Simulation of a five qubits convolutional code
1Aziz Mouzali, 2Fatiha Merazka
1Departement of physics, ENPEI, Rouiba, Algeria.
2 Computer Engineering Faculty, USTHB, Algeria.
Introduction
We describe in this work a five-qubit quantum convolutional error correcting code and its implementation on a classical computer. The encoding and decoding circuits and an error correction procedure are presented. We will verify that if any X, Y, Z error or any product of them occurs on one or two qubit, this correction always allows to recover the useful information or to obtain a list of possible errors. The originality in this correction is the winning time obtained by looking at only the required syndromes for each error, thus avoiding the decoherence phenomenon. Also, we give the average fidelity for double errors recovered as single errors having same syndrome. This work is described in more detail in convo2012.pdf , included with this application.
NOTE: a) This application requires the use of the Feynman package, by T.Radtke and S.Fritzsche. This Maple package is available from the CPC Program Library from Queen’s University in Belfast, as http://cpc.cs.qub.ac.uk/summaries/ADWE_v4_0.html. You must install that package first before running this application. b) The running time is very short for the coding circuit but can be much more higher for the orthers parts of the application.
Acknowledgement
We would like to thank very much Mrs S.Fritzsch and T.Radtke for providing us with the version 4 (2008) of Feynman Program.
Coding Circuit
This circuit is schematized in figure 11 , page 22 of the ref[4] (see the included file convo2012.pdf).
Stabilisation of the coded state Q25 We verify that the obtained coded state Q25 is correct.
Channel error We introduce here an example of error affecting some qubits during the transmission in the channel.
Syndrome extraction The syndrome extraction is explained in convo2012.pdf. It is a non-destructive measure on qubits which allow us to know if the qubits states are affected or not whithout changing them. For example, the symbol "c" in the instruction Q281a means that the operator X is applied on qubits 2 only if the first one has the value 1. We have added the symbol # next to the instruction which are not rewuired for detection the error introduced in Q27 as example. One can change the position of this symbol next to the required instructions upon on the generators M to be measured to detect the choosen error. This procedure allow us to reduce the running time.
Error detection and correction The syndrome extraction allow us to identify the channel error and then to correct it. See table 2 in convo2012.pdf to understand the procedure.
Decoding circuit It is the inversed coding circuit. This part of the simulation could have a very high running time or to get stuck . Then it is advisable to run the application part by part. We do not know if Feynman program needs some modifications or if it contains others properties that we ignore and allowing a better decoding procedure with a smaller execution time.
