DOCSLIB.ORG

Quantum Error Correction

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Quantum Error Correction Indian Institute of Technology, Kanpur CS682A Quantum Computation Quantum Error Correction Author: Roll Numbers: Pranav Bisht 17111268 Samik Some 17111271 November 15, 2017 Contents 1 Introduction . .1 2 Bit flip code . .2 2.1 Encoding: . .2 2.2 Error Detection: . .3 2.3 Error Recovery: . .4 3 Phase Flip Code . .4 4 Alternate Look At Syndrome Measurements . .5 5 The Shor Code [Sho95] . .6 5.1 Bit Flip Errors . .7 5.2 Phase Flip Errors . .7 5.3 Combined Bit and Phase Flip Errors . .8 6 Classical Linear Codes . .9 6.1 Generator Matrix . .9 6.2 Parity Check Matrix . .9 6.3 Error Correction . 10 7 CSS codes . 11 7.1 Error Detection and Correction . 12 8 The Steane Code [Ste96] . 14 9 Conclusion . 15 1 Abstract This document serves as a project report on Quantum Error Correcting codes (QEC) done as part of the course Quantum Computation. Efficient and reliable communication has posed a challenge to mankind since ages, eventually leading to birth of Information Technology. This field received a great boost after the advent of Internet with high demand for time and space efficient error correcting codes. Now, at the dawn of Quantum computation, which with every day is coming closer to being a feasible reality, researchers have explored error correcting schemes for reliable transfer of qubits. We primarily focus on Quantum codes and study the classical codes briefly to the extent required for the presentation here. We start with simple repetition code and its quantum equivalent, the bit flip and phase flip codes. We show how Shor code combines both of these. We also look at a broader class of QECs called CSS codes and study Steane code in its light. Pranav Bisht Quantum Error Correction Samik Some 1 Introduction In the classical world, error correcting codes have been extensively studied and developed to ensure reliable communication across a noisy channel. Variety of error correcting schemes exist catering to different requirements like Parity codes, Hamming Codes, Reed Solomon codes et cetera. The primary requirements these codes try to achieve are that of time efficiency with respect to encoding and decoding a message, while also trying to save on the number of extra bits used in the transfer of message. However, in this report our primary focus is to explore Error Correcting algorithms in the Quantum world, highlighting the potential challenges they face and how they are cleverly handled. Quantum error correcting codes derive main idea from their classical counterparts. We will briefly discuss simple repetition code and classical linear codes [6], in order to understand QECs. We will be discussing qubit flip codes [2], phase flip codes [3], Shor's code [5], Steane's code [8] and the general framework of Calderbank-Shor-Steane (CSS) codes [7] in detail in this report. Every error correction scheme has 3 parts - Message Encoding, Error Detection, Error Correc- tion. In all these codes, we will try to describe all the 3 parts in one flow. On the way we will state and assume certain mathematical properties but we will prove them at the end of each section as lemmas, for ease of reading. The setting is as follows: Alice wants to send Bob a message. We can assume the message to be composed of bits (qubits in case of QEC). In real world the communication channel (air, wire) is noisy. Physical factors tend to corrupt the bits. Bob receives a corrupted or original message, but he does not know. How can Bob recover the original message? It is definitely not possible if the message was n bits and Alice sends exactly n bits. However, if Alice sends extra bits, Bob may be able to recover the message. How? Consider Alice wants to send a single bit 0 to Bob. Suppose we are assured that only a single bit error can occur during transmission. If Alice sends 000 as message. Then Bob can receive one of the following: 000 // no error 100 // error on 1st bit 010 // error on 2nd bit 001 // error on 3rd bit On receiving the message, Bob simply takes the majority (max of 0,1) in order to get the original message (in this case 0). Similarly, if Alice wished to send 1 she would instead send 111, and Bob could decode the message appropriately. This simple idea is exploited in Repetition codes. The key idea in any kind of error correction scheme is to append “sufficient” number of redundant bits to the message. Researchers are always trying to optimize the extra bits needed. In fact, there is a dedicated field known as Coding Theory which studies these codes in detail providing theoretical bounds alongside. 