New Methods in Quantum Error Correction and Fault-Tolerant Quantum Computing
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Conversion of a General Quantum Stabilizer Code to an Entanglement
Conversion of a general quantum stabilizer code to an entanglement distillation protocol∗ Ryutaroh Matsumoto Dept. of Communications and Integrated Systems Tokyo Institute of Technology, 152-8552 Japan Email: [email protected] April 4, 2003 Abstract We show how to convert a quantum stabilizer code to a one-way or two- way entanglement distillation protocol. The proposed conversion method is a generalization of those of Shor-Preskill and Nielsen-Chuang. The recurrence protocol and the quantum privacy amplification protocol are equivalent to the protocols converted from [[2, 1]] stabilizer codes. We also give an example of a two-way protocol converted from a stabilizer bet- ter than the recurrence protocol and the quantum privacy amplification protocol. The distillable entanglement by the class of one-way protocols converted from stabilizer codes for a certain class of states is equal to or greater than the achievable rate of stabilizer codes over the channel cor- responding to the distilled state, and they can distill asymptotically more entanglement from a very noisy Werner state than the hashing protocol. 1 Introduction arXiv:quant-ph/0209091v4 4 Apr 2003 In many applications of quantum mechanics to communication, the sender and the receiver have to share a maximally entangled quantum state of two particles. When there is a noiseless quantum communication channel, the sender can send one of two particles in a maximally entangled state to the receiver and sharing of it is easily accomplished. However, the quantum communication channel is usually noisy, that is, the quantum state of the received particle changes probabilistically from the original state of a particle. -
Painstakingly-Edited Typeset Lecture Notes
Physics 239: Topology from Physics Winter 2021 Lecturer: McGreevy These lecture notes live here. Please email corrections and questions to mcgreevy at physics dot ucsd dot edu. Last updated: May 26, 2021, 14:36:17 1 Contents 0.1 Introductory remarks............................3 0.2 Conventions.................................8 0.3 Sources....................................9 1 The toric code and homology 10 1.1 Cell complexes and homology....................... 21 1.2 p-form ZN toric code............................ 23 1.3 Some examples............................... 26 1.4 Higgsing, change of coefficients, exact sequences............. 32 1.5 Independence of cellulation......................... 36 1.6 Gapped boundaries and relative homology................ 39 1.7 Duality.................................... 42 2 Supersymmetric quantum mechanics and cohomology, index theory, Morse theory 49 2.1 Supersymmetric quantum mechanics................... 49 2.2 Differential forms consolidation...................... 61 2.3 Supersymmetric QM and Morse theory.................. 66 2.4 Global information from local information................ 74 2.5 Homology and cohomology......................... 77 2.6 Cech cohomology.............................. 80 2.7 Local reconstructability of quantum states................ 86 3 Quantum Double Model and Homotopy 89 3.1 Notions of `same'.............................. 89 3.2 Homotopy equivalence and cohomology.................. 90 3.3 Homotopy equivalence and homology................... 91 3.4 Morse theory and homotopy -
(CST Part II) Lecture 14: Fault Tolerant Quantum Computing
Quantum Computing (CST Part II) Lecture 14: Fault Tolerant Quantum Computing The history of the universe is, in effect, a huge and ongoing quantum computation. The universe is a quantum computer. Seth Lloyd 1 / 21 Resources for this lecture Nielsen and Chuang p474-495 covers the material of this lecture. 2 / 21 Why we need fault tolerance Classical computers perform complicated operations where bits are repeatedly \combined" in computations, therefore if an error occurs, it could in principle propagate to a huge number of other bits. Fortunately, in modern digital computers errors are so phenomenally unlikely that we can forget about this possibility for all practical purposes. Errors do, however, occur in telecommunications systems, but as the purpose of these is the simple transmittal of some information, it suffices to perform error correction on the final received data. In a sense, quantum computing is the worst of both of these worlds: errors do occur with significant frequency, and if uncorrected they will propagate, rendering the computation useless. Thus the solution is that we must correct errors as we go along. 3 / 21 Fault tolerant quantum computing set-up For fault tolerant quantum computing: We use encoded qubits, rather than physical qubits. For example we may use the 7-qubit Steane code to represent each logical qubit in the computation. We use fault tolerant quantum gates, which are defined such thata single error in the fault tolerant gate propagates to at most one error in each encoded block of qubits. By a \block of qubits", we mean (for example) each block of 7 physical qubits that represents a logical qubit using the Steane code. -
