Simple scheme for encoding and decoding a qubit in unknown state for various topological codes.

Justyna Łodyga,Michał Horodecki,Paweł Mazurek,Andrzej Grudka

doi:10.1038/srep08975

Abstract

We present a scheme for encoding and decoding an unknown state for CSS codes, based on syndrome measurements. We illustrate our method by means of Kitaev toric code, defected-lattice code, topological subsystem code and 3D Haah code. The protocol is local whenever in a given code the crossings between the logical operators consist of next neighbour pairs, which holds for the above codes. For subsystem code we also present scheme in a noisy case, where we allow for bit and phase-flip errors on qubits as well as state preparation and syndrome measurement errors. Similar scheme can be built for two other codes. We show that the fidelity of the protected qubit in the noisy scenario in a large code size limit is of , where p is a probability of error on a single qubit per time step. Regarding Haah code we provide noiseless scheme, leaving the noisy case as an open problem.

Highlights

We present a scheme for encoding and decoding an unknown state for CSS codes, based on syndrome measurements
Quantum information is stored here in the codespace of total Hilbert space
An example of topological stabilizer code is a surface Kitaev code embedded on a square lattice[11], which uses topological properties to provide protection against noise whenever the qubit error rate is below some threshold value

Summary

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We present a scheme for encoding and decoding an unknown state for CSS codes, based on syndrome measurements. We present an extension of the encoding/decoding procedure to topological stabilizer CSS codes where logical operators cross at more than one qubit This case is illustrated schematically, where ZL,i, XL,i operators act nontrivially on line of qubits. We discuss how to modify the procedure in order to encode an unknown state into topological CSS codes where both logical operators act nontrivially on larger (odd) number of qubits. The general idea is to prepare all qubits as described, measure Xs and Zp stabilizers many times in the area confined by the whole lattice (except for the last time step where X and Z operators are measured), store all error syndromes and use them to apply error correcting procedure. 12, where pi stands for the probability of error on the i-th qubit

For qubits in the green bottom triangle pi

Conclusions

Additional information