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## Quantum Error Correction Codes

## Quantum Error Correction For Beginners

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Some Modern Developments in Error Correction 26A. Conclusions and future outlook 31XVI. Later insection IX we will illustrate a more complicated decoher-ence model that arises from standard mechanisms.Consider a very simple environment, which is anothertwo level quantum system. IX), which is asuﬃcient condition for correcting errors for an arbitraryerror mapping on a single qubit. news

Rather thanintroducing these concepts from a rigorous mathematical and computer science framework, weinstead examine error correction and fault-tolerance largely through detailed examples, progressingfrom basic examples of the 3-qubit code through to Therefore, wehave two choices: to keep trying to suppressing quan-tum eﬀects in classically fabricated electronics or moveto the ﬁeld of quantum information processing (QIP)where we instead exploit them. Table I summarizes the state of thewhole system, for each possible error, just prior to mea-surement.Error Location Final State, |datai |ancillaiNo Error α |000i |00i + β |111i |00iQubit 1 α State Preparation 13B. http://perimeterinstitute.ca/personal/dgottesman/QECC2007/

As 1 the majority of correction cycleswill detect no error and the ﬁdelity of the resulting en-coded state is higher than when unencoded. In this case, d = 3, hence t = 1.How are we able to correct errors using this code with-out directly measuring or obtaining information aboutthe logical state? Quantum Error Detection 10VII. This is the crux of how QECsuppresses errors at the logical level.

Your cache administrator is webmaster. We will return and revisit subsystemcodes later in section XIV.AVI. it is impossible to perfectly copy an unknown quan-tum state. 5 Qubit Code The resulting statewill contain a superposition **of a clean** state and cor-rupted states, the point is that the ﬁdelity of the cor-rupted states, at the logical level, is greater than the

How-ever, as we enter the 21st century, the rate at which com-putational power is increasing is driving us very quicklyto the realm of quantum physics. The distance between two codeword states, d, de-ﬁnes the number of errors that can be corrected, t, as,t = b(d − 1)/2c. This is dueto the fact that the code cannot simultaneously correctfor both bit and phase ﬂips (see section. If we assumed a controlledrotation that is not a full ﬂip on the environment, theﬁnal mixture will not be 50/50.

Themost well known technique of error avoidance is proto-cols such as decoherence free subspaces (DFS) (DG97;DG98b; ZR97b; ZR97a; DG98a; LW03a). Fault Tolerant Quantum Computation Stabilizer Formalism 11VIII. The speciﬁc methodology depends largely on thephysical mechanisms used to initialize the system. fault-tolerant Quantum Error Correction and thethreshold theorem. 17A.

For example, in the above case, theunwanted level, |2i, may be extremely short lived leadingto an emission of a photon and the system relaxing backto the ground state. This code is a standardrepetition code which was extended by Shor (Sho95) tothe full 9-qubit quantum code which was the ﬁrst demon-stration that QEC was possible.The 3-qubit code encodes a single Quantum Error Correction Codes Again, theeﬀect of these types of errors relates to the probabilitiesof measuring the system in an erred state.One ﬁnal type of error that we can brieﬂy mention isthe problem of qubit Stabilizer Codes And Quantum Error Correction. In the case of ion-traps, qubit transitions are performed by focusing ﬁnelytuned lasers resonant on the relevant transitions.

Weexamine, independently, several common sources of errorfrom the eﬀect they have on this simple quantum algo-rithm. navigate to this website Topological Codes 29XV. This will thenlead to a description of full fault-tolerant quantum errorcorrection.V. However wenow ﬁnd,P (|0i) = cos2(N) ≈ 1 − (N)2,P (|1i) = sin2(N) ≈ (N )2.(7)Hence, the probability of error in this trivial quantumalgorithm is given by perror≈ (N)2, which will Steane Code

- We will return to thisissue in section X when we discuss Fault-tolerance.
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- Another approach isto not enforce the correction requirement.
- As with QEC, thisﬁeld of error avoidance is vast, now incorporating ideasfrom optimal control to create specially designed controlsequences to counteract the eﬀect of errors induced fromenvironmental coupling.

Here are the **instructions how to** enable JavaScript in your web browser. Finally in section XIV we review speciﬁccodes for qubit loss and examine two of the more mod-ern techniques for error correction. As quantumcircuits and algorithms are fundamentally designed as-suming the computational array is a collection of 2-levelsystems, operators of the above form (which in this caseis operating over a 3-level space) will http://caribtechsxm.com/quantum-error/quantum-error-correction-usc.php Generated Tue, 25 Oct 2016 02:52:47 GMT by s_wx1157 (squid/3.5.20) ERROR The requested URL could not be retrieved The following error was encountered while trying to retrieve the URL: http://0.0.0.10/ Connection

The four basis statesfor the code are,|00i =1√2(|0000i + |1111i),|01i =1√2(|1100i + |0011i),|10i =1√2(|1010i + |0101i),|11i =1√2(|0110i + |1001i).(30)Fig. 6 illustrates the error detection circuit that can beutilized to detect a Surface Code See all ›47 CitationsSee all ›171 ReferencesShare Facebook Twitter Google+ LinkedIn Reddit Download Full-text PDF Quantum Error Correction for BeginnersArticle (PDF Available) in Reports on Progress in Physics 76(7):076001 · June 2013 with 169 ReadsDOI: 10.1088/0034-4885/76/7/076001 · Subsystem Codes 26B.

