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Li Z, Roy T, Rodríguez Pérez D, Lee KH, Kapit E, Schuster DI. Autonomous error correction of a single logical qubit using two transmons. Nat Commun 2024; 15:1681. [PMID: 38395989 PMCID: PMC10891116 DOI: 10.1038/s41467-024-45858-z] [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: 11/14/2023] [Accepted: 02/05/2024] [Indexed: 02/25/2024] Open
Abstract
Large-scale quantum computers will inevitably need quantum error correction to protect information against decoherence. Traditional error correction typically requires many qubits, along with high-efficiency error syndrome measurement and real-time feedback. Autonomous quantum error correction instead uses steady-state bath engineering to perform the correction in a hardware-efficient manner. In this work, we develop a new autonomous quantum error correction scheme that actively corrects single-photon loss and passively suppresses low-frequency dephasing, and we demonstrate an important experimental step towards its full implementation with transmons. Compared to uncorrected encoding, improvements are experimentally witnessed for the logical zero, one, and superposition states. Our results show the potential of implementing hardware-efficient autonomous quantum error correction to enhance the reliability of a transmon-based quantum information processor.
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Affiliation(s)
- Ziqian Li
- James Franck Institute, University of Chicago, Chicago, IL, 60637, USA
- Department of Physics, University of Chicago, Chicago, IL, 60637, USA
- Department of Applied Physics, Stanford University, Stanford, CA, 94305, USA
| | - Tanay Roy
- James Franck Institute, University of Chicago, Chicago, IL, 60637, USA
- Department of Physics, University of Chicago, Chicago, IL, 60637, USA
| | | | - Kan-Heng Lee
- James Franck Institute, University of Chicago, Chicago, IL, 60637, USA
- Department of Physics, University of Chicago, Chicago, IL, 60637, USA
| | - Eliot Kapit
- Department of Physics, Colorado School of Mines, Golden, CO, 80401, USA
| | - David I Schuster
- James Franck Institute, University of Chicago, Chicago, IL, 60637, USA.
- Department of Physics, University of Chicago, Chicago, IL, 60637, USA.
- Department of Applied Physics, Stanford University, Stanford, CA, 94305, USA.
- Pritzker School of Molecular Engineering, University of Chicago, Chicago, IL, 60637, USA.
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2
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Ahmed S, Quijandría F, Kockum AF. Gradient-Descent Quantum Process Tomography by Learning Kraus Operators. PHYSICAL REVIEW LETTERS 2023; 130:150402. [PMID: 37115870 DOI: 10.1103/physrevlett.130.150402] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/11/2022] [Revised: 02/17/2023] [Accepted: 03/16/2023] [Indexed: 06/19/2023]
Abstract
We perform quantum process tomography (QPT) for both discrete- and continuous-variable quantum systems by learning a process representation using Kraus operators. The Kraus form ensures that the reconstructed process is completely positive. To make the process trace preserving, we use a constrained gradient-descent (GD) approach on the so-called Stiefel manifold during optimization to obtain the Kraus operators. Our ansatz uses a few Kraus operators to avoid direct estimation of large process matrices, e.g., the Choi matrix, for low-rank quantum processes. The GD-QPT matches the performance of both compressed-sensing (CS) and projected least-squares (PLS) QPT in benchmarks with two-qubit random processes, but shines by combining the best features of these two methods. Similar to CS (but unlike PLS), GD-QPT can reconstruct a process from just a small number of random measurements, and similar to PLS (but unlike CS) it also works for larger system sizes, up to at least five qubits. We envisage that the data-driven approach of GD-QPT can become a practical tool that greatly reduces the cost and computational effort for QPT in intermediate-scale quantum systems.