Arxiv:Quant-Ph/9705031V3 26 Aug 1997 Eddt Rvn Unu Optrfo Crashing
CALT-68-2112 QUIC-97-030 quant-ph/9705031 Reliable Quantum Computers John Preskill1 California Institute of Technology, Pasadena, CA 91125, USA Abstract The new field of quantum error correction has developed spectacularly since its origin less than two years ago. Encoded quantum information can be protected from errors that arise due to uncontrolled interactions with the environment. Recovery from errors can work effectively even if occasional mistakes occur during the recovery procedure. Furthermore, encoded quantum information can be processed without serious propagation of errors. Hence, an arbitrarily long quantum computation can be performed reliably, provided that the average probability of error per quantum gate is less than a certain critical value, the accuracy threshold. A quantum computer storing about 106 qubits, with a probability of error per quantum gate of order 10−6, would be a formidable factoring engine. Even a smaller, less accurate quantum computer would be able to perform many useful tasks. This paper is based on a talk presented at the ITP Conference on Quantum Coherence and Decoherence, 15-18 December 1996. 1 The golden age of quantum error correction Many of us are hopeful that quantum computers will become practical and useful computing devices some time during the 21st century. It is probably fair to say, though, that none of us can now envision exactly what the hardware of that machine of the future will be like; surely, it will be much different than the sort of hardware that experimental physicists are investigating these days. But of one thing we can be quite confident—that a practical quantum computer will incorporate some type of error correction into its operation. -
Superconducting Quantum Simulator for Topological Order and the Toric Code
Superconducting Quantum Simulator for Topological Order and the Toric Code Mahdi Sameti,1 Anton Potoˇcnik,2 Dan E. Browne,3 Andreas Wallraff,2 and Michael J. Hartmann1 1Institute of Photonics and Quantum Sciences, Heriot-Watt University Edinburgh EH14 4AS, United Kingdom 2Department of Physics, ETH Z¨urich,CH-8093 Z¨urich,Switzerland 3Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, United Kingdom (Dated: March 20, 2017) Topological order is now being established as a central criterion for characterizing and classifying ground states of condensed matter systems and complements categorizations based on symmetries. Fractional quantum Hall systems and quantum spin liquids are receiving substantial interest because of their intriguing quantum correlations, their exotic excitations and prospects for protecting stored quantum information against errors. Here we show that the Hamiltonian of the central model of this class of systems, the Toric Code, can be directly implemented as an analog quantum simulator in lattices of superconducting circuits. The four-body interactions, which lie at its heart, are in our concept realized via Superconducting Quantum Interference Devices (SQUIDs) that are driven by a suitably oscillating flux bias. All physical qubits and coupling SQUIDs can be individually controlled with high precision. Topologically ordered states can be prepared via an adiabatic ramp of the stabilizer interactions. Strings of qubit operators, including the stabilizers and correlations along non-contractible loops, can be read out via a capacitive coupling to read-out resonators. Moreover, the available single qubit operations allow to create and propagate elementary excitations of the Toric Code and to verify their fractional statistics. -
LI-DISSERTATION-2020.Pdf
FAULT-TOLERANCE ON NEAR-TERM QUANTUM COMPUTERS AND SUBSYSTEM QUANTUM ERROR CORRECTING CODES A Dissertation Presented to The Academic Faculty By Muyuan Li In Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the School of Computational Science and Engineering Georgia Institute of Technology May 2020 Copyright c Muyuan Li 2020 FAULT-TOLERANCE ON NEAR-TERM QUANTUM COMPUTERS AND SUBSYSTEM QUANTUM ERROR CORRECTING CODES Approved by: Dr. Kenneth R. Brown, Advisor Department of Electrical and Computer Dr. C. David Sherrill Engineering School of Chemistry and Biochemistry Duke University Georgia Institute of Technology Dr. Edmond Chow Dr. Richard Vuduc School of Computational Science and School of Computational Science and Engineering Engineering Georgia Institute of Technology Georgia Institute of Technology Dr. T.A. Brian Kennedy Date Approved: March 19, 2020 School of Physics Georgia Institute of Technology I think it is safe to say that no one understands quantum mechanics. R. P. Feynman To my family and my friends. ACKNOWLEDGEMENTS I would like to thank my advisor, Ken Brown, who has guided me through my graduate studies with his patience, wisdom, and generosity. He has always been supportive and helpful, and always makes himself available when I needed. I have been constantly inspired by his depth of knowledge in research, as well as his immense passion for life. I would also like to thank my committee members, Professors Edmond Chow, T.A. Brian Kennedy, C. David Sherrill, and Richard Vuduc, for their time and helpful sugges- tions. One half of my graduate career was spent at Georgia Tech and the other half at Duke University. -