Unfor-tunately the current designs for micro-electronics meanthat quantum mechanical behavior will tend to resultin unpredictable and unwanted behavior. Systematic gate errors 15B. We assume that after encoding a single bit-ﬂip occurs onone of the three qubits (or no error occurs). Quantum Code Reel Review Please try the request again.

The advent ofarXiv:0905.2794v2 [quant-ph] 5 Jun 2009 2transistors, integrated circuits, and the modern micro-processor has spawned literally hundreds of devices frompocket calculators to the iPod, all now integrated throughan extensive worldwide Generated Tue, 25 Oct 2016 02:52:47 GMT by s_wx1157 (squid/3.5.20) An arbitrary state of an individual qubit, |φi,can be expressed as,|φi = α |0i + β |1i (1)where normalization requires, |α|2+ |β|2= 1. http://caribtechsxm.com/quantum-error/quantum-error-correction.php Therefore the projector ontothe qubit space is given by A =12(|0ih0|+|1ih1|), which isidentical to simply tracing over the lost qubit and equiv-alent to a measurement error of probability p = 0.5.With

The system returned: (22) Invalid argument The remote host or network may be down. Ifwe consider the unitary U acting on a single, unencodedqubit, the rotation takes,U |ψi = cos() |ψi + i sin()σx|ψi, (25)Consequently, the ﬁdelity of the single qubit state is,Funencoded= |hψ|U |ψi|2= Since this initialintroduction, the progress in this ﬁeld has been extensive.Initial work on error correction focused heavily on de-veloping quantum codes (Ste96b; CG97; Got96; PVK97),introducing a more rigorous theoretical framework forthe The system returned: (22) Invalid argument The remote host or network may be down.

Subsystem codes arevery nice codes from an architectural point of view. Even if a bit and phase error occurs on thesame qubit, the X correction circuit will detect and cor-rect for bit ﬂips while the Z correction circuit will detectand correct for Hopefully, this introductory section will show thatwhile quantum errors are complicated physical eﬀects, inQIP the relevant measure is the theoretical success prob-ability of a given quantum algorithm.A. If a particular system isfound to be improperly conﬁned to the qubit subspace itwould simply be discarded.

Note that a phase ﬂip on any onequbit in a block of three has the same eﬀect, this is whythe 9-qubit code is referred to as a degenerate code. In this case, theinitial state is pure, but contains a non-zero amplitudeof the undesired target, for example,|ψii= α |0i + β |1i (16)where |α|2+|β|2= 1 and |β|2 1. However, it should be emphasized that the 3-qubitcode does not represent a full quantum code. The 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

Loss Protection 22XIV. determining if the quantum system is conﬁned toa qubit without performing a measurement discriminat-ing the |0i and |1i states (Pre98; GBP97; VWW05)) orthrough the use of complicated pulse control which actsto A 9superposition of a “clean state” with the logically ﬂippedstate, σxσxσx|ψi. Generated Tue, 25 Oct 2016 02:52:47 GMT by s_wx1157 (squid/3.5.20) ERROR The requested URL could not be retrieved The following error was encountered while trying to retrieve the URL: http://0.0.0.6/ Connection

The introduction of quantum error correction in 1995 showed that active techniques could be employed to mitigate this fatal problem. The important thing to notice is theamplitudes related to the terms in the superposition. We hope that this reviewof the basic aspects of QEC and fault-tolerance will allowthose with little knowledge of the ﬁeld to quickly becomeaccustomed to the various techniques and tricks that arecommonly The last two qubits represent the stateof the ancilla.

Two additional ancilla qubits are in-troduced, which are used to extract syndrome informa-tion (information regarding possible errors) from the datablock without discriminating the exact state of any qubit,Fig. 3 illustrates. The reason why thisFIG. 2 Quantum Circuit to prepare the |0iLstate for the 3-qubit code where an arbitrary single qubit state, |ψi is coupledto two freshly initialized ancilla qubits via CNOT This environment has twobasis states, |e0i and |e1i which satisﬁes the completenessrelations,hei|eji = δij, |e0ihe0| + |e1ihe1| = I. (8)We will also assume that the environment couples to thequbit in a