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Quantum error correction
KTH, School of Engineering Sciences (SCI), Applied Physics, Quantum Electronics and Quantum Optics, QEO.ORCID iD: 0000-0002-8721-3580
2012 (English)Licentiate thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis intends to familiarise the reader with quantum error correction, and also show some relations to the well known concept of information - and the lesser known quantum information. Quantum information describes how information can be carried by quantum states, and how interaction with other systems give rise to a full set of quantum phenomena, many of which have no correspondence in classical information theory. These phenomena include decoherence, as a consequence of entanglement. Decoherence can also be understood as "information leakage", i.e., knowledge of an event is transferred to the reservoir - an effect that in general destroys superpositions of pure states.

It is possible to protect quantum states (e.g., qubits) from interaction with the environment - but not by amplification or duplication, due to the "no-cloning" theorem. Instead, this is done using coding, non-demolition measurements, and recovery operations. In a typical scenario, however, not all types of destructive events are likely to occur, but only those allowed by the information carrier, the type of interaction with the environment, and how the environment "picks up" information of the error events. These characteristics can be incorporated into a code, i.e., a channel-adapted quantum error-correcting code. Often, it is assumed that the environment's ability to distinguish between error events is small, and I will denote such environments "memory-less".

 This assumption is not always valid, since the ability to distinguish error events is related to the \emph{temperature} of the environment, and in the particular case of information coded onto photons,  typically holds, and one must then assume that the environment has a "memory". In this thesis, I describe a short quantum error-correcting code (QECC), adapted for photons interacting with a cold environment, i.e., this code protects from an environment that continuously records which error occurred in the coded quantum state.

Also, it is of interest to compare the performance of different QECCs - But which yardstick should one use? We compare two such figures of merit, namely the quantum mutual information and the quantum fidelity, and show that they can not, in general, be simultaneously maximised in an error correcting procedure. To show this, we have used a five-qubit perfect code, but assumed a channel that only cause bit-flip errors. It appears that quantum mutual information is the better suited yardstick of the two, however more tedious to calculate than quantum fidelity - which is more commonly used.

Abstract [sv]

Denna avhandling är en introduktion till kvantfelrättning, där jag undersöker släktskapet med teorin om klassisk information - men också det mindre välkända området kvantinformation. Kvantinformation beskriver hur information kan bäras av kvanttillstånd, och hur växelverkan med andra system ger upphov till åtskilliga typer av fel och effekter, varav många saknar motsvarighet i den klassiska informationsteorin. Bland dessa effekter återfinns dekoherens - en konsekvens av s.k. sammanflätning. Dekoherens kan också förstås som "informationsläckage", det vill säga att kunskap om en händelse överförs till omgivningen - en effekt som i allmänhet förstör superpositioner i rena kvanttillstånd.

 Det är möjligt att med hjälp av kvantfelrättning skydda kvanttillstånd (t.ex. qubitar) från omgivningens påverkan, dock kan sådana tillstånd aldrig förstärkas eller dupliceras, p.g.a icke-kloningsteoremet. Tillstånden skyddas genom att införa redundans, varpå tillstånden interagerar med omgivningen. Felen identifieras m.h.a. icke-förstörande mätningar och återställs med unitära grindar och ancilla-tillstånd.Men i realiteten kommer inte alla tänkbara fel att inträffa, utan dessa begränsas av vilken informationsbärare som används, vilken interaktion som uppstår med omgivningen, samt hur omgivningen "fångar upp" information om felhändelserna. Med kunskap om sådan karakteristik kan man bygga koder, s.k. kanalanpassade kvantfelrättande koder. Vanligtvis antas att omgivningens förmåga att särskilja felhändelser är liten, och man kan då tala om en minneslös omgivning.

Antagandet gäller inte alltid, då denna förmåga bestäms av reservoirens temperatur, och i det speciella fall då fotoner används som informationsbärare gäller typiskt , och vi måste anta att reservoiren faktiskt har ett "minne". I avhandlingen beskrivs en kort, kvantfelrättande kod som är anpassad för fotoner i växelverkan med en "kall" omgivning, d.v.s. denna kod skyddar mot en omgivning som kontinuerligt registrerar vilket fel som uppstått i det kodade tillståndet.