1 Pranav Bisht Quantum Error Correction Samik Some Let us now think about QECs. Alice wants to send Bob qubits. Noise in the channel corrupts qubits. How does Bob recover the original qubits? A careful thinker will be stumped by these challenges first: • No cloning: The moment we hear repeated copies, it should raise an alarm in the quantum world, because of the no cloning theorem. The laws of quantum mechanics don't allow copying of qubits. So we cannot simply send 3 copies of qubits to send a single qubit reliably. • Continuous errors: In the classical world, the only kind of error is bit flip. But here a continuum of different errors may occur, due to infinite superposition states possible. Deciphering the order of errors seems to require infinite precision. • Measurement destroys the state: In classical error correction, Bob observes the message received and adopts decoding procedure accordingly. But in the case of qubits observing or measuring perturbs the quantum state. Despite these potential challenges, we will see how researchers deftly overcome these in the error correcting codes to follow. Also, in this writing, we assume that no errors are introduced while encoding or decoding which in the real world does happen. Nonetheless, the theory of fault tolerant quantum computation ensures that they are not a significant setback. 2 Bit flip code As of now, assume that there are only two types of qubit errors: bit flip and phase flip, described by unitary matrices X and Z respectively. Throughout this report, we will concern ourselves with reliable transmission of just a single qubit from Alice to Bob. In this section, we will show how to correct single qubit bit flip error and in the next section we will show the same for qubit phase flip error. Later we will discuss why we can correct any kind of quantum error if we are able to correct both bit flip and phase flip errors. Single bit flip error is described by: j0i ! j1i j1i ! j0i This is the same as action of Pauli matrix X. 2.1 Encoding: Suppose Alice wants to send qubit Ψ = a j0i + b j1i. Similar to the repetition code, we will append two qubits j00i to Ψ, giving us Ψ1. Note that we bypassed the first challenge as we did not copy the state Ψ. We merely appended two ancilla bits which is always possible. 2 Pranav Bisht Quantum Error Correction Samik Some Figure 1: Bit Flip Code Encoding [Wik17] Ψ0 ≡ (a j0i + b j1i) j0i j0i ≡ a j000i + b j100i Ψ1 ≡ a j000i + b j111i 2.2 Error Detection: Assuming that only a single qubit flip can occur during transmission, Bob will receive one of these 4 possible states: Φ0 ≡ a j000i + b j111i // No error st Φ1 ≡ a j100i + b j011i // 1 qubit flip nd Φ2 ≡ a j010i + b j101i // 2 qubit flip rd Φ3 ≡ a j001i + b j110i // 3 qubit flip Let us define the following 4 Projection operators: P0 ≡ j000i h000j + j111i h111j // No error st P1 ≡ j100i h100j + j011i h011j // 1 qubit flip nd P2 ≡ j010i h010j + j101i h101j // 2 qubit flip rd P3 ≡ j001i h001j + j110i h110j // 3 qubit flip Define measurement M = 0P0 + 1P1 + 2P2 + 3P3. Lemma 2.1 proves that fP0;P1;P2;P3g indeed form a valid measurement. Observe that MΦi = i 8i = 0 ::: 3, since ( 1 if i = j Pi jΦji = 0 otherwise This detects which qubit was flipped during the transmission. Note that we chose our pro- jections for measurement cleverly so that the state before and after the measurement is same jPiΦii Φi = 2 . In this way the third challenge is taken care of by measuring in superposition. kPiΦik 3 Pranav Bisht Quantum Error Correction Samik Some Figure 2: Bit Flip Code Encoding and Decoding [Wik17] 2.3 Error Recovery: Once the error syndrome is detected, error recovery is very simple: • No error (0)) Do nothing. • 1st qubit flip (1)) Flip 1st qubit again. • 2nd qubit flip (2)) Flip 2nd qubit again. • 3rd qubit flip (3)) Flip 3rd qubit again. The overall encoding, decoding quantum circuit is shown in Figure 2 Lemma 2.1. The four projectors P0;P1;P2;P3 defined in section 2.2 form a valid measurement Proof. It is easy to verify that 3 X Pi = I8×8 i=0 Pi · Pj = 0 ; 8i 6= j Hence, they form a valid projective measurement. 3 Phase Flip Code Now, we will see how to handle only a single qubit phase flip which might creep in during transmission. Phase flip error is equivalent to action of Pauli matrix Z. a j0i + b j1i ! a j0i − b j1i Here Hadamard gate comes to our rescue again. The neat idea is that phase flip error in the j+i ; |−i basis is same as bit flip error: j+i ! |−i |−i ! j+i This simple idea gives the whole algorithm for phase flip errors: 4 Pranav Bisht Quantum Error Correction Samik Some Figure 3: Phase Flip Code Encoding [Wik17] Figure 4: Phase Flip Encoding and Decoding [Wik17] • Append ancilla bits: a j0i + b j1i ! a j000i + b j111i • Apply Hadamard: a j000i + b j111i ! a j+ + +i + b |− − −i • Apply bit flip algorithm to detect and correct a bit flip error in the j+i ; |−i basis.
