# Quantum Error Correction With Imperfect Gates

The weight wt(P) of a Pauli **operator P ∈ Pn is** the number of qubits on which it acts as X, Y, or Z (i.e., not as the identity). Rev. 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). Any qubit stored unprotected or one transmitted through a communications channel will inevitably come out at least slightly changed. weblink

The Shor code[edit] The error channel may induce either a bit flip, a sign flip, or both. Generated Tue, 25 Oct 2016 02:49:25 GMT by s_wx1157 (squid/3.5.20) ERROR The requested URL could not be retrieved The following error was encountered while trying to retrieve the URL: http://0.0.0.9/ Connection Let E phase {\displaystyle E_{\text{phase}}} be a quantum channel that can cause at most one phase flip. Martin-Delgado, Strong Resilience of Topological Codes to Depolarization, Phys.

Yu. However, in order to have a stabilizer code at all, the generators produced by the above procedure must commute. Sijher, Protected Gates for Topological Quantum Field Theories, arXiv:1409.3898.H.

- 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.
- Browne, Fault Tolerance with the Gauge Color Code, arXiv:1503.08217.B. J.
- Calderbank, Peter Shor and Andrew Steane.
- Rev.
- His current research interests include optical networks, error control coding, constrained coding, coded modulation, turbo equalization, OFDM applications, and quantum error correction.

Phys. 15, 055023 (2013).J. K. B. All of them share some basic features: they involve creation and verification of specialized ancilla states, and use transversal gates which interact the data block with the ancilla state.

Shor, “Algorithms for quantum computation: discrete log and factoring”, Proceedings of the 35th Annual Symposium on the Foundations of Computer Science (IEEE Computer Society Press, Los Alamitos, CA, 1994), p. 124.[2]P. Rev. 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 http://libra.msra.cn/Publication/3635279/quantum-error-correction-with-imperfect-gates Math.

Support For full functionality of ResearchGate it is necessary to enable JavaScript. A 54, 1862 (1996).D. Finally, because the behavior of our decoder's performance for two of the noise models we consider maps onto an order-disorder phase transition in the three-body random-bond Ising model in 2D and Then we complete the operation with a further transversal gate which depends on the outcome of the measurement.

A 5-qubit code is the smallest possible code which protects a single logical qubit against single-qubit errors. try this Duclos-Cianci and D. WeinsteinRead full-textQuantum Error Correction Implementation after Multiple Gates Full-text · Article · May 2013 Yaakov S. If we perform a Hadamard transform and then measure each qubit in the ancilla, we get either a random even weight string (for eigenvalue + 1) or an odd weight string (for

Yu. http://vealcine.com/quantum-error/quantum-error-correction-usc.php Note that for P, Q ∈ Pn, wt(PQ) ≤ wt(P) + wt(Q). The errors are assumed to be independent and uncorrelated between qubits except when a gate connects them. Preskill, Sufficient Condition on Noise Correlations for Scalable Quantum Computing, Quantum Inf.

We therefore have to resort to more complicated techniques. Rev. Bombin, R. W. http://vealcine.com/quantum-error/quantum-error-correction-ppt.php Preskill, Topological Quantum Memory, J.

Gottesman, Class of Quantum Error-Correcting Codes Saturating the Quantum Hamming Bound, Phys. Cerf, Ulrik L. Knill, R.

## The product is in the Clifford group, and is only performed if the measurement outcome is 1.

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. Nielsen and Isaac L. König, F. Some stabilizer codes have interesting symmetries under the action of certain Clifford group elements, and these symmetries result in transversal gate operations.

As of late 2004, estimates for this threshold indicate that it could be as high as 1-3% [4], provided that there are sufficiently many qubits available. Inf. Indeed, the set S ⊥ \ S of undetectable errors is a boon in this case, as it allows us to perform these gates. this content Cambridge University Press. ^ W.Shor, Peter (1995). "Scheme for reducing decoherence in quantum computer memory".

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 A 52 2493–2496 (1995).ADS[3]A. Holevo (2) C. They would therefore appear to be those errors which cannot be detected by the code.

Let C1 be an [n, k1, d1] code and let C2 be an [n, k2, d2] code. Fowler, S. Nickerson, and D. E. Rev.

Phys. 14, 123011 (2012).A. J. Rev. Dyn. 17, 1 (2010).B. J.