 Det är också av stort intresse att kunna jämföra prestanda hos kvantfelrättande koder, utifrån någon slags "måttstock" - men vilken? Jag jämför två sådana mått, nämligen ömsesidig kvantinformation, samt kvantfidelitet, och visar att dessa i allmänhet inte kan maximeras samtidigt i en felrättningsprocedur. För att visa detta har en 5-qubitarskod använts i en tänkt kanal där bara bitflip-fel uppstår, och utrymme därför finns att detektera fel. Ömsesidig kvantinformation framstår som det bättre måttet, dock är detta mått betydligt mer arbetskrävande att beräkna, än kvantfidelitet - som är det mest förekommande måttet.

Place, publisher, year, edition, pages
Stockholm: KTH Royal Institute of Technology, 2012. , xviii, 66 p.
Series
Trita-FYS, ISSN 0280-316X ; 2012:19
Keyword [en]
quantum error correction, dissipative channel, decoherence, fidelity, mutual information
National Category
Physical Sciences
Identifiers
URN: urn:nbn:se:kth:diva-106795ISBN: 978-91-7501-317-6 (print)OAI: oai:DiVA.org:kth-106795DiVA: diva2:574131
Presentation
2012-12-19, sal FB53, AlbaNova Universitetscentrum, Kungl Tekniska högskolan, Roslagstullsbacken 21, Stockholm, 14:00 (English)
Opponent
Supervisors
Note

QC 20121206

Available from: 2012-12-06 Created: 2012-12-04 Last updated: 2012-12-06Bibliographically approved
List of papers
1. A short and efficient error correcting code for polarization coded photonic qubits in a dissipative channel
Open this publication in new window or tab >>A short and efficient error correcting code for polarization coded photonic qubits in a dissipative channel
2011 (English)In: Optics Communications, ISSN 0030-4018, E-ISSN 1873-0310, Vol. 284, no 1, 550-554 p.Article in journal (Refereed) Published
Abstract [en]

We propose a short and efficient non-degenerate quantum error correcting code that is adapted for qubits encoded on two orthogonal, single-photon states (e.g., horizontally and vertically polarized) subject to a dissipative channel. The proposed code draws its strength from the fact that it is adapted to the physical characteristics of the information-carrying basis states under the action of the channel. The code combines different energy manifolds and consists of only 3 spatio-temporal modes and on average 2 photons per code word.

Keyword
Quantum error correcting code, Photonic qubit, Dissipative channel
National Category
Atom and Molecular Physics and Optics
Identifiers
urn:nbn:se:kth:diva-30519 (URN)10.1016/j.optcom.2010.09.006 (DOI)000285893200100 ()2-s2.0-78649652279 (Scopus ID)
Note
QC 20110304Available from: 2011-03-04 Created: 2011-02-28 Last updated: 2017-12-11Bibliographically approved
2. Fidelity as a figure of merit in quantum error correction
Open this publication in new window or tab >>Fidelity as a figure of merit in quantum error correction
2013 (English)In: Quantum information & computation, ISSN 1533-7146, Vol. 13, no 1-2, 0009-0020 p.Article in journal (Refereed) Published
Abstract [en]

We discuss the fidelity as a figure of merit in quantum error correction schemes. We show that when identifiable but uncorrectable errors occur as a result of the action of the channel, a common strategy that improves the fidelity actually decreases the transmitted mutual information. The conclusion is that while the fidelity is simple to calculate and therefore often used, it is perhaps not always a recommendable figure of merit for quantum error correction. The reason is that while it roughly speaking encourages optimisation of the "mean probability of success", it gives no incentive for a protocol to indicate exactly where the errors lurk. For small error probabilities, the latter information is more important for the integrity of the information than optimising the mean probability of success.

Keyword
Fidelity, Mutual information, Quantum error correction
National Category
Atom and Molecular Physics and Optics
Identifiers
urn:nbn:se:kth:diva-107042 (URN)000315304700002 ()2-s2.0-84871693736 (Scopus ID)
Funder
Swedish Research Council
Note

QC 20130205. Updated from accepted to published.

Available from: 2012-12-06 Created: 2012-12-06 Last updated: 2017-12-07Bibliographically approved

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