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Affiliation(s)
- Shahnawaz Ahmed
- Department of Microtechnology and Nanoscience, Chalmers University of Technology, 412 96 Gothenburg, Sweden
| | - Fernando Quijandría
- Department of Microtechnology and Nanoscience, Chalmers University of Technology, 412 96 Gothenburg, Sweden
- Quantum Machines Unit, Okinawa Institute of Science and Technology Graduate University, Onna-son, Okinawa 904-0495, Japan
| | - Anton Frisk Kockum
- Department of Microtechnology and Nanoscience, Chalmers University of Technology, 412 96 Gothenburg, Sweden
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3
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Ho LB. Systematic errors in direct state measurements with quantum controlled measurements. JOURNAL OF PHYSICS B: ATOMIC, MOLECULAR AND OPTICAL PHYSICS 2020; 53:115501. [DOI: 10.1088/1361-6455/ab7881] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 09/01/2023]
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4
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Stephenson LJ, Nadlinger DP, Nichol BC, An S, Drmota P, Ballance TG, Thirumalai K, Goodwin JF, Lucas DM, Ballance CJ. High-Rate, High-Fidelity Entanglement of Qubits Across an Elementary Quantum Network. PHYSICAL REVIEW LETTERS 2020; 124:110501. [PMID: 32242699 DOI: 10.1103/physrevlett.124.110501] [Citation(s) in RCA: 9] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/03/2019] [Accepted: 02/06/2020] [Indexed: 06/11/2023]
Abstract
We demonstrate remote entanglement of trapped-ion qubits via a quantum-optical fiber link with fidelity and rate approaching those of local operations. Two ^{88}Sr^{+} qubits are entangled via the polarization degree of freedom of two spontaneously emitted 422 nm photons which are coupled by high-numerical-aperture lenses into single-mode optical fibers and interfere on a beam splitter. A novel geometry allows high-efficiency photon collection while maintaining unit fidelity for ion-photon entanglement. We generate heralded Bell pairs with fidelity 94% at an average rate 182 s^{-1} (success probability 2.18×10^{-4}).
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Affiliation(s)
- L J Stephenson
- Department of Physics, University of Oxford, Clarendon Laboratory, Parks Road, Oxford OX1 3PU, United Kingdom
| | - D P Nadlinger
- Department of Physics, University of Oxford, Clarendon Laboratory, Parks Road, Oxford OX1 3PU, United Kingdom
| | - B C Nichol
- Department of Physics, University of Oxford, Clarendon Laboratory, Parks Road, Oxford OX1 3PU, United Kingdom
| | - S An
- Department of Physics, University of Oxford, Clarendon Laboratory, Parks Road, Oxford OX1 3PU, United Kingdom
| | - P Drmota
- Department of Physics, University of Oxford, Clarendon Laboratory, Parks Road, Oxford OX1 3PU, United Kingdom
| | - T G Ballance
- Department of Physics, University of Oxford, Clarendon Laboratory, Parks Road, Oxford OX1 3PU, United Kingdom
| | - K Thirumalai
- Department of Physics, University of Oxford, Clarendon Laboratory, Parks Road, Oxford OX1 3PU, United Kingdom
| | - J F Goodwin
- Department of Physics, University of Oxford, Clarendon Laboratory, Parks Road, Oxford OX1 3PU, United Kingdom
| | - D M Lucas
- Department of Physics, University of Oxford, Clarendon Laboratory, Parks Road, Oxford OX1 3PU, United Kingdom
| | - C J Ballance
- Department of Physics, University of Oxford, Clarendon Laboratory, Parks Road, Oxford OX1 3PU, United Kingdom
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5
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Métillon V, Gerlich S, Brune M, Raimond JM, Rouchon P, Dotsenko I. Benchmarking Maximum-Likelihood State Estimation with an Entangled Two-Cavity State. PHYSICAL REVIEW LETTERS 2019; 123:060404. [PMID: 31491182 DOI: 10.1103/physrevlett.123.060404] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/10/2019] [Indexed: 06/10/2023]
Abstract
The efficient quantum state reconstruction algorithm described by Six et al. [Phys. Rev. A 93, 012109 (2016)PLRAAN2469-992610.1103/PhysRevA.93.012109] is experimentally implemented on the nonlocal state of two microwave cavities entangled by a circular Rydberg atom. We use information provided by long sequences of measurements performed by resonant and dispersive probe atoms over timescales involving the system decoherence. Moreover, we benefit from the consolidation, in the same reconstruction, of different measurement protocols providing complementary information. Finally, we obtain realistic error bars for the matrix elements of the reconstructed density operator. These results demonstrate the pertinence and precision of the method, directly applicable to any complex quantum system.
