# Quantum Error Correction Pdf

## Contents |

It was wellknown even before the introduction of quantum informa-tion that coherent quantum states were extremely fragileand many believed that to maintain large, multi-qubit,coherent quantum states for a long enough time 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. Unlikeclassical information processing, conservation of proba-bility for quantum states require that all operations bereversible and hence unitary.When describing a quantum gate on an individualqubit, any dynamical operation, G, is a member The interpretation ofthese two types of initialization models is identical to thecoherent and incoherent models presented. weblink

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 way, errors only perturbcodeword states by small amounts which can then bedetected and corrected, without directly measuring thequantum state of any qubit.III. The result of the measurement will **then dictate ifan** X correction gate needs to be applied to a speciﬁcqubit, i.e.Ancilla Measurement: |00i, Collapsed State: α |000i+ β |111i ∴ Clean StateAncilla Insection IX we then brieﬂy return to the noise modelsand relate the abstract analysis of QEC, where errorsare assumed to be discrete and probabilistic, to some ofthe physical mechanisms which can

## Quantum Error Correction Codes

Sec-tions X and XI introduces the concept of fault-toleranterror correction, the threshold theorem and how logi-cal gate operations can be applied directly to quantumdata. In the case of ion-traps, qubit transitions are performed by focusing ﬁnelytuned lasers resonant on the relevant transitions. The Nine Qubit Code: The First Full Quantumcode 9VI. Please **try the request again. **

- For ex-ample, a phase ﬂip on qubits one, two, four and ﬁve willleave both logical states unchanged.
- Single Qubit Operations 19B.
- Instead the subroutine canbe reset and re-run.VII.
- 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
- As with the previous algorithm weshould measure the state |0i with probability one.
- Subsystem codes arevery nice codes from an architectural point of view.

fault-tolerant circuit design for logical statepreparation 21XIII. Once again we willtake a very basic example of a decoherence model andexamine how it inﬂuences our trivial algorithm. Error Correction 14IX. 5 Qubit Code Please try the request again.

Topological Codes 29XV. 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 Although the ﬁeld is largely basedon classical coding theory, there are several issues thatneed to be considered when transferring classical errorcorrection to the quantum regime.First, coding based on data-copying, which is https://www.researchgate.net/publication/240309280_Quantum_Error_Correction_for_Beginners Please try the request again.

The temporal interval of our identitygate in the above algorithm is long enough to enact thisfull controlled-ﬂip operation. Fault Tolerant Quantum Computation If one error has already been correctedthen the failure rate of the logical system is conditionalon experiencing one further error (which will be propor-tional to 2). Errorcorrection circuits and gates are **generally simpler thanfor non-subsystem codes,** allowing for circuit structuresmore amenable to the physical restrictions of a computerarchitecture (AC07). In general, leakage induced errors need to be cor-rected via the non-demolition detection of a leakage event(i.e.

## Quantum Error Correction For Beginners

The Shor code is a degeneratesingle error correcting code able to correct a logical qubitfrom one discrete bit ﬂip, one discrete phase ﬂip or one ofeach on any of the nine 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 Quantum Error Correction Codes Hencein loss correction protocols, an initial non-demolition de-tection method must be employed (which determines ifthe qubit is actually present without performing a pro-jective measurement on the computational state) beforestandard correction can Stabilizer Codes And Quantum Error Correction. The basic idea in post-selected schemes is to en-code the computer with error detecting circuits and if er-rors are detected, the relevant subroutine of the quantum 11algorithm is reset and run

If a particular system isfound to be improperly conﬁned to the qubit subspace itwould simply be discarded. http://vealcine.com/quantum-error/quantum-error-correction-usc.php When the qubit is in the |1istate, the coupling ﬂips the environmental state while ifthe qubit is in the |0i state nothing happens to the envi-ronment. These qubitsare then measured to obtain the classical syndrome re-sult. The two logical basisstates |0iLand |1iLare deﬁned as,|0iL= |000i, |1iL= |111i, (18)such that an arbitrary single qubit state |ψi = α |0i+β |1iis mapped to,α |0i + β |1i → α Steane Code

Employing characterizationat this stage would then eliminate the need to implementpulse control of leakage, shortening gate times and ulti-mately reducing error rates in the computer.In this section we introduced the basic 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 · While this model of qubit loss is equiva-lent to environmental coupling, correcting this type oferror requires additional machinery on top of standardQEC protocols. http://vealcine.com/quantum-error/quantum-error-correction-ppt.php Thisis not a necessary requirement, however more general in-coherent mappings would require us to move to densitymatrices.We assume that each qubit experiences the same error,hence the error operator acting on the

As mentioned, the X errorcorrection does have the ability to correct for up to threeindividual bit ﬂips (provided each bit ﬂip occurs in a dif-ferent block of three). Surface Code 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 II.

## Decoherence: The devil is in the environmentEnvironmental decoherence is another importantsource of errors in quantum systems.

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. We will return to thisissue in section X when we discuss Fault-tolerance. Section VII then introduces the stabilizerformalism (Got97), demonstrating how QEC circuits aresynthesized once the structure of the code is known. Quantum Code Reel Review In section IIIwe review some basic noise models from the context ofhow they inﬂuence quantum algorithms.

The two basis states forthe code are,|0iL=1√8(|000i + |111i)(|000i + |111i)(|000i + |111i)|1iL=1√8(|000i − |111i)(|000i− |111i)(|000i − |111i)(29)and the circuit to perform the encoding is shown in Fig. 4.Correction for X Thestructure of this ion is **clearly not a** 2-level quantum system.Hence leakage into non-qubit states is an important factor toconsider.eﬀects the system. Please try the request again. this content 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

Threshold Theorem 18XI. Your cache administrator is webmaster. This leads to a paradigmshift in the way we view and process information andhas lead to considerable interest from physicists, en-gineers, computer scientists and mathematicians. INTRODUCTIONThe micro-computer revolution of the late 20th centuryhas arguably been of greater impact to the world that anyother technological revolution in history.

This was the last theoretical aspectneeded to convince the general community that quantumcomputation was indeed a possibility. Consequentlythe probability of measuring the correct result at the endof a speciﬁc algorithm increases when the system is en-coded.This example shows the basic principles of error cor-rection.