Load more

Recommended publications
  • A New Family of Fault Tolerant Quantum Reed-Muller Codes
    Clemson University TigerPrints All Theses Theses December 2020 A New Family of Fault Tolerant Quantum Reed-Muller Codes Harrison Beam Eggers Clemson University, [email protected] Follow this and additional works at: https://tigerprints.clemson.edu/all_theses Recommended Citation Eggers, Harrison Beam, "A New Family of Fault Tolerant Quantum Reed-Muller Codes" (2020). All Theses. 3463. https://tigerprints.clemson.edu/all_theses/3463 This Thesis is brought to you for free and open access by the Theses at TigerPrints. It has been accepted for inclusion in All Theses by an authorized administrator of TigerPrints. For more information, please contact [email protected]. A New Family of Fault Tolerant Quantum Reed-Muller Codes A Thesis Presented to the Graduate School of Clemson University In Partial Fulfillment of the Requirements for the Degree Master of Science Mathematics by Harrison Eggers December 2020 Accepted by: Dr. Felice Manganiello, Committee Chair Dr. Shuhong Gao Dr. Kevin James Abstract Fault tolerant quantum computation is a critical step in the development of practical quan- tum computers. Unfortunately, not every quantum error correcting code can be used for fault tolerant computation. Rengaswamy et. al. define CSS-T codes, which are CSS codes that admit the transversal application of the T gate, which is a key step in achieving fault tolerant computation. They then present a family of quantum Reed-Muller fault tolerant codes. Their family of codes admits a transversal T gate, but the asymptotic rate of the family is zero. We build on their work by reframing their CSS-T conditions using the concept of self-orthogonality.
    [Show full text]
  • (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.
    [Show full text]
  • 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.
    [Show full text]
  • Fault-Tolerant Quantum Gates Ph/CS 219 2 February 2011
    Fault-tolerant quantum gates Ph/CS 219 2 February 2011 Last time we considered the requirements for fault-tolerant quantum gates that act nontrivially on the codespace of a quantum error-correcting code. In the special case of a code that corrects t=1 error, the requirements are: -- if the gate gadget is ideal (has no faults) and its input is a codeword, then the gadget realizes the encoded operation U acting on the code space. -- if the gate gadget is ideal and its input has at most one error (is one-deviated from the codespace), then the output has at most one error in each output block. -- if the gate has one fault and its input has no errors, then the output has at most one error in each block (the errors are correctable). We considered the Clifford group, the finite subgroup of the m-qubit unitary group generated by the Hadamard gate H, the phase gate P (rotation by Pi/2 about the z-axis) and the CNOT gate. For a special class of codes, the generators of the Clifford group can be executed transversally (i.e., bitwise). The logical U can be done by applying a product of n U (or inverse of U) gates in parallel (where n is the code's length). If we suppose that the number of encoded qubits is k=1, then: -- the CNOT gate is transversal for any CSS code. -- the H gate is transversal for a CSS code that uses the same classical code to correct X errors and Z errors.
