#QUANTUM ERROR CORRECTION THRESHOLD FOR SURFACE CODE CODE#
The minimum weight of the logical operator defines the code distance (d), and \(d=5\) surface code with boundaries is shown in Fig. Let us denote Logical X by \(X_L\) and Logical Z by \(Z_L\). By applying physical X(Z) operation on the edges in logical operators, logical operators can be performed. Logical X (Logical Z) connects smooth (rough) boundaries. Logical operators that are homologically non-trivial chains are operators that connect smooth or rough boundaries. Section 4 describes the simulation results, and we conclude in Sect. 3, we present the rectangular surface codes. 2, we review some backgrounds of the surface code and introduce the noise model. The remainder of this paper is organized as follows: In Sect. Edmonds’ MWPM algorithm counts the weight of noise that causes the observed syndrome and performs error correction using the minimum weight error chain. We applied the Edmonds’ minimum weight perfect matching (MWPM) algorithm to decode the surface codes however, the expected alternative algorithms such as the machine learning (ML) decoder could be applied. We simulated the performance of the optimal lattice size surface code to verify whether the code has achieved a given logical failure. As a result, the optimal lattice size for the required logical error rate could be derived. The number of qubits for rectangular surface code was calculated to minimize the overhead. Secondly, we analyzed an overhead as a function of the single-qubit physical error rate and the logical failure rate. We scaled the reduced logical failure rate using the lattice size and the physical error rate as parameters. Thereafter, we analyzed the impact of the large weight of the logical Z operator on logical X error. We have proposed a larger weight of logical Z operator than logical X operator because logical Z(X) error occurs only due to Z(X) physical error. In other words, Pauli Z errors occurred at a higher rate than Pauli X errors. In this study, we first explored a method for reducing the logical failure rate of the surface code with a non-periodic boundary when the noise was biased. This kind of biased error arises in superconducting qubits, quantum dots, and trapped-ion qubit systems. proposed an effective machine learning decoder in the surface code under a biased noise error channel by using X, Y stabilizer instead of X, Z stabilizer and thus obtained more information regarding Z errors.
By employing the phase flip code as an inner code, the number of Z errors that induce logical Z error increases, and the logical operation can be performed with an outer code, such as an RM code or a topological code. proposed a concatenated phase flip QEC code. A framework that applies machine learning techniques for decoding and improves the logical error rate in the depolarizing noise channel is also proposed in. Recent QEC research focuses on the implementation and construction of the QEC codes by considering a biased noise error channel and other channels, by designing the efficient decoder using the machine learning techniques, and by improving the threshold below which the logical failure rate can be decreased. In both cases, the number of physical qubits scales as O( \(L^2\)). The toric code can encode two logical qubits, and surface codes with non-periodic boundaries can encode one logical qubit using physical qubits. Surface codes with non-periodic boundaries which allow planar qubit location have been developed. The earlier topological codes of Kitaev called toric codes have periodic boundaries and qubits that are located on the torus. In particular, because of the lower stabilizer weight than circuit-based codes and the nearest stabilizer measurement operation of topological codes, surface codes such as toric codes and color codes have become the main topics in several researches. Two types of QEC codes such as circuit-based codes (Shor code, Steane code, RM codes, etc.) and topological codes have been developed to protect the quantum states. For the realization of quantum computing, errors that are induced when interacted with the environment should be detected using quantum error correction (QEC) codes.
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