Fault-tolerant quantum computation with a soft-decision decoder for error correction and detection by teleportation

Hayato Goto, Hironori Uchikawa
2013 Scientific Reports  
Fault-tolerant quantum computation with quantum error-correcting codes has been considerably developed over the past decade. However, there are still difficult issues, particularly on the resource requirement. For further improvement of fault-tolerant quantum computation, here we propose a softdecision decoder for quantum error correction and detection by teleportation. This decoder can achieve almost optimal performance for the depolarizing channel. Applying this decoder to Knill's C 4 /C 6
more » ... eme for fault-tolerant quantum computation, which is one of the best schemes so far and relies heavily on error correction and detection by teleportation, we dramatically improve its performance. This leads to substantial reduction of resources. Q uantum computers 1,2 are expected to outperform current classical computers. Many problems intractable for classical computers are believed to be solved by quantum computers more efficiently 1,3-11 . The most famous one is the prime number factoring problem 3 , the difficulty of which ensures today's internet security. The origin of the speed of quantum computation is quantum superposition of physical states. This enables us to perform a great number of calculations in parallel (quantum parallelism). Unfortunately, the quantum superposition is very fragile. The destruction of the superposition is called decoherence. The decoherence induces errors in quantum computation 12 and makes quantum computers difficult to be realized. The standard approaches to this problem are based on quantum error correction. Using quantum errorcorrecting codes, we can make quantum computation fault-tolerant 1,13 . If the error probabilities of elementary operations are lower than a threshold, we can, in principle, perform arbitrarily long quantum computation reliably. This fact is known as the threshold theorem. The threshold has gone up to about 1% 14-19 as a result of theoretical advances over the past decade. Although this value is comparable to error probabilities in state-of-the-art experiments 2,20,21 , this does not mean that the realization of quantum computers is within reach. There are still difficult issues, particularly on resource requirement. First, the threshold is the value at which necessary resources become infinite. Therefore, the error probabilities should be much lower than the threshold. Second, even if the error probabilities become as low as 0.1%, the resources required for practical quantum computation will still be enormous 22,23 . Thus, further improvement of fault-tolerant quantum computation has been desired. Towards more efficient fault-tolerant quantum computation, here we propose a new decoder using softdecision decoding. Decoding is a crucial part of error correction in both quantum and classical situations. In the history of classical error correction, the use of soft-decision decoding based on probabilistic inference, instead of conventional hard-decision decoding based on algebraic techniques, was a key step to achieve the theoretical limit 24 . This is natural because decoding is, in essence, a problem of probabilistic inference. In general, such a problem is computationally hard. In the case of classical error correction, clever algorithms and approximations with appropriate error-correcting codes have enabled efficient soft-decision decoding. In the case of quantum error correction, an efficient soft-decision (optimal) decoding is possible for quantum concatenated codes, which has been shown by Poulin 25 . The decoding has displayed high performance on a simple quantum channel called the depolarizing channel. To the best of our knowledge, however, this has not been applied to fault-tolerant quantum computation. The reason for this is probably as follows: this algorithm is based on conventional syndrome measurements, which require many iterative fault-tolerant measurements 13,26 and consequently may
doi:10.1038/srep02044 pmid:23784512 pmcid:PMC3687222 fatcat:bgcanhzncnhanakayfinop6keu