    [Show full text]
  • 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.
    [Show full text]
  • Arxiv:2104.09539V1 [Quant-Ph] 19 Apr 2021
    Practical quantum error correction with the XZZX code and Kerr-cat qubits Andrew S. Darmawan,1, 2 Benjamin J. Brown,3 Arne L. Grimsmo,3 David K. Tuckett,3 and Shruti Puri4, 5 1Yukawa Institute for Theoretical Physics (YITP), Kyoto University, Kitashirakawa Oiwakecho, Sakyo-ku, Kyoto 606-8502, Japan∗ 2JST, PRESTO, 4-1-8 Honcho, Kawaguchi, Saitama 332-0012, Japan 3Centre for Engineered Quantum Systems, School of Physics, University of Sydney, Sydney, New South Wales 2006, Australia 4 Department of Applied Physics, Yale University, New Haven, Connecticut 06511, USAy 5Yale Quantum Institute, Yale University, New Haven, Connecticut 06511, USA (Dated: April 21, 2021) The development of robust architectures capable of large-scale fault-tolerant quantum computa- tion should consider both their quantum error-correcting codes, and the underlying physical qubits upon which they are built, in tandem. Following this design principle we demonstrate remarkable error correction performance by concatenating the XZZX surface code with Kerr-cat qubits. We contrast several variants of fault-tolerant systems undergoing different circuit noise models that reflect the physics of Kerr-cat qubits. Our simulations show that our system is scalable below a threshold gate infidelity of pCX 6:5% within a physically reasonable parameter regime, where ∼ pCX is the infidelity of the noisiest gate of our system; the controlled-not gate. This threshold can be reached in a superconducting circuit architecture with a Kerr-nonlinearity of 10MHz, a 6:25 photon cat qubit, single-photon lifetime of > 64µs, and thermal photon population < 8%.∼ Such parameters are routinely achieved in superconducting∼ circuits. ∼ I. INTRODUCTION qubit [22, 34, 35].
    [Show full text]
  • 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.
    [Show full text]
  • 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]]
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • Graphical and Programming Support for Simulations of Quantum Computations
    AGH University of Science and Technology in Kraków Faculty of Computer Science, Electronics and Telecommunications Institute of Computer Science Master of Science Thesis Graphical and programming support for simulations of quantum computations Joanna Patrzyk Supervisor: dr inż. Katarzyna Rycerz Kraków 2014 OŚWIADCZENIE AUTORA PRACY Oświadczam, świadoma odpowiedzialności karnej za poświadczenie nieprawdy, że niniejszą pracę dyplomową wykonałam osobiście i samodzielnie, i nie korzystałam ze źródeł innych niż wymienione w pracy. ................................... PODPIS Akademia Górniczo-Hutnicza im. Stanisława Staszica w Krakowie Wydział Informatyki, Elektroniki i Telekomunikacji Katedra Informatyki Praca Magisterska Graficzne i programowe wsparcie dla symulacji obliczeń kwantowych Joanna Patrzyk Opiekun: dr inż. Katarzyna Rycerz Kraków 2014 Acknowledgements I would like to express my sincere gratitude to my supervisor, Dr Katarzyna Rycerz, for the continuous support of my M.Sc. study, for her patience, motivation, enthusiasm, and immense knowledge. Her guidance helped me a lot during my research and writing of this thesis. I would also like to thank Dr Marian Bubak, for his suggestions and valuable advices, and for provision of the materials used in this study. I would also thank Dr Włodzimierz Funika and Dr Maciej Malawski for their support and constructive remarks concerning the QuIDE simulator. My special thank goes to Bartłomiej Patrzyk for the encouragement, suggestions, ideas and a great support during this study. Abstract The field of Quantum Computing is recently rapidly developing. However before it transits from the theory into practical solutions, there is a need for simulating the quantum computations, in order to analyze them and investigate their possible applications. Today, there are many software tools which simulate quantum computers.
    [Show full text]
  • 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.
    [Show full text]