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# Quantum Error Code

## 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 Found. 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 The readout resonator frequencies are ωRi/2π{6.494,6.695,6.491,6.693} GHz, while the four bus resonators, unmeasured, are designed to be at ωBii/2π={8,7.5,8,7.5} GHz for ij∈{12,23,34,41}. https://en.wikipedia.org/wiki/Quantum_error_correction

## Quantum Error Correction For Beginners

If the three bit flip group (1,2,3), (4,5,6), and (7,8,9) are considered as three inputs, then the Shor code circuit can be reduced as a sign flip code. If one of the qubits turns out to disagree with the other two, it can be reset to their value. 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.

• Fault-Tolerant Measurement and Error Correction Since all our gates are unreliable, including those used to correct errors, we will need some sort of fault-tolerant quantum error correction procedure.
• The stabilizer S {\displaystyle {\mathcal {S}}} has a minimal representation in terms of n − k {\displaystyle n-k} independent generators { g 1 , … , g n − k
• The extension of the above definitions and mapping N {\displaystyle N} to multiple qubits is straightforward.
• This entanglement is achieved in our architecture with one CNOT and one SWAP gate (a).

A 71, 052332 (2005), doi:10.1103/PhysRevA.71.052332 ^ J. Leibfried, T. The Pauli group Pk, however, can be performed transversally on any stabilizer code. 5-qubit Quantum Error Correction The first step of the three qubit bit flip code is to entangle the qubit with two other qubits using two CNOT gates with input | 0 ⟩ {\displaystyle |0\rangle }

Suppose [ A ] {\displaystyle \left[A\right]} is a set of equivalence classes of an operator A {\displaystyle A} that have the same phase: [ A ] = { β A   Stabilizer Codes And Quantum Error Correction. A classical [n, k, d] linear code (n physical bits, k logical bits, classical distance d) can be defined in terms of an (n − k) × n binary parity check matrix H --- every classical codeword The computational complexity of the encoder is frequently a great deal lower than that of the decoder. https://en.wikipedia.org/wiki/Stabilizer_code Definition Let us define an [ n , k ] {\displaystyle \left[n,k\right]} stabilizer quantum error-correcting code to encode k {\displaystyle k} logical qubits into n {\displaystyle n} physical qubits.

The Shor code The error channel may induce either a bit flip, a sign flip, or both. Quantum Error Correction Book Note that the Gilbert-Varshamov bound simply states that codes at least this good exist; it does not suggest that better codes cannot exist. Theorem 2 Let S be a stabilizer with n − k generators, and let S ⊥  = {E ∈ Pn s.t. [E, M] = 0 ∀M ∈ S}. Characterization of addressability by simultaneous randomized benchmarking.

## Stabilizer Codes And Quantum Error Correction.

In other words, we need the computation to be fault-tolerant. http://news.mit.edu/2015/quantum-error-correction-0526 Barreiro, T. Quantum Error Correction For Beginners A stabilizer quantum error-correcting code appends ancilla qubits to qubits that we want to protect. 5 Qubit Code For instance, if Cab = δab, then the various erroneous subspaces are orthogonal to each other.

Since the shots producing the error syndromes are evenly distributed throughout the different unitary rotation pulses on the code qubits, the effect on the code state is to reduce the number have a peek at these guys A.D.C. However, in a quantum channel, it is no longer possible, due to the no-cloning theorem, which forbids the creation of identical copies of an arbitrary unknown quantum state. 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. Steane Code

Niset, G. The solution is to use transversal gates whenever possible. Code qubits in the SC are placed at the vertices of a two-dimensional array and each stabilizer involves four neighbouring code qubits. check over here Sloane ([2], [3]); these are also called additive codes.

The sign flip code Quantum circuit of the phase flip code Flipped bits are the only kind of error in classical computer, but there is another possibility of an error with Quantum Code 7 For an error induced by a unitary operation, the data is explained by cosines (solid lines in Fig. 3). Cory, M.

## Freedman, Michael H.; Meyer, David A.: Projective plane and planar quantum codes.

Phys. We can similarly define the phase-free n {\displaystyle n} -qubit Pauli group [ Π n ] = { [ A ]   |   A ∈ Π n } {\displaystyle \left[\Pi A, vol. 54, no. 2, pp. 1098–1105, Aug 1996. Bit Flip Memory Error Histograms are fitted to the integrated shots and thresholds for each channel are set at the point of maximum distance between cumulative distributions of the histograms.The assignment fidelity of each channel

However, any measurement of the superposition will collapse the quantum state into one of its component classical states. Lett. 108, 070502 (2012).CASPubMedArticleDownload referencesAcknowledgementsWe thank M. The results are consistent with a higher uncertainty in the phase-flip error detection, likely due to decoherence during the full sequence and the order of syndrome detection.Full size imageDiscussionWe have provided http://vealcine.com/quantum-error/quantum-error-code-2401.php The final state represents the result of the computation.

A number of different techniques have been developed. Because the ancillas in Steane and Knill error correction are more complicated than the cat state, it is especially important to verify the ancillas before using them. M. & Cleland, A. The theory of quantum error-correcting codes has been developed to counteract noise introduced in this way.

Shor, “Scheme for reducing decoherence in quantum computer memory,” Phys. Chwalla, M. It may be possible to implement the researchers’ scheme without actually duplicating banks of qubits. R is a quantum operation and (R ∘ Ea)(∣ψ⟩) = ∣ψ⟩ for all Ea ∈ E, ∣ψ⟩ ∈ C.

Itano, J. When this is true, C1 and C2 define an [[n, k1 + k2 − n, d]] stabilizer code, where d ≥ min(d1, d2). These error detection experiments constitute a key milestone for SC implementation, as our operations now extend into the plane of the two-dimensional surface and we show the ability to concurrently perform The disadvantage is that it is not transversal, and thus not fault-tolerant.

Schindler, J.