse
Affiliation(s)
- S Touzard
- Department of Applied Physics and Physics, Yale University, New Haven, Connecticut 06520, USA
| | - A Kou
- Department of Applied Physics and Physics, Yale University, New Haven, Connecticut 06520, USA
| | - N E Frattini
- Department of Applied Physics and Physics, Yale University, New Haven, Connecticut 06520, USA
| | - V V Sivak
- Department of Applied Physics and Physics, Yale University, New Haven, Connecticut 06520, USA
| | - S Puri
- Yale Quantum Institute, Yale University, New Haven, Connecticut 06520, USA
| | - A Grimm
- Department of Applied Physics and Physics, Yale University, New Haven, Connecticut 06520, USA
| | - L Frunzio
- Department of Applied Physics and Physics, Yale University, New Haven, Connecticut 06520, USA
| | - S Shankar
- Department of Applied Physics and Physics, Yale University, New Haven, Connecticut 06520, USA
| | - M H Devoret
- Department of Applied Physics and Physics, Yale University, New Haven, Connecticut 06520, USA
| |
Collapse
|
22
|
Serniak K, Hays M, de Lange G, Diamond S, Shankar S, Burkhart LD, Frunzio L, Houzet M, Devoret MH. Hot Nonequilibrium Quasiparticles in Transmon Qubits. PHYSICAL REVIEW LETTERS 2018; 121:157701. [PMID: 30362798 DOI: 10.1103/physrevlett.121.157701] [Citation(s) in RCA: 28] [Impact Index Per Article: 4.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/02/2018] [Revised: 07/27/2018] [Indexed: 06/08/2023]
Abstract
Nonequilibrium quasiparticle excitations degrade the performance of a variety of superconducting circuits. Understanding the energy distribution of these quasiparticles will yield insight into their generation mechanisms, the limitations they impose on superconducting devices, and how to efficiently mitigate quasiparticle-induced qubit decoherence. To probe this energy distribution, we systematically correlate qubit relaxation and excitation with charge-parity switches in an offset-charge-sensitive transmon qubit, and find that quasiparticle-induced excitation events are the dominant mechanism behind the residual excited-state population in our samples. By itself, the observed quasiparticle distribution would limit T_{1} to ≈200 μs, which indicates that quasiparticle loss in our devices is on equal footing with all other loss mechanisms. Furthermore, the measured rate of quasiparticle-induced excitation events is greater than that of relaxation events, which signifies that the quasiparticles are more energetic than would be predicted from a thermal distribution describing their apparent density.
Collapse
Affiliation(s)
- K Serniak
- Department of Applied Physics, Yale University, New Haven, Connecticut 06520, USA
| | - M Hays
- Department of Applied Physics, Yale University, New Haven, Connecticut 06520, USA
| | - G de Lange
- Department of Applied Physics, Yale University, New Haven, Connecticut 06520, USA
- QuTech and Kavli Institute of Nanoscience, Delft University of Technology, 2600 GA Delft, Netherlands
| | - S Diamond
- Department of Applied Physics, Yale University, New Haven, Connecticut 06520, USA
| | - S Shankar
- Department of Applied Physics, Yale University, New Haven, Connecticut 06520, USA
| | - L D Burkhart
- Department of Applied Physics, Yale University, New Haven, Connecticut 06520, USA
| | - L Frunzio
- Department of Applied Physics, Yale University, New Haven, Connecticut 06520, USA
| | - M Houzet
- Univ. Grenoble Alpes, CEA, INAC-Pheliqs, F-38000 Grenoble, France
| | - M H Devoret
- Department of Applied Physics, Yale University, New Haven, Connecticut 06520, USA
| |
Collapse
|
23
|
Opremcak A, Pechenezhskiy IV, Howington C, Christensen BG, Beck MA, Leonard E, Suttle J, Wilen C, Nesterov KN, Ribeill GJ, Thorbeck T, Schlenker F, Vavilov MG, Plourde BLT, McDermott R. Measurement of a superconducting qubit with a microwave photon counter. Science 2018; 361:1239-1242. [DOI: 10.1126/science.aat4625] [Citation(s) in RCA: 44] [Impact Index Per Article: 7.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/28/2018] [Accepted: 07/17/2018] [Indexed: 11/02/2022]
Affiliation(s)
- A. Opremcak
- Department of Physics, University of Wisconsin–Madison, Madison, WI 53706, USA
| | - I. V. Pechenezhskiy
- Department of Physics, University of Wisconsin–Madison, Madison, WI 53706, USA
| | - C. Howington
- Department of Physics, Syracuse University, Syracuse, NY 13244, USA
| | - B. G. Christensen
- Department of Physics, University of Wisconsin–Madison, Madison, WI 53706, USA
| | - M. A. Beck
- Department of Physics, University of Wisconsin–Madison, Madison, WI 53706, USA
| | - E. Leonard
- Department of Physics, University of Wisconsin–Madison, Madison, WI 53706, USA
| | - J. Suttle
- Department of Physics, University of Wisconsin–Madison, Madison, WI 53706, USA
| | - C. Wilen
- Department of Physics, University of Wisconsin–Madison, Madison, WI 53706, USA
| | - K. N. Nesterov
- Department of Physics, University of Wisconsin–Madison, Madison, WI 53706, USA
| | - G. J. Ribeill
- Department of Physics, University of Wisconsin–Madison, Madison, WI 53706, USA
| | - T. Thorbeck
- Department of Physics, University of Wisconsin–Madison, Madison, WI 53706, USA
| | - F. Schlenker
- Department of Physics, University of Wisconsin–Madison, Madison, WI 53706, USA
| | - M. G. Vavilov
- Department of Physics, University of Wisconsin–Madison, Madison, WI 53706, USA
| | - B. L. T. Plourde
- Department of Physics, Syracuse University, Syracuse, NY 13244, USA
| | - R. McDermott
- Department of Physics, University of Wisconsin–Madison, Madison, WI 53706, USA
| |
Collapse
|
24
|
Magnard P, Kurpiers P, Royer B, Walter T, Besse JC, Gasparinetti S, Pechal M, Heinsoo J, Storz S, Blais A, Wallraff A. Fast and Unconditional All-Microwave Reset of a Superconducting Qubit. PHYSICAL REVIEW LETTERS 2018; 121:060502. [PMID: 30141638 DOI: 10.1103/physrevlett.121.060502] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/17/2018] [Indexed: 06/08/2023]
Abstract
Active qubit reset is a key operation in many quantum algorithms, and particularly in quantum error correction. Here, we experimentally demonstrate a reset scheme for a three-level transmon artificial atom coupled to a large bandwidth resonator. The reset protocol uses a microwave-induced interaction between the |f,0⟩ and |g,1⟩ states of the coupled transmon-resonator system, with |g⟩ and |f⟩ denoting the ground and second excited states of the transmon, and |0⟩ and |1⟩ the photon Fock states of the resonator. We characterize the reset process and demonstrate reinitialization of the transmon-resonator system to its ground state in less than 500 ns and with 0.2% residual excitation. Our protocol is of practical interest as it has no additional architectural requirements beyond those needed for fast and efficient single-shot readout of transmons, and does not require feedback.
