# Quantum Error Correction Tutorial

## Contents |

Therefore, we must be careful and use some sort of technique to verify the cat state, for instance by checking if random pairs of qubits are the same. Therefore, if p is below the threshold pt, we can achieve an arbitrarily good error rate ε per logical gate or timestep using only poly(logε) resources, which is excellent theoretical scaling. AT&T Bell Laboratories. ^ A.R.Calderbank E.M.Rains P.W.Shor and N.J.A.Sloane "Quantum Error Correction Via Codes Over GF(4)"IEEE.Transactions on Information Theory,Vol.44,No.4,July 1998 ^ D. In particular, each coset S ⊥ /S corresponds to a different logical Pauli operator (with S itself corresponding to the identity). weblink

Cory, **M. **The salient point in these error-correction conditions is that the matrix element Cab does not depend on the encoded basis states i and j, which roughly speaking indicates that neither the L. Frank Gaitan (2008). "Quantum Error Correction and Fault Tolerant Quantum Computing". https://en.wikipedia.org/wiki/Quantum_error_correction

## Quantum Error Correction For Beginners

Copying quantum information is not possible due to the no-cloning theorem. A generalisation of this concept are the CSS codes, named for their inventors: A. Peter Shor first discovered this method of formulating a quantum error correcting code by storing the information of one qubit onto a highly entangled state of nine qubits.

A quantum error correcting code protects quantum information against errors of a limited form. Please try the request again. In the case of certain codes, such as the 7-qubit code, a number of different gates can be performed transversally. Steane Code Through the transmission in a channel the relative sign between | 0 ⟩ {\displaystyle |0\rangle } and | 1 ⟩ {\displaystyle |1\rangle } can become inverted.

When this is true, C1 and C2 define an [[n, k1 + k2 − n, d]] stabilizer code, where d ≥ min(d1, d2). Quantum Error Correction Codes To determine whether a given subspace **is able to correct a** given set of errors, we can apply the quantum error-correction conditions: Theorem 1 A QECC C corrects the set of The system returned: (22) Invalid argument The remote host or network may be down. Then we complete the operation with a further transversal gate which depends on the outcome of the measurement.

Define the dual C ⊥ of a classical code C as the set of vectors w s.t. Quantum Error Correction Book R. If this condition is **satisfied, t separate single-qubit or** single-gate failures are required for a distance 2t + 1 code to fail. For lower physical error rates, overhead requirements are more modest, particularly if we only attempt to optimize for calculations of a given size, but are still larger than one would like.

- Furthermore, these calculations make a number of assumptions about the physical properties of the computer.
- Then by comparing the definition of distance with the quantum error-correction conditions, we immediately see that a QECC corrects t general errors iff its distance d > 2t.
- Some stabilizer codes have interesting symmetries under the action of certain Clifford group elements, and these symmetries result in transversal gate operations.
- Steane (Submitted on 2 Apr 2003 (v1), last revised 3 Apr 2003 (this version, v2)) Abstract: The main ideas of quantum error correction are introduced.
- In general, a gate coupling pairs of qubits allows errors to spread in both directions across the coupling.
- Classical error correcting codes use a syndrome measurement to diagnose which error corrupts an encoded state.
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## Quantum Error Correction Codes

Found. see here Category:Introductory Tutorials Category:Quantum Error Correction Category:Handbook of Quantum InformationLast modified:Monday, October 26, 2015 - 17:56 For full functionality of ResearchGate it is necessary to enable JavaScript. Quantum Error Correction For Beginners C. Stabilizer Codes And Quantum Error Correction. Unfortunately, the practical requirements for this result are not nearly so good.

Math. have a peek at these guys Please try the request again. In addition, CSS codes **have some** very useful properties which make them excellent choices for fault-tolerant quantum computation. Barrett, R. 5-qubit Quantum Error Correction

MoussaRead full-textSimulation of Quantum Error Correction with Mathematica[Show abstract] [Hide abstract] ABSTRACT: In this paper we consider the problem of quantum error correction and its simulation with the computer algebra system To further lower the logical error rate, we turn to a family of codes known as concatenated codes. The basic design principle of a fault-tolerant protocol is that an error in a single location --- either a faulty gate or noise on a quiescent qubit --- should not be http://vealcine.com/quantum-error/quantum-error-correction-usc.php A transversal operation has the virtue that an error occurring on the 3rd qubit in a block, say, can only ever propagate to the 3rd qubit of other blocks of the

The advantage of this procedure is that it measures just M and nothing more. 5 Qubit Code Chwalla, **M. **Then the stabilizer for a code becomes a pair of (n − k) × n binary matrices, and most interesting properties can be determined by an appropriate linear algebra exercise.

## It could be used in conjunction with the above tutorial article.

The simplest method, due to Shor, is very general but also requires the most overhead and is frequently the most susceptible to noise. Pittman, B. A number of different techniques have been developed. Fault-tolerant Quantum Computation the Shor code, encodes 1 logical qubit in 9 physical qubits and can correct for arbitrary errors in a single qubit.

Definition 2 The distance d of an ((n, K)) is the smallest weight of a nontrivial Pauli operator E ∈ Pn s.t. Sundar RajanRead full-textData provided are for informational purposes only. Quantum circuit of the bit flip code Let | ψ ⟩ = α 0 | 0 ⟩ + α 1 | 1 ⟩ {\displaystyle |\psi \rangle =\alpha _{0}|0\rangle +\alpha _{1}|1\rangle } this content H.

The procedure is transversal, so an error on a single qubit in the initial cat state or in a single gate during the interaction will only produce one error in the Comments: 12 pages, a very simple tutorial introduction to error correction, with emphasis on avoiding misconceptions regarding the treatment of noise Subjects: Quantum Physics (quant-ph) Journalreference: A. Here R {\displaystyle {\mathcal {R}}} is known as the correction operation. In either case, the final state still tells us nothing about the data beyond the eigenvalue of M.

The first (3 bit) example in the tutorial is worked through in this (web-converted) powerpoint presentation (very basic level, final year undergraduate or early graduate). The Pauli group Pk, however, can be performed transversally on any stabilizer code. A classical [n, k, d] linear code (n physical bits, k logical bits, classical distance d) can be defined in terms of an (n − k) × n binary parity check matrix H --- every classical codeword Thus, it is sufficient in general to check that the error-correction conditions hold for a basis of errors.

A 5-qubit code is the smallest possible code which protects a single logical qubit against single-qubit errors. Theorem 2 Let S be a stabilizer with n − k generators, and let S ⊥ = {E ∈ Pn s.t. [E, M] = 0 ∀M ∈ S}. Note that the square brackets specify that the code is a stabilizer code, and that the middle term k refers to the number of encoded qubits, and not the dimension 2k Privacy policy About Wikipedia Disclaimers Contact Wikipedia Developers Cookie statement Mobile view Cornell University Library We gratefully acknowledge support fromthe Simons Foundation and member institutions arXiv.org > quant-ph > arXiv:quant-ph/0304016 Search

Text is available under the Creative Commons Attribution-ShareAlike License; additional terms may apply. The product is in the Clifford group, and is only performed if the measurement outcome is 1. It is frequently useful to consider other representations of stabilizer codes. Therefore, given the ability to perform fault-tolerant Clifford group operations, fault-tolerant measurements, and to prepare the encoded ∣ψπ/8⟩ state, we have universal fault-tolerant quantum computation.

Schaetz, M.