Assessing the Progress of Trapped-Ion Processors Towards Fault-Tolerant Quantum Computation
PHYSICAL REVIEW X 7, 041061 (2017) Assessing the Progress of Trapped-Ion Processors Towards Fault-Tolerant Quantum Computation A. Bermudez,1,2 X. Xu,3 R. Nigmatullin,4,3 J. O’Gorman,3 V. Negnevitsky,5 P. Schindler,6 T. Monz,6 U. G. Poschinger,7 C. Hempel,8 J. Home,5 F. Schmidt-Kaler,7 M. Biercuk,8 R. Blatt,6,9 S. Benjamin,3 and M. Müller1 1Department of Physics, College of Science, Swansea University, Singleton Park, Swansea SA2 8PP, United Kingdom 2Instituto de Física Fundamental, IFF-CSIC, Madrid E-28006, Spain 3Department of Materials, University of Oxford, Oxford OX1 3PH, United Kingdom 4Complex Systems Research Group, Faculty of Engineering and IT, The University of Sydney, Sydney, Australia 5Institute for Quantum Electronics, ETH Zürich, Otto-Stern-Weg 1, 8093 Zürich, Switzerland 6Institute for Experimental Physics, University of Innsbruck, 6020 Innsbruck, Austria 7Institut für Physik, Universität Mainz, Staudingerweg 7, 55128 Mainz, Germany 8ARC Centre for Engineered Quantum Systems, School of Physics, The University of Sydney, Sydney, NSW 2006, Australia 9Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences, A-6020 Innsbruck, Austria (Received 24 May 2017; revised manuscript received 11 August 2017; published 13 December 2017) A quantitative assessment of the progress of small prototype quantum processors towards fault-tolerant quantum computation is a problem of current interest in experimental and theoretical quantum information science. We introduce a necessary and fair criterion for quantum error correction (QEC), which must be achieved in the development of these quantum processors before their sizes are sufficiently big to consider the well-known QEC threshold. -
Quantum Error-Correcting Codes by Concatenation QEC11
Quantum Error-Correcting Codes by Concatenation QEC11 Second International Conference on Quantum Error Correction University of Southern California, Los Angeles, USA December 5–9, 2011 Quantum Error-Correcting Codes by Concatenation Markus Grassl joint work with Bei Zeng Centre for Quantum Technologies National University of Singapore Singapore Markus Grassl – 1– 07.12.2011 Quantum Error-Correcting Codes by Concatenation QEC11 Why Bei isn’t here Jonathan, November 24, 2011 Markus Grassl – 2– 07.12.2011 Quantum Error-Correcting Codes by Concatenation QEC11 Overview Shor’s nine-qubit code revisited • The code [[25, 1, 9]] • Concatenated graph codes • Generalized concatenated quantum codes • Codes for the Amplitude Damping (AD) channel • Conclusions • Markus Grassl – 3– 07.12.2011 Quantum Error-Correcting Codes by Concatenation QEC11 Shor’s Nine-Qubit Code Revisited Bit-flip code: 0 000 , 1 111 . | → | | → | Phase-flip code: 0 + ++ , 1 . | → | | → | − −− Effect of single-qubit errors on the bit-flip code: X-errors change the basis states, but can be corrected • Z-errors at any of the three positions: • Z 000 = 000 | | “encoded” Z-operator Z 111 = 111 | −| = Bit-flip code & error correction convert the channel into a phase-error ⇒ channel = Concatenation of bit-flip code and phase-flip code yields [[9, 1, 3]] ⇒ Markus Grassl – 4– 07.12.2011 Quantum Error-Correcting Codes by Concatenation QEC11 The Code [[25, 1, 9]] The best single-error correcting code is = [[5, 1, 3]] • C0 Re-encoding each of the 5 qubits with yields = [[52, 1, 32]] -
Entangled Many-Body States As Resources of Quantum Information Processing
ENTANGLED MANY-BODY STATES AS RESOURCES OF QUANTUM INFORMATION PROCESSING LI YING A thesis submitted for the Degree of Doctor of Philosophy CENTRE FOR QUANTUM TECHNOLOGIES NATIONAL UNIVERSITY OF SINGAPORE 2013 DECLARATION I hereby declare that the thesis is my original work and it has been written by me in its entirety. I have duly acknowledged all the sources of information which have been used in the thesis. This thesis has also not been submitted for any degree in any university previously. LI YING 23 July 2013 Acknowledgments I am very grateful to have spent about four years at CQT working with Leong Chuan Kwek. He always brings me new ideas in science and has helped me to establish good collaborative relationships with other scien- tists. Kwek helped me a lot in my life. I am also very grateful to Simon C. Benjamin. He showd me how to do high quality researches in physics. Simon also helped me to improve my writing and presentation. I hope to have fruitful collaborations in the near future with Kwek and Simon. For my project about the ground-code MBQC (Chapter2), I am thank- ful to Tzu-Chieh Wei, Daniel E. Browne and Robert Raussendorf. In one afternoon, Tzu-Chieh showed me the idea of his recent paper in this topic in the quantum cafe, which encouraged me to think about the ground- code MBQC. Dan and Robert have a high level of comprehension on the subject of the MBQC. And we had some very interesting discussions and communications. I am grateful to Sean D. -
