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# Quantum Error Correction With Degenerate Codes For Correlated Noise

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 Rev. 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. pt is the threshold for fault-tolerant quantum computation. http://vealcine.com/quantum-error/quantum-error-correction-continuous-variable-states-against-gaussian-noise.php

Another useful representation is to map the single-qubit Pauli operators I, X, Y, Z to the finite field GF(4), which sets up a connection between stabilizer codes and a subset of 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. We prove that degenerate codes can outperform nondegenerate ones in the presence of correlated noise, by exhibiting examples where the quantum packing bound is violated.

1. Any qubit stored unprotected or one transmitted through a communications channel will inevitably come out at least slightly changed.
2. These include EPR and GHZ states.
3. If errors occur on the physical qubits independently at random with probability p per gate or timestep, the fault-tolerant protocol has probability of logical error for a single logical gate or
4. 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⟩.
5. A procedure due to Steane uses (forCSS codes) one ancilla in a logical $\left|\overline{0}\right\rangle$ state of the same code and one ancilla in a logical $\left|\overline{0}\right\rangle + \left|\overline{1}\right\rangle$ state.
6. If we are instead interested in erasure errors, when the location of the error is known but not its precise nature, a distance d code corrects d − 1 erasure errors.
7. We could repeat this procedure to get an n3-qubit code, and so forth.
8. We convert H1 into stabilizer generators as above, replacing each 0 with I and each 1 with Z.

The phenomenon of degeneracy has no analogue for classical error correcting codes, and makes the study of quantum codes substantially more difficult than the study of classical error correction. We can imagine the various possible errors taking the subspace C into other subspaces of Hn, and we want those subspaces to be isomorphic to C, and to be distinguishable from Mod. Fluids Phys.

About INIS Repository INIS Repository contains 377,500 full texts and 3,970,000 bibliographic records Updates In 2016 more than 95,000 bibliographic records and 4,200 full texts have been added INIS Newsletter Search Conversely, we are also interested in the problem of setting upper bounds on achievable values of (logK)/n and d/n. Phys. https://inis.iaea.org/search/search.aspx?orig_q=RN:43025640 The errors are assumed to be independent and uncorrelated between qubits except when a gate connects them.

Each measurement gives us one bit of the error syndrome, which we then decipher classically to determine the actual error. When p < pt = 1/C, the fault-tolerance helps, decreasing the logical error rate. the equation ⟨ψi∣E∣ψj⟩ = C(E)δij fails. A transversal operation is one in which the ith qubit in each block of a QECC interacts only with the ith qubit of other blocks of the code or of special

We recommend that Javascript be enabled to use all the functionalities offered by INIS Repository Search website. http://iicqi.sharif.edu/node/96 Lett. Phys. A 83, 052305 (2011) DOI: 10.1103/PhysRevA.83.052305 Citeas: arXiv:1007.3655 [quant-ph] (or arXiv:1007.3655v2 [quant-ph] for this version) Submission history From: Michele Dall'Arno [view email] [v1] Wed, 21 Jul 2010 13:40:44 GMT (11kb)

The computational complexity of the encoder is frequently a great deal lower than that of the decoder. have a peek at these guys Note that the error syndrome does not tell us anything about the encoded state, only about the error that has occurred. 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. Your cache administrator is webmaster.

We prove that degenerate codes can outperform nondegenerate ones in the presence of correlated noise, by exhibiting examples where the quantum packing bound is violated.Primary SubjectCLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS 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. Res. check over here 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.

The codewords of the QECC are by definition in the  + 1-eigenspace of all elements of the stabilizer, but an error E acting on a codeword will move the state into the Two Pauli operators P = (pX∣pZ) and Q = (qX∣qZ) commute iff pX ⋅ qZ + pZ ⋅ qX = 0. The set of stabilizer codes is exactly the set of codes which can be created by a Clifford group encoder circuit using ∣0⟩ ancilla states.

## Also, note that a single phase error in the cat state will cause the final measurement outcome to be wrong (even and odd switch places), so we should repeat the measurement

If the classical distance d = 2t + 1, the quantum code can correct t bit flip (X) errors, just as could the classical code. 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. The set of such eigenvalues can be represented as an (n − k)-dimensional binary vector known as the error syndrome. A Phys.

We use the notation [[n, k, d]] to a refer to such a stabilizer code. By adding extra qubits and carefully encoding the quantum state we wish to protect, a quantum system can be insulated to great extent against errors. We get a similar result in the case where the noise is a general quantum operation on each qubit which differs from the identity by something of size O(ε). this content Phys.