Home > Quantum Error > Quantum Error Correction With Degenerate Codes For Correlated Noise# Quantum Error Correction With Degenerate Codes For Correlated Noise

## If this condition is satisfied, t separate single-qubit or single-gate failures are required for a distance 2t + 1 code to fail.

Cornell University Library We gratefully acknowledge **support fromthe Simons Foundation** and member institutions arXiv.org > quant-ph > arXiv:1007.3655 Search or Article-id (Help | Advanced search) All papers Titles Authors Abstracts Fault-Tolerant Gates We will focus on stabilizer codes. Despite being efficiently simulable, most stabilizer states on a large number of qubits exhibit maximal bipartite entanglement[Dahlsten and Plenio, QIC 2006]. The system returned: (22) Invalid argument The remote host or network may be down. news

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. However, the effects of relaxing these assumptions on the threshold value and overhead requirements have not been well-studied. Using this or another verification procedure, we can check a non-fault-tolerant construction. Mucciolo; Harold U.

Accel. Optimized simulations of fault-tolerant protocols suggest the true threshold may be as high as 5%, but to tolerate this much error, existing protocols require enormous overhead, perhaps increasing the number of Phys. It is frequently useful to consider other representations of stabilizer codes.

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- Let C1 be an [n, k1, d1] code and let C2 be an [n, k2, d2] code.
- Frequently, we are interested in codes that correct any error affecting t or fewer physical qubits.

Stabilizer Codes In order to better manipulate and discover quantum error-correcting codes, it is helpful to have a more detailed mathematical structure to work with. One of the central problems in the theory of quantum error correction is to find codes which maximize the ratios (logK)/n and d/n, so they can encode as many qubits as CSS Codes CSS codes are a very useful class of stabilizer codes invented by Calderbank and Shor, and by Steane. When this is **true, C1 and C2 define** an [[n, k1 + k2 − n, d]] stabilizer code, where d ≥ min(d1, d2).

Universal fault-tolerance is known to be possible for any stabilizer code, but in most cases the more complicated type of construction is needed for all but a few gates. Rev. It is assumed that measurements and classical computations can be performed quickly and reliably, and that quantum gates can be performed between arbitrary pairs of qubits in the computer, irrespective of https://inis.iaea.org/search/search.aspx?orig_q=RN:43025640 Rev.

Thus, measuring the eigenvalues of the generators of S tells us information about the error that has occurred. Then S encodes k qubits and has distance d, where d is the smallest weight of an operator in S ⊥ \ S. Unfortunately, the practical requirements for this result are not nearly so good. Since these two operations are **completely separate, it can also** correct Y errors as both a bit flip and a phase error.

Definition 3 Let S ⊂ Pn be an Abelian subgroup of the Pauli group that does not contain − 1 or ± i, and let C(S) = {∣ψ⟩ s.t. The errors are assumed to be independent and uncorrelated between qubits except when a gate connects them. The most widely-used structure gives a class of codes known as stabilizer codes. Definition 2 The distance d of an ((n, K)) is the smallest weight of a nontrivial Pauli operator E ∈ Pn s.t.

Phys. navigate to this website The cat state contains as many qubits as the operator M to be measured, and we perform the controlled-X, -Y, or -Z operations transversally from the appropriate qubits of the cat Please try the request again. Include: All words Exact Phrase ------------------- Abstract Author Country of publication Descriptors DEC DEI INIS Issue INIS Volume Journal Language Library Subject Primary Subject Record Type Reference Number Related Record Report

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. E Phys. The fault-tolerant procedures concatenate as well, and after L levels of concatenation, the effective logical error rate is pt(p/pt)2L (for a base code correcting 1 error). http://caribtechsxm.com/quantum-error/quantum-error-correction-codes.php Top 20 searches for the past 30 days Loading...

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. Phys. We recommend that Javascript be enabled to use all the functionalities offered by INIS Repository Search website.

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. Phys. Rev. Rev.

For instance, if Cab = δab, then the various erroneous subspaces are orthogonal to each other. These include EPR and GHZ states. Then measure the ancilla in the basis of ∣ + ⟩ and ∣ − ⟩ = ∣0⟩ − ∣1⟩. click site Given a codeword of a particular [[n, 1]] QECC, we can take each physical qubit and again encode it using the same code, producing an [[n2, 1]] QECC.

If the classical distance d = 2t + 1, the quantum code can correct t bit flip (X) errors, just as could the classical code. In particular, the task of determining what error has occured can be computationally difficult (NP-hard, in fact), and designing codes with efficient decoding algorithms is an important task in quantum correction Rev. (Series I) Physics Volume: Article: × Looks like Javascript is disabled on your browser. For instance, P ∈ Pn can be represented by a pair of n-bit binary vectors (pX∣pZ) where pX is 1 for any location where P has an X or Y tensor factor and

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 set of such eigenvalues can be represented as an (n − k)-dimensional binary vector known as the error syndrome. Then the stabilizer code generated by these operators is precisely a quantum version of the classical error-correcting code given by H. Rev.

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 Rev.