# Quantum Error Correction Codes

## Contents |

The Clifford group on n qubits **is defined as the** set of unitary operations which conjugate the Pauli group Pn into itself; Cn can be generated by the Hadamard transform, the Classical error correction employs redundancy. The following circuit performs a π/8 rotation, given an ancilla state ∣ψπ/8⟩ = ∣0⟩ + exp(iπ/4)∣1⟩: Here P is the π/4 phase rotation diag(1, i), and X is the bit flip. Schoelkopf, "Realization of Three-Qubit Quantum Error Correction with Superconducting Circuits," Nature 482, 382-385 (2012), doi:10.1038/nature10786, arXiv:1109.4948 ^ M. have a peek at this web-site

Due to linearity, it follows that the Shor code can correct arbitrary 1-qubit errors.[clarification needed] General codes[edit] In general, a quantum code for a quantum channel E {\displaystyle {\mathcal {E}}} is Note that the error syndrome does not tell us anything about the encoded state, only about the error that has occurred. 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. Rev. https://en.wikipedia.org/wiki/Quantum_error_correction

## Quantum Error Correction For Beginners

Knill, C. Pittman, B. 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

For instance, a qubit in the state | − ⟩ = ( | 0 ⟩ − | 1 ⟩ ) / 2 {\displaystyle |-\rangle =(|0\rangle -|1\rangle )/{\sqrt {2}}} may have its Jost, E. Sun, L. 5 Qubit Code Contents 1 The bit flip code **2 The sign flip code** 3 The Shor code 4 General codes 5 Models 6 Experimental realization 7 See also 8 References 9 Bibliography 10

Then by comparing qubits within blocks of three, we can detect bit flip errors, and by comparing the signs of the three blocks, we can detect sign errors. Stabilizer Codes And Quantum Error Correction. If U = σ z {\displaystyle U=\sigma _{z}} , a sign flip error occurs. It is possible that a double-bit error occurs and the transmitted message is equal to three zeros, but this outcome is less likely than the above outcome. More Bonuses 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.

A quantum error correcting code protects quantum information against errors of a limited form. Fault-tolerant Quantum Computation G. The code will be able to correct bit flip (X) errors as if it had a distance d1 and to correct phase (Z) errors as if it had a distance d2. If U = i σ y {\displaystyle U=i\sigma _{y}} then both a bit flip error and a sign flip error occur.

## Stabilizer Codes And Quantum Error Correction.

Conversely, we are also interested in the problem of setting upper bounds on achievable values of (logK)/n and d/n. https://www.perimeterinstitute.ca/personal/dgottesman/QECC.html Category:Introductory Tutorials Category:Quantum Error Correction Category:Handbook of Quantum InformationLast modified:Monday, October 26, 2015 - 17:56 ERROR The requested URL could not be retrieved The following error was encountered while trying to Quantum Error Correction For Beginners Cambridge University Press. Steane Code If U = σ x {\displaystyle U=\sigma _{x}} , a bit flip error occurs.

Nebendahl, D. Check This Out In addition, CSS codes have some very useful properties which make them excellent choices for fault-tolerant quantum computation. According to the quantum Hamming bound, encoding a single logical qubit and providing for arbitrary error correction in a single qubit requires a minimum of 5 physical qubits. The set of such eigenvalues can be represented as an (n − k)-dimensional binary vector known as the error syndrome. Quantum Error Correction Book

Calderbank, Peter Shor and Andrew Steane. Jacobs and J. Define the dual C ⊥ of a classical code C as the set of vectors w s.t. Source 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.

Comments: 4 pages (including figures), latex file using RevTex Subjects: Quantum Physics (quant-ph) Citeas: arXiv:quant-ph/9602019 (or arXiv:quant-ph/9602019v1 for this version) Submission history From: Raymond LaFlamme [view email] [v1] Tue, 27 Quantum Code 7 Mikael Lassen, Metin Sabuncu, Alexander Huck, Julien Niset, Gerd Leuchs, Nicolas J. A non-degenerate code is one for which different elements of the set of correctable errors produce linearly independent results when applied to elements of the code.

## Nebendahl, D.

Niset, G. The latter is counter-intuitive at first sight: Since noise is arbitrary, how can the effect of noise be one of only few distinct possibilities? For example, in the case where the first qubit is flipped, the result would be | ψ r ′ ⟩ = α 0 | 100 ⟩ + α 1 | 011 Bit Flip Memory Error The Shor code[edit] The error channel may induce either a bit flip, a sign flip, or both.

R. A 71, 052332 (2005), doi:10.1103/PhysRevA.71.052332 ^ J. Sun, L. have a peek here Despite being efficiently simulable, most stabilizer states on a large number of qubits exhibit maximal bipartite entanglement[Dahlsten and Plenio, QIC 2006].

But we didn’t have constructive examples of getting here. It is possible that a double-bit error occurs and the transmitted message is equal to three zeros, but this outcome is less likely than the above outcome. The decoder is sometimes also taken to map Hn into an unencoded Hilbert space HlogK isomorphic to C. Raymond Laflamme and collaborators found a class of 5-qubit codes which do the same, which also have the property of being fault-tolerant.