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Affiliation(s)
- V Métillon
- Laboratoire Kastler Brossel, Collège de France, CNRS, ENS-PSL University, Sorbonne Université, 11 place Marcelin Berthelot, F-75231 Paris, France
| | - S Gerlich
- Laboratoire Kastler Brossel, Collège de France, CNRS, ENS-PSL University, Sorbonne Université, 11 place Marcelin Berthelot, F-75231 Paris, France
| | - M Brune
- Laboratoire Kastler Brossel, Collège de France, CNRS, ENS-PSL University, Sorbonne Université, 11 place Marcelin Berthelot, F-75231 Paris, France
| | - J M Raimond
- Laboratoire Kastler Brossel, Collège de France, CNRS, ENS-PSL University, Sorbonne Université, 11 place Marcelin Berthelot, F-75231 Paris, France
| | - P Rouchon
- Centre Automatique et Systèmes, Mines-ParisTech, PSL Research University, 60 Boulevard Saint-Michel, 75006 Paris, France
- INRIA Paris, 2 rue Simone Iff, 75012 Paris, France
| | - I Dotsenko
- Laboratoire Kastler Brossel, Collège de France, CNRS, ENS-PSL University, Sorbonne Université, 11 place Marcelin Berthelot, F-75231 Paris, France
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6
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Oh C, Teo YS, Jeong H. Probing Bayesian Credible Regions Intrinsically: A Feasible Error Certification for Physical Systems. PHYSICAL REVIEW LETTERS 2019; 123:040602. [PMID: 31491280 DOI: 10.1103/physrevlett.123.040602] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/10/2019] [Revised: 06/20/2019] [Indexed: 06/10/2023]
Abstract
Standard computation of size and credibility of a Bayesian credible region for certifying any point estimator of an unknown parameter (such as a quantum state, channel, phase, etc.) requires selecting points that are in the region from a finite parameter-space sample, which is infeasible for a large dataset or dimension as the region would then be extremely small. We solve this problem by introducing the in-region sampling theory to compute both region qualities just by sampling appropriate functions over the region itself using any Monte Carlo sampling method. We take in-region sampling to the next level by understanding the credible-region capacity (an alternative description for the region content to size) as the average l_{p}-norm distance (p>0) between a random region point and the estimator, and present analytical formulas for p=2 to estimate both the capacity and credibility for any dimension and a sufficiently large dataset without Monte Carlo sampling, thereby providing a quick alternative to Bayesian certification. All results are discussed in the context of quantum-state tomography.
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Affiliation(s)
- Changhun Oh
- Department of Physics and Astronomy, Seoul National University, 08826 Seoul, Korea
| | - Yong Siah Teo
- Department of Physics and Astronomy, Seoul National University, 08826 Seoul, Korea
| | - Hyunseok Jeong
- Department of Physics and Astronomy, Seoul National University, 08826 Seoul, Korea
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7
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Wang J, Scholz VB, Renner R. Confidence Polytopes in Quantum State Tomography. PHYSICAL REVIEW LETTERS 2019; 122:190401. [PMID: 31144942 DOI: 10.1103/physrevlett.122.190401] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/29/2018] [Indexed: 06/09/2023]
Abstract
Quantum state tomography is the task of inferring the state of a quantum system from measurement data. A reliable tomography scheme should not only report an estimate for that state, but also well-justified error bars. These may be specified in terms of confidence regions, i.e., subsets of the state space which contain the system's state with high probability. Here, building upon a quantum generalization of Clopper-Pearson confidence intervals-a notion known from classical statistics-we present a simple and reliable scheme for generating confidence regions. These have the shape of a polytope and can be computed efficiently. We provide several examples to demonstrate the practical usability of the scheme in experiments.