Lecture 6: Quantum Error Correction and Quantum Capacity
Lecture 6: Quantum error correction and quantum capacity Mark M. Wilde∗ The quantum capacity theorem is one of the most important theorems in quantum Shannon theory. It is a fundamentally \quantum" theorem in that it demonstrates that a fundamentally quantum information quantity, the coherent information, is an achievable rate for quantum com- munication over a quantum channel. The fact that the coherent information does not have a strong analog in classical Shannon theory truly separates the quantum and classical theories of information. The no-cloning theorem provides the intuition behind quantum error correction. The goal of any quantum communication protocol is for Alice to establish quantum correlations with the receiver Bob. We know well now that every quantum channel has an isometric extension, so that we can think of another receiver, the environment Eve, who is at a second output port of a larger unitary evolution. Were Eve able to learn anything about the quantum information that Alice is attempting to transmit to Bob, then Bob could not be retrieving this information|otherwise, they would violate the no-cloning theorem. Thus, Alice should figure out some subspace of the channel input where she can place her quantum information such that only Bob has access to it, while Eve does not. That the dimensionality of this subspace is exponential in the coherent information is perhaps then unsurprising in light of the above no-cloning reasoning. The coherent information is an entropy difference H(B) − H(E)|a measure of the amount of quantum correlations that Alice can establish with Bob less the amount that Eve can gain. -
2013 Industry Canada Report
Industry Canada Report 2012/13 Reporting Period Institute for Quantum Computing University of Waterloo June 15, 2013 Industry Canada Report | 2 Note from the Executive Director Harnessing the quantum world will lead to new technologies and applications that will change the world. The quantum properties of nature allow the accomplishment of tasks which seem intractable with today’s technologies, offer new means of securing private information and foster the development of new sensors with precision yet unseen. In a short 10 years, the Institute for Quantum Computing (IQC) at the University of Waterloo has become a world-renowned institute for research in the quantum world. With more than 160 researchers, we are well on our way to reaching our goal of 33 faculty, 60 post doctoral fellows and 165 students. The research has been world class and many results have received international attention. We have recruited some of the world’s leading researchers and rising stars in the field. 2012 has been a landmark year. Not only did we celebrate our 10th anniversary, but we also expanded into our new headquarters in the Mike and Ophelia Lazaridis Quantum-Nano Centre in the heart of the University of Waterloo campus. This 285,000 square foot facility provides the perfect environment to continue our research, grow our faculty complement and attract the brightest students from around the globe. In this report, you will see many examples of the wonderful achievements we’ve celebrated this year. IQC researcher Andrew Childs and his team proposed a new computational model that has the potential to become an architecture for a scalable quantum computer. -
Arxiv:1803.00026V4 [Cond-Mat.Str-El] 8 Feb 2019 3 Variational Quantum-Classical Simulation (VQCS) 5
SciPost Physics Submission Efficient variational simulation of non-trivial quantum states Wen Wei Ho1*, Timothy H. Hsieh2,3, 1 Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA 2 Kavli Institute for Theoretical Physics, University of California, Santa Barbara, California 93106, USA 3 Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada *[email protected] February 12, 2019 Abstract We provide an efficient and general route for preparing non-trivial quantum states that are not adiabatically connected to unentangled product states. Our approach is a hybrid quantum-classical variational protocol that incorporates a feedback loop between a quantum simulator and a classical computer, and is experimen- tally realizable on near-term quantum devices of synthetic quantum systems. We find explicit protocols which prepare with perfect fidelities (i) the Greenberger- Horne-Zeilinger (GHZ) state, (ii) a quantum critical state, and (iii) a topologically ordered state, with L variational parameters and physical runtimes T that scale linearly with the system size L. We furthermore conjecture and support numeri- cally that our protocol can prepare, with perfect fidelity and similar operational costs, the ground state of every point in the one dimensional transverse field Ising model phase diagram. Besides being practically useful, our results also illustrate the utility of such variational ans¨atzeas good descriptions of non-trivial states of matter. Contents 1 Introduction 2 2 Non-trivial quantum states