On Quantum Codes and Networks
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
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. -
(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. -
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. -
Classical Zero-Knowledge Arguments for Quantum Computations
Classical zero-knowledge arguments for quantum computations Thomas Vidick∗ Tina Zhangy Abstract We show that every language in BQP admits a classical-verifier, quantum-prover zero-knowledge ar- gument system which is sound against quantum polynomial-time provers and zero-knowledge for classical (and quantum) polynomial-time verifiers. The protocol builds upon two recent results: a computational zero-knowledge proof system for languages in QMA, with a quantum verifier, introduced by Broadbent et al. (FOCS 2016), and an argument system for languages in BQP, with a classical verifier, introduced by Mahadev (FOCS 2018). 1 Introduction The paradigm of the interactive proof system is a versatile tool in complexity theory. Although traditional complexity classes are usually defined in terms of a single Turing machine|NP, for example, can be defined as the class of languages which a non-deterministic Turing machine is able to decide|many have reformulations in the language of interactive proofs, and such reformulations often inspire natural and fruitful variants on the traditional classes upon which they are based. (The class MA, for example, can be considered a natural extension of NP under the interactive-proof paradigm.) Intuitively speaking, an interactive proof system is a model of computation involving two entities, a verifier and a prover, the former of whom is computationally efficient, and the latter of whom is unbounded but untrusted. The verifier and the prover exchange messages, and the prover attempts to `convince' the verifier that a certain problem instance is a yes-instance. We can define some particular complexity class as the set of languages for which there exists an interactive proof system that 1) is complete, 2) is sound, and 3) has certain other properties which vary depending on the class in question. -
G53NSC and G54NSC Non Standard Computation Research Presentations
G53NSC and G54NSC Non Standard Computation Research Presentations March the 23rd and 30th, 2010 Tuesday the 23rd of March, 2010 11:00 - James Barratt • Quantum error correction 11:30 - Adam Christopher Dunkley and Domanic Nathan Curtis Smith- • Jones One-Way quantum computation and the Measurement calculus 12:00 - Jack Ewing and Dean Bowler • Physical realisations of quantum computers Tuesday the 30th of March, 2010 11:00 - Jiri Kremser and Ondrej Bozek Quantum cellular automaton • 11:30 - Andrew Paul Sharkey and Richard Stokes Entropy and Infor- • mation 12:00 - Daniel Nicholas Kiss Quantum cryptography • 1 QUANTUM ERROR CORRECTION JAMES BARRATT Abstract. Quantum error correction is currently considered to be an extremely impor- tant area of quantum computing as any physically realisable quantum computer will need to contend with the issues of decoherence and other quantum noise. A number of tech- niques have been developed that provide some protection against these problems, which will be discussed. 1. Introduction It has been realised that the quantum mechanical behaviour of matter at the atomic and subatomic scale may be used to speed up certain computations. This is mainly due to the fact that according to the laws of quantum mechanics particles can exist in a superposition of classical states. A single bit of information can be modelled in a number of ways by particles at this scale. This leads to the notion of a qubit (quantum bit), which is the quantum analogue of a classical bit, that can exist in the states 0, 1 or a superposition of the two. A number of quantum algorithms have been invented that provide considerable improvement on their best known classical counterparts, providing the impetus to build a quantum computer. -
Performance of Various Low-Level Decoder for Surface Codes in the Presence of Measurement Error
Air Force Institute of Technology AFIT Scholar Theses and Dissertations Student Graduate Works 3-2021 Performance of Various Low-level Decoder for Surface Codes in the Presence of Measurement Error Claire E. Badger Follow this and additional works at: https://scholar.afit.edu/etd Part of the Computer Sciences Commons Recommended Citation Badger, Claire E., "Performance of Various Low-level Decoder for Surface Codes in the Presence of Measurement Error" (2021). Theses and Dissertations. 4885. https://scholar.afit.edu/etd/4885 This Thesis is brought to you for free and open access by the Student Graduate Works at AFIT Scholar. It has been accepted for inclusion in Theses and Dissertations by an authorized administrator of AFIT Scholar. For more information, please contact [email protected]. Performance of Various Low-Level Decoders for Surface Codes in the Presence of Measurement Error THESIS Claire E Badger, B.S.C.S., 2d Lieutenant, USAF AFIT-ENG-MS-21-M-008 DEPARTMENT OF THE AIR FORCE AIR UNIVERSITY AIR FORCE INSTITUTE OF TECHNOLOGY Wright-Patterson Air Force Base, Ohio DISTRIBUTION STATEMENT A APPROVED FOR PUBLIC RELEASE; DISTRIBUTION UNLIMITED. The views expressed in this document are those of the author and do not reflect the official policy or position of the United States Air Force, the United States Department of Defense or the United States Government. This material is declared a work of the U.S. Government and is not subject to copyright protection in the United States. AFIT-ENG-MS-21-M-008 Performance of Various Low-Level Decoders for Surface Codes in the Presence of Measurement Error THESIS Presented to the Faculty Department of Electrical and Computer Engineering Graduate School of Engineering and Management Air Force Institute of Technology Air University Air Education and Training Command in Partial Fulfillment of the Requirements for the Degree of Master of Science in Computer Science Claire E Badger, B.S.C.S., B.S.E.E. -
Arxiv:0905.2794V4 [Quant-Ph] 21 Jun 2013 Error Correction 7 B
Quantum Error Correction for Beginners Simon J. Devitt,1, ∗ William J. Munro,2 and Kae Nemoto1 1National Institute of Informatics 2-1-2 Hitotsubashi Chiyoda-ku Tokyo 101-8340. Japan 2NTT Basic Research Laboratories NTT Corporation 3-1 Morinosato-Wakamiya, Atsugi Kanagawa 243-0198. Japan (Dated: June 24, 2013) Quantum error correction (QEC) and fault-tolerant quantum computation represent one of the most vital theoretical aspect of quantum information processing. It was well known from the early developments of this exciting field that the fragility of coherent quantum systems would be a catastrophic obstacle to the development of large scale quantum computers. The introduction of quantum error correction in 1995 showed that active techniques could be employed to mitigate this fatal problem. However, quantum error correction and fault-tolerant computation is now a much larger field and many new codes, techniques, and methodologies have been developed to implement error correction for large scale quantum algorithms. In response, we have attempted to summarize the basic aspects of quantum error correction and fault-tolerance, not as a detailed guide, but rather as a basic introduction. This development in this area has been so pronounced that many in the field of quantum information, specifically researchers who are new to quantum information or people focused on the many other important issues in quantum computation, have found it difficult to keep up with the general formalisms and methodologies employed in this area. Rather than introducing these concepts from a rigorous mathematical and computer science framework, we instead examine error correction and fault-tolerance largely through detailed examples, which are more relevant to experimentalists today and in the near future.