# Quantum Error Rate

## Contents |

**Rev. **Then we interact the ancilla with the encoded data qubits using gates from our stock of transversal gates and perform a fault-tolerant measurement. In addition, if we write any superoperator S in terms of its operator-sum representation S(ρ) ↦ ∑AkρAk † , a QECC that corrects the set of errors {Ak} automatically corrects S as well. Use of the American Physical Society websites and journals implies that the user has read and agrees to our Terms and Conditions and any applicable Subscription Agreement. have a peek at this web-site

Rev. Instead of the unencoded ∣ + ⟩ state, we must use a more complex ancilla state ∣00…0⟩ + ∣11…1⟩ known as a 'cat' state. A newer idea is Alexei Kitaev's topological quantum codes and the more general idea of a topological quantum computer. The solution is to use transversal gates whenever possible. https://en.wikipedia.org/wiki/Quantum_error_correction

## Quantum Error Correction For Beginners

A + 1 eigenstate in the data therefore leaves us with ∣00…0⟩ + ∣11…1⟩ in the ancilla and a − 1 eigenstate leaves us with ∣00…0⟩ + ∣11…1⟩. Note **that for P, Q ∈ Pn, wt(PQ) ≤ wt(P) + wt(Q).** Copying quantum information is not possible due to the no-cloning theorem. Poulin, and T. M.

D Phys. Nielsen and Isaac L. Lett. 81, 2152–2155 (1998), doi:10.1103/PhysRevLett.81.2152 ^ T. Bit Flip Memory Error D.

It is possible to correct for both types of errors using one code, and the Shor code does just that. Particular caution is necessary, as computational gates can cause errors to propagate from their original location onto qubits that were previously correct. Accel. https://arxiv.org/abs/0904.2557 We discuss optimization of the binomial codes and demonstrate that by relaxing the parity structure, codes with even lower unrecoverable error rates can be achieved.

But it is possible to spread the information of one qubit onto a highly entangled state of several (physical) qubits. Fault-tolerant Quantum Computation This approach is powerful, but its realization is a daunting engineering challenge. Girvin and R. Different colors correspond to different code sizes, k=nr; shading indicates ±1σ.

## Quantum Error Correction Codes

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 If the state is a + 1 eigenvector of M, the ancilla will be ∣ + ⟩, and if the state is a − 1 eigenvector, the ancilla will be ∣ − ⟩. Quantum Error Correction For Beginners StacePhys. Stabilizer Codes And Quantum Error Correction. The set of such eigenvalues can be represented as an (n − k)-dimensional binary vector known as the error syndrome.

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 Check This Out In that case, let us consider tensor products of the Pauli matrices $I=\begin{pmatrix}1&0\\0&1\end{pmatrix}, X=\begin{pmatrix}0&1\\1&0\end{pmatrix}, Y=\begin{pmatrix}0&-i\\i&0\end{pmatrix}, Z=\begin{pmatrix}1&0\\0&-1\end{pmatrix}$ Define the Pauli group Pn as the group consisting of tensor products of I, X, Leibfried, T. Skip to main content Quantiki Toggle navigation Browse News Forums Video Abstracts Journal Articles RSS Feeds Journal Articles News Positions Video Abstracts Events Past events Groups Positions Wiki Index Popular Recent Quantum Code 7

Current technology has made remarkable progress in developing superconducting cavities that are controllable on the quantum level, with lifetimes longer than the best corresponding physical quantum bits. Accel. Authorization RequiredLog InOther OptionsBuy Article »Find an Institution with the Article »×Download & SharePDFExportReuse & PermissionsTweet×ImagesFigure 1Examples of progenitor clusters for clusterized CSS codes. (a) Clusterized Steane code. (b) Clusterized Shor code. (c) Clusterized http://johnlautner.net/quantum-error/quantum-physics-error.html Poulin2, and T. M.

With the Shor code, a qubit state | ψ ⟩ = α 0 | 0 ⟩ + α 1 | 1 ⟩ {\displaystyle |\psi \rangle =\alpha _{0}|0\rangle +\alpha _{1}|1\rangle } will Quantum Error Correction Book Our codes are similar in spirit to “cat codes” based on superpositions of the coherent states but offer several advantages such as smaller mean boson number, exact rather than approximate orthonormality Thus, measuring the eigenvalues of the generators of S tells us information about the error that has occurred.

## Now these qubits will be sent through a channel E bit {\displaystyle E_{\text{bit}}} where we assume that at most one bit flip may occur.

Ozeri and D. All rights reserved. Beams Phys. Steane Code Nigg, L.

Mod. Rev. 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. have a peek here The construction takes two binary classical linear codes and produces a quantum code, and can therefore take advantage of much existing knowledge from classical coding theory.

In general, a quantum error-correcting code is a subspace of a Hilbert space designed so that any of a set of possible errors can be corrected by an appropriate quantum operation. Then the bit flip code from above can recover | ψ ⟩ {\displaystyle |\psi \rangle } by transforming into the Hadamard basis before and after transmission through E phase {\displaystyle E_{\text{phase}}} Log In CancelForgot your username/password?Create an account×SearchAll Fields Author Abstract Abstract/Title Title Cited Author Affiliation Collaboration Article LookupPaste a citation or DOIEnter a citationJournal: Phys. Phys.

pt is the threshold for fault-tolerant quantum computation. Pittman, B. E Phys.