Collapse
Affiliation(s)
- P Magnard
- Department of Physics, ETH Zürich, CH-8093 Zürich, Switzerland
| | - P Kurpiers
- Department of Physics, ETH Zürich, CH-8093 Zürich, Switzerland
| | - B Royer
- Institut Quantique and Département de Physique, Université de Sherbrooke, Sherbrooke, Québec J1K 2R1, Canada
| | - T Walter
- Department of Physics, ETH Zürich, CH-8093 Zürich, Switzerland
| | - J-C Besse
- Department of Physics, ETH Zürich, CH-8093 Zürich, Switzerland
| | - S Gasparinetti
- Department of Physics, ETH Zürich, CH-8093 Zürich, Switzerland
| | - M Pechal
- Department of Physics, ETH Zürich, CH-8093 Zürich, Switzerland
| | - J Heinsoo
- Department of Physics, ETH Zürich, CH-8093 Zürich, Switzerland
| | - S Storz
- Department of Physics, ETH Zürich, CH-8093 Zürich, Switzerland
| | - A Blais
- Institut Quantique and Département de Physique, Université de Sherbrooke, Sherbrooke, Québec J1K 2R1, Canada
- Canadian Institute for Advanced Research, Toronto, Ontario M5G IZ8, Canada
| | - A Wallraff
- Department of Physics, ETH Zürich, CH-8093 Zürich, Switzerland
| |
Collapse
|
25
|
Wendin G. Quantum information processing with superconducting circuits: a review. REPORTS ON PROGRESS IN PHYSICS. PHYSICAL SOCIETY (GREAT BRITAIN) 2017; 80:106001. [PMID: 28682303 DOI: 10.1088/1361-6633/aa7e1a] [Citation(s) in RCA: 107] [Impact Index Per Article: 15.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/28/2023]
Abstract
During the last ten years, superconducting circuits have passed from being interesting physical devices to becoming contenders for near-future useful and scalable quantum information processing (QIP). Advanced quantum simulation experiments have been shown with up to nine qubits, while a demonstration of quantum supremacy with fifty qubits is anticipated in just a few years. Quantum supremacy means that the quantum system can no longer be simulated by the most powerful classical supercomputers. Integrated classical-quantum computing systems are already emerging that can be used for software development and experimentation, even via web interfaces. Therefore, the time is ripe for describing some of the recent development of superconducting devices, systems and applications. As such, the discussion of superconducting qubits and circuits is limited to devices that are proven useful for current or near future applications. Consequently, the centre of interest is the practical applications of QIP, such as computation and simulation in Physics and Chemistry.
Collapse
Affiliation(s)
- G Wendin
- Department of Microtechnology and Nanoscience-MC2, Chalmers University of Technology, SE-41296 Gothenburg, Sweden
| |
Collapse
|
26
|
Harrington PM, Monroe JT, Murch KW. Quantum Zeno Effects from Measurement Controlled Qubit-Bath Interactions. PHYSICAL REVIEW LETTERS 2017; 118:240401. [PMID: 28665648 DOI: 10.1103/physrevlett.118.240401] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/24/2017] [Indexed: 06/07/2023]
Abstract
The Zeno and anti-Zeno effects are features of measurement-driven quantum evolution where frequent measurement inhibits or accelerates the decay of a quantum state. Either type of evolution can emerge depending on the system-environment interaction and measurement method. In this experiment, we use a superconducting qubit to map out both types of Zeno effect in the presence of structured noise baths and variable measurement rates. We observe both the suppression and acceleration of qubit decay as repeated measurements are used to modulate the qubit spectrum causing the qubit to sample different portions of the bath. We compare the Zeno effects arising from dispersive energy measurements and purely dephasing "quasimeasurements," showing energy measurements are not necessary to accelerate or suppress the decay process.
Collapse
Affiliation(s)
- P M Harrington
- Department of Physics, Washington University, Saint Louis, Missouri 63130, USA
| | - J T Monroe
- Department of Physics, Washington University, Saint Louis, Missouri 63130, USA
| | - K W Murch
- Department of Physics, Washington University, Saint Louis, Missouri 63130, USA
- Institute for Materials Science and Engineering, Saint Louis, Missouri 63130, USA
| |
Collapse
|