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Affiliation(s)
- Jinzhao Wang
- Institute for Theoretical Physics, ETH Zurich, 8093 Zürich, Switzerland
| | - Volkher B Scholz
- Institute for Theoretical Physics, ETH Zurich, 8093 Zürich, Switzerland
| | - Renato Renner
- Institute for Theoretical Physics, ETH Zurich, 8093 Zürich, Switzerland
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8
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Butucea C, Guţă M, Nussbaum M. Local asymptotic equivalence of pure states ensembles and quantum Gaussian white noise. Ann Stat 2018. [DOI: 10.1214/17-aos1672] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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9
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Cavaillès A, Le Jeannic H, Raskop J, Guccione G, Markham D, Diamanti E, Shaw MD, Verma VB, Nam SW, Laurat J. Demonstration of Einstein-Podolsky-Rosen Steering Using Hybrid Continuous- and Discrete-Variable Entanglement of Light. PHYSICAL REVIEW LETTERS 2018; 121:170403. [PMID: 30411932 DOI: 10.1103/physrevlett.121.170403] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/20/2018] [Indexed: 06/08/2023]
Abstract
Einstein-Podolsky-Rosen steering is known to be a key resource for one-sided device-independent quantum information protocols. Here we demonstrate steering using hybrid entanglement between continuous- and discrete-variable optical qubits. To this end, we report on suitable steering inequalities and detail the implementation and requirements for this demonstration. Steering is experimentally certified by observing a violation by more than 5 standard deviations. Our results illustrate the potential of optical hybrid entanglement for applications in heterogeneous quantum networks that would interconnect disparate physical platforms and encodings.
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Affiliation(s)
- A Cavaillès
- Laboratoire Kastler Brossel, Sorbonne Université, CNRS, ENS-Université PSL, Collège de France, 4 Place Jussieu, 75005 Paris, France
| | - H Le Jeannic
- Laboratoire Kastler Brossel, Sorbonne Université, CNRS, ENS-Université PSL, Collège de France, 4 Place Jussieu, 75005 Paris, France
| | - J Raskop
- Laboratoire Kastler Brossel, Sorbonne Université, CNRS, ENS-Université PSL, Collège de France, 4 Place Jussieu, 75005 Paris, France
| | - G Guccione
- Laboratoire Kastler Brossel, Sorbonne Université, CNRS, ENS-Université PSL, Collège de France, 4 Place Jussieu, 75005 Paris, France
| | - D Markham
- Laboratoire d'Informatique de Paris 6, Sorbonne Université, CNRS, 4 Place Jussieu, 75005 Paris, France
| | - E Diamanti
- Laboratoire d'Informatique de Paris 6, Sorbonne Université, CNRS, 4 Place Jussieu, 75005 Paris, France
| | - M D Shaw
- Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, Pasadena, California 91109, USA
| | - V B Verma
- National Institute of Standards and Technology, 325 Broadway, Boulder, Colorado 80305, USA
| | - S W Nam
- National Institute of Standards and Technology, 325 Broadway, Boulder, Colorado 80305, USA
| | - J Laurat
- Laboratoire Kastler Brossel, Sorbonne Université, CNRS, ENS-Université PSL, Collège de France, 4 Place Jussieu, 75005 Paris, France
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10
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Pfister C, Rol MA, Mantri A, Tomamichel M, Wehner S. Capacity estimation and verification of quantum channels with arbitrarily correlated errors. Nat Commun 2018; 9:27. [PMID: 29295975 PMCID: PMC5750239 DOI: 10.1038/s41467-017-00961-2] [Citation(s) in RCA: 13] [Impact Index Per Article: 2.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/09/2016] [Accepted: 08/08/2017] [Indexed: 11/09/2022] Open
Abstract
The central figure of merit for quantum memories and quantum communication devices is their capacity to store and transmit quantum information. Here, we present a protocol that estimates a lower bound on a channel's quantum capacity, even when there are arbitrarily correlated errors. One application of these protocols is to test the performance of quantum repeaters for transmitting quantum information. Our protocol is easy to implement and comes in two versions. The first estimates the one-shot quantum capacity by preparing and measuring in two different bases, where all involved qubits are used as test qubits. The second verifies on-the-fly that a channel's one-shot quantum capacity exceeds a minimal tolerated value while storing or communicating data. We discuss the performance using simple examples, such as the dephasing channel for which our method is asymptotically optimal. Finally, we apply our method to a superconducting qubit in experiment.
