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## Quantum Error Correction Codes

## Stabilizer Codes And Quantum Error Correction

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Lett., **80:4084, 1998. [CG97]** R. Consequently, the density matrixevolution is given by,∂tρ = −i~[H, ρ] + ΓZ(ZρZ − ρ) +Γ2(XρX + Y ρY −2ρ).(56)If it is assumed that the qubit is not undergoing anycoherent evolution (H QEC, A GOOD STARTING POINT: THE 3-QUBITCODEThe 3-qubit bit-ﬂip code is traditionally used as a basicintroduction to the concept of Quantum Error Correc-tion. Phys. More about the author

This type of error is typically associatedwith bad system control and/or characterization whereimprecise manipulation of the qubit introduces inaccu-rate Hamiltonian dynamics. IX), which is asuﬃcient condition for correcting errors for an arbitraryerror mapping on a single qubit. As an introduction we will fo-cus on arguably the most well known quantum code, the7-qubit Steane code, ﬁrst proposed in 1996 (Ste96a).The 7-qubit code represents a full quantum code thatencodes seven quant-ph/9705052, 1997. [Got98] D.

Benjamin, and A. Quantum Error Suppression There are a few ways in which to suppress errors that occur during a quantum computation. The qubit in this sys-tem is deﬁned only with two electronic states, howeverthe system itself contains many more levels (includingsome which are used for qubit readout and initializationthrough optical pumping and fault-tolerant Quantum Error Correction and thethreshold theorem. 17A.

- Fault-Tolerant LogicalRev.
- Lidar, L.-A.
- Instead the subroutine canbe reset and re-run.VII.

See all ›47 CitationsSee all ›171 **ReferencesShare Facebook Twitter** Google+ LinkedIn Reddit Download Full-text PDF Quantum Error Correction for BeginnersArticle (PDF Available) in Reports on Progress in Physics 76(7):076001 · June 2013 with 171 ReadsDOI: 10.1088/0034-4885/76/7/076001 · Another approach isto not enforce the correction requirement. American Association for the Advancement of Science; 2013 Nov 15 [cited 2016 Jan 26];342(6160):830‐3.Available from: http://science.sciencemag.org/content/342/6160/830.abstract 3. Surface Code Rev.

This is a problem as the environment can be seen as measuring the quantum system, which collapses the quantum state and irreversibly destroys the quantum information. Stabilizer Codes And Quantum Error Correction It can make the exact same computation give different results. Knill and R. DiVincenzo, and B.M.

A., 54:1862, 1996. [Got97] D. Bit Flip Error arxiv:0811.3616 - Loock - 2008 1 SelfConsistent non-Markovian theory of a quantum-state evolution for quantum-information-processing. Qubits are subject to bit flips ‐ like classical bits ‐ and also phase errors \( |1 \rangle\) \(\rightarrow\) \(-|1 \rangle \). Quantum Computa-edition, 2000. [Neu55] J.

Twamley. During a round oferror correction, if no error is detected (which if the er-ror rate is small, occurs with high probability), the erroron the resulting state is suppressed from O(2) to Quantum Error Correction Codes Phys. 5 Qubit Code Section VII then introduces the stabilizerformalism (Got97), demonstrating how QEC circuits aresynthesized once the structure of the code is known.

Hence the erred state, E |ψiL,satisﬁes,KiE |ψiL= (−1)mEKi|ψiL= (−1)mE |ψiL. (52)where m is deﬁned as, m = 0, if [E, Ki] = 0 and m = 1,if {E, Ki} = 0. http://caribtechsxm.com/quantum-error/quantum-error-correction.php A., 67:032313, 2003. [MCM+05] J.M. Close This Message CREATE AN ACCOUNT Name: Username: Password: Verify Password: E-mail: Verify E-mail: *All Fields Are Required. Phys. Steane Code

Notice that the stabilizer setseparates into X and Z sectors which deﬁnes the codeas a Calderbank-Shor-Steane (CSS) code. It was well known from the early developments of this exciting field that the fragility of coherent quantum systems would be a catastrophic obstacle to the development of large scale quantum This way, errors only perturbcodeword states by small amounts which can then bedetected and corrected, without directly measuring thequantum state of any qubit.III. http://caribtechsxm.com/quantum-error/quantum-error-correction-usc.php Gottesman.

For these reasons, leakage is oneof the most problematic error channels to correct usingQEC. Hadamard Gate In more recent years QEC pro-tocols have been developed for various systems, such ascontinuous variables (LS98; Bra98; vL08; ATK+08), ion-traps and other systems containing motional degrees offreedom (LW03b; ST98), adiabatic computation For some arbi-MHHUFIG. 7 Quantum Circuit required to project an arbitrarystate, |ψiIinto a ±1 eigenstate of the Hermitian operator,U = U†.

QuantumPhys. Fault-Tolerant quantum computation.Proc. 37th IEEE Symp. A,258:77, 1999. [ZR97a] P. Quantum Code If a single bit and/or phase ﬂip oc-curs on one of the four qubits then the ancilla qubits willbe measured in the |1i state.

Rasetti. This is when Peter Shor, a mathematics professor at MIT and the brain behind the encryption … Runtime Error R6205 Read Error Rate On Hard Drive On C Is 1 In White, S.D.without destroying it. navigate to this website Steane.

Introduction To Quantum Computation.World Scienti c, Singapore, 1998. [PVK97] M.B. Par-ticularly in superconducting systems where decoherencecan be caused by small numbers of ﬂuctuating charges.In this case more speciﬁc decoherence models need to beconsidered.Using this formalism, the evolution of the density ma-trix Subsystem Codes 26B. The state of the system, just prior to themeasurement of the ancilla is in table.

Lett. Lidar and L.-A. A., 69:032314, 2004. [Fow05] A.G. quant-ph/0506126, 2005. [FSG08] A.G.

The technique required to stop this from happening is called quantum error correction. The input state |0i⊗7is projected into an eigenstate ofeach of the X stabilizers shown in Eq. 44. Whaley, and D.S. Fowler, and L.C.L.

Quant. It is also impossible to use any sort of direct measurement as it would destroy the quantum superposition. This coupling follows the de-coherence analysis shown earlier, where a 50/50 mixedstate of the qubit results. Phys.

Jozsa. arXiv:0904.2557, 2009. [Gro97] L.K.