# Quantum Error Syndrome

## Contents |

Demonstration of sufficient control **for two rounds** of quantum error correction in a solid state ensemble quantum information processor. A. & Laflamme, R. The experiments combine a variety of key components required for scaling quantum systems up to larger numbers of qubits: high-fidelity one- and two-qubit gates, high single-shot assignment fidelities allowing for non-demolition Comput. weblink

M. Superconducting quantum circuits at the surface code threshold for fault tolerance. Your **cache administrator is webmaster. **Suppose further that a noisy error corrupts the three-bit state so that one bit is equal to zero but the other two are equal to one. https://en.wikipedia.org/wiki/Quantum_error_correction

## Quantum Error Correction Codes

The ECRij gate plus four single-qubit rotations as depicted in Fig. 8c correspond to a CNOT operation between Qi and Qj in our device.Tracking bit- and phase-flip errorsAs introduced in the Thus, for the 7-qubit code, the full logical Clifford group is accessible via transversal operations. Keefe for fabricating devices.

- Freedman, Michael H.; Meyer, David A.; Luo, Feng: Z2-Systolic freedom and quantum codes.
- Simple pulses for elimination of leakage in weakly nonlinear qubits.
- In this arrangement, we use Q1 and Q3 (Fig. 1b; purple) as code qubits, Q2 as the Z-syndrome qubit (Fig. 1b; green) and Q4 as the X-syndrome qubit (Fig. 1b; yellow).

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. 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. Lett. 109, 240504 (2012).CASPubMedArticle24.Saira, O.-P. 5 Qubit Code The theory of quantum error-correcting codes has been developed to counteract noise introduced in this way.

Denoting the conditional states ρdd, ρdu, ρud and ρuu and we have full tomographic information of the state of the code qubits for each of the four bins. Quantum Error Correction For Beginners Cerf, Ulrik L. Surface codes: towards practical large-scale quantum computation. https://quantiki.org/wiki/quantum-error-correction-and-fault-tolerance-0 Then the Pauli operators of weight t or less form a basis for the set of all errors acting on t or fewer qubits, so a QECC which corrects these Pauli

The most widely-used structure gives a class of codes known as stabilizer codes. Quantum Error Correction Book Thus, the distance of the quantum code is at least min(d1, d2), but might be higher because of the possibility of degeneracy. Steffen, Jay M. Conditioned on {0,+}, state tomography of the code qubits is performed, with a reconstructed final state (Pauli vector shown in Fig. 2a), commensurate with the originally prepared codeword state with a

## Quantum Error Correction For Beginners

Among the syndrome qubits, a distinction is made between bit-flip syndromes (or Z-syndromes) and phase-flip syndromes (or X-syndromes). So, for example, there are only 6 stabilizer states on one qubit. Quantum Error Correction Codes B. Stabilizer Codes And Quantum Error Correction. By encoding both the XX and the ZZ stabilizers in the four-qubit lattice, we can protect a maximally entangled state of the two-code qubits against an arbitrary error.To demonstrate the SC

Classical error correction employs redundancy. http://vealcine.com/quantum-error/quantum-of-error.php M., Rebentrost, P. & Wilhelm, F. Lett. **109, 050507 (2012).CASPubMedArticle22.Corcoles, A. **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 Steane Code

Blatt, "Experimental Repetitive Quantum Error Correction," Science 332, 1059-1061 (2011), doi:10.1126/science.1203329 ^ M. et al. H. check over here Taylor & Francis.

Quantum error correction is essential if one is to achieve fault-tolerant quantum computation that can deal not only with noise on stored quantum information, but also with faulty quantum gates, faulty 5-qubit Quantum Error Correction A 87, 030301 (2013).CASArticle23.Gambetta, J. Similarly for the phase-flip operation ,and for Since the state after the SWAP gate is (the qubits are ordered and Q1, Q3 are the code qubits), the error syndromes are given

## If we have a channel which causes errors independently with probability O(ε) on each qubit in the QECC, then the code will allow us to decode a correct state except with

Your cache administrator is webmaster. J. Of these, only the assumption of independent errors is at all necessary, and that can be considerably relaxed to allow short-range correlations and certain kinds of non-Markovian environments. Quantum Code 7 Phys.

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 measured populations are renormalized by the observed contrast at θ∼0 in Fig. 3, and in the equivalent plots for X and Z errors shown in the Methods, to account for The observed contrast between the different syndrome qubit state populations, near 0.6 in Fig. 3, is commensurate with a master-equation simulation that takes into account the measured coherence times of our http://vealcine.com/quantum-error/quantum-error.php A distance d stabilizer code which has nontrivial P ∈ S with wt(P) < d is called degenerate, whereas one which does not is non-degenerate.

If we want to make a QECC that can also correct phase (Z) errors, we should choose two classical codes C1 and C2, with parity check matrices H1 and H2. Lett. 109, 060501 (2012).CASPubMedArticle30.Magesan, E., Gambetta, J. Particular caution is necessary, as computational gates can cause errors to propagate from their original location onto qubits that were previously correct. Dark blue bars represent the ideal outcome for each ɛ and teal bars are measurements calibrated by the full X, Y and Z error rotation curves.

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 construction takes two binary classical linear codes and produces a quantum code, and can therefore take advantage of much existing knowledge from classical coding theory. Srinivasan, Andrew W. A SWAP gate operation is equivalent to three CNOT gates alternating direction (Fig. 8b).

This implies we expect that, to first order in θ, state tomography is robust to over-under rotation errors.We can model and verify this effect by directly applying a unitary error of Realization of three-qubit quantum error correction with superconducting circuits.