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Affiliation(s)
- Corsin Pfister
- QuTech, Delft University of Technology, Lorentzweg 1, 2628 CJ, Delft, The Netherlands.,Centre for Quantum Technologies, 3 Science Drive 2, Singapore, 117543, Singapore
| | - M Adriaan Rol
- QuTech, Delft University of Technology, Lorentzweg 1, 2628 CJ, Delft, The Netherlands.,Kavli Institute of Nanoscience, Delft University of Technology, P.O. Box 5046, 2600 GA, Delft, The Netherlands
| | - Atul Mantri
- Singapore University of Technology and Design, 20 Dover Drive, Singapore, 138682, Singapore
| | - Marco Tomamichel
- Centre for Quantum Software and Information, University of Technology Sydney, Broadway, NSW, 2007, Australia
| | - Stephanie Wehner
- QuTech, Delft University of Technology, Lorentzweg 1, 2628 CJ, Delft, The Netherlands.
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11
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Silva GB, Glancy S, Vasconcelos HM. Investigating bias in maximum-likelihood quantum-state tomography. PHYSICAL REVIEW. A 2017; 95:022107. [PMID: 30997437 PMCID: PMC6463874 DOI: 10.1103/physreva.95.022107] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/09/2023]
Abstract
Maximum-likelihood quantum-state tomography yields estimators that are consistent, provided that the likelihood model is correct, but the maximum-likelihood estimators may have bias for any finite data set. The bias of an estimator is the difference between the expected value of the estimate and the true value of the parameter being estimated. This paper investigates bias in the widely used maximum-likelihood quantum-state tomography. Our goal is to understand how the amount of bias depends on factors such as the purity of the true state, the number of measurements performed, and the number of different bases in which the system is measured. For this, we perform numerical experiments that simulate optical homodyne tomography of squeezed thermal states under various conditions, perform tomography, and estimate bias in the purity of the estimated state. We find that estimates of higher purity states exhibit considerable bias, such that the estimates have lower purities than the true states.
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Affiliation(s)
- G. B. Silva
- Departamento de Engenharia de Teleinformática, Universidade Federal do Ceará, Fortaleza, Ceará, Brazil
| | - S. Glancy
- Applied and Computational Mathematics Division, National Institute of Standards and Technology, Boulder, Colorado 80305, USA
| | - H. M. Vasconcelos
- Departamento de Engenharia de Teleinformática, Universidade Federal do Ceará, Fortaleza, Ceará, Brazil
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12
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Lerch S, Stefanov A. Ultrashort pulses characterization by quantum state tomography. OPTICS EXPRESS 2016; 24:30088-30099. [PMID: 28059287 DOI: 10.1364/oe.24.030088] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/06/2023]
Abstract
We apply the method of quantum state tomography for the reconstruction of classical laser pulses. The scheme is based on linear inversion, has no need for iterative inversion algorithm or deconvolution, and accounts for partial coherence. The reconstruction protocol is successfully tested on amplitude and phase shaped femtosecond pulses.
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