Matching algorithms can be used for identifying errors in quantum systems, being the most famous the Blossom algorithm. Recent works have shown that small distance quantum error correction codes can be efficiently decoded by employing machine learning techniques based on neural networks (NN). Various NN-based decoders have been proposed to enhance the decoding performance and the decoding time. Their implementation differs in how the decoding is performed, at logical or physical level, as well as in several neural network related parameters. In this work, we implement and compare two NN-based decoders, a low level decoder and a high level decoder, and study how different NN parameters affect their decoding performance and execution time. Crucial parameters such as the size of the training dataset, the structure and the type of the neural network, and the learning rate used during training are discussed. After performing this comparison, we conclude that the high level decoder based on a Recurrent NN shows a better balance between decoding performance and execution time and it is much easier to train. We then test its decoding performance for different code distances, probability datasets and under the depolarizing and circuit error models.
Matching algorithms can be used for identifying errors in quantum systems, being the most famous the Blossom algorithm. Recent works have shown that small distance quantum error correction codes can be efficiently decoded by employing machine learning techniques based on neural networks (NN). Various NN-based decoders have been proposed to enhance the decoding performance and the decoding time. Their implementation differs in how the decoding is performed, at logical or physical level, as well as in several neural network related parameters. In this work, we implement and compare two NN-based decoders, a low level decoder and a high level decoder, and study how different NN parameters affect their decoding performance and execution time. Crucial parameters such as the size of the training dataset, the structure and the type of the neural network, and the learning rate used during training are discussed. After performing this comparison, we conclude that the high level decoder based on a Recurrent NN shows a better balance between decoding performance and execution time and it is much easier to train. We then test its decoding performance for different code distances, probability datasets and under the depolarizing and circuit error models.
Quantum error correction (QEC) is key to have reliable quantum computation and storage, due to the fragility of qubits in current quantum technology and the imperfect application of quantum operations. In order to have efficient quantum computation and storage, active QEC is required. QEC consists of an encoding and a decoding process. The way that encoding protects quantum information is through grouping many unreliable physical qubits into one more reliable logical qubit. Then, computation occurs based on the logical qubits, however, errors still occur on the physical qubits. Decoding is the process of identifying the location and type of errors occurring on the physical qubits. The decoder proposes corrections against the errors that have been identified. In this thesis, we are exploring novel ways to design decoders for QEC codes, focusing on the surface code. We began our investigation by implementing a rule-based decoder for the smallest surface code, which consists of 17 qubits. We incorporated this decoder to a platform that we created, called Quantum Platform Development Framework (QPDO), in order to study the working principles of a Pauli frame and to quantify its potential effect on the decoding performance. The Pauli frame unit keeps track of errors on physical qubits without the need to apply corrections constantly. We quantified through simulation the benefits in terms of the decoding performance and the execution schedule of QEC, minimizing the idle time. Minimizing the execution time is critical, due to the limited time budget of quantum error correction, thus requiring a high speed decoder capable of still reaching high decoding performance. We show that when the decoding time is equal to the time required to run a surface code cycle, the decoder reaches its maximum performance. However, such a rule-based decoder cannot easily scale to larger quantum systems, therefore other decoding approaches should be considered. Most of the classical decoders that have been developed so far, do not have a good balance between short execution time and high decoding performance. Therefore, we proposed decoders that incorporate neural networks to keep the execution time small, while keeping the decoding performance high. We designed a two-module decoder, which included a classical module and a neural network. We named this configuration neural network based decoder (NNbD). We compare different designs of NNbDs with classical decoders and prove that NNbDs can reach similar or better decoding performance compared to classical decoders while having constant execution time. Furthermore, we quantified the execution time of a NNbD and argued about the speed that can be achieved in a hardware chip like a Field Programmable Gate Array (FPGA) or an Application-Specific Integrated Circuit (ASIC). Both the classical module and the neural network are highly parallelizable and fast modules by construction, leading to constant execution time for a given code distance. We proved that neural network based decoders can adapt to any noise model, since the neural network functionality is based on creating a map between the input and output data, requiring no knowledge about the underlying error model. Following that, a comparison between different NNbD design approaches was performed. We show that it is advantageous to start with a classical decoding module and improve on its decoding performance with a neural network rather than having a neural network perform the decoding on its own. Also, in the latter case, the execution time of such a decoder is non-constant and on average larger than the decoder containing a classical module and a neural network. Moreover, we show that for the design containing a classical module and a neural network, the execution time is increasing linearly as the code distance increased, which was mainly attributed to the increase of the size of the neural network. However, there is a fundamental difference between NNbDs and classical decoders in that NNbDs require sampling and training based on data obtained from the problem, unlike classical decoders. As the code distance increases, the amount of data required to be gathered and trained are exponentially increasing, imposing a limit to the size of the quantum system that can be efficiently decoded. We proposed as a solution to have a distributed decoding approach that divides the code into small regions and then decodes each region locally. We show that using such a distributed decoding approach for small code distances does not lead to significant loss in decoding performance, while simultaneously providing a way to decode large code distances. Thus, we were able to create a decoder that can achieve high decoding performance with constant execution time. However, there are still some issues to keep in mind with such kind of decoders. The main challenge of NNbDs is that they are a dedicated decoder for a given problem. Every time that some aspect of the problem changes (quantum error correcting code, code distance, error model), sampling, training and evaluating the decoder needs to be repeated. Moreover, there is a large number of neural network parameters that need to be specifically tuned when the problem changes. A careful study of the design choices is required to maximize the performance of the decoder. We envision that when sampling and training are performed in hardware, the time required for these processes will be decreased compared to the time required in software. Finally, if the hardware resources allow us to include multiple neural networks, then this can potentially increase the decoding performance. As we presented, dividing the task of decoding to smaller tasks that are distributed to many neural networks can be beneficial.
Abstract Surface codes reach high error thresholds when decoded with known algorithms, but the decoding time will likely exceed the available time budget, especially for near-term implementations. To decrease the decoding time, we reduce the decoding problem to a classification problem that a feedforward neural network can solve. We investigate quantum error correction and fault tolerance at small code distances using neural network-based decoders, demonstrating that the neural network can generalize to inputs that were not provided during training and that they can reach similar or better decoding performance compared to previous algorithms. We conclude by discussing the time required by a feedforward neural network decoder in hardware.
Errors in surface code have typically been decoded by Minimum Weight Perfect Matching (MWPM) based method. Recently, neural-network-based Machine Learning (ML) techniques have been employed for this purpose. Here we propose a two-level (low and high) ML-based decoding scheme, where the first level corrects errors on physical qubits and the second one corrects any existing logical errors, for different noise models. Our results show that our proposed decoding method achieves ∼10× and ∼2× higher values of pseudo-threshold and threshold respectively, than for MWPM. We show that usage of more sophisticated ML models with higher training/testing time, do not provide significant improvement in the decoder performance. Finally, data generation for training the ML decoder requires significant overhead hence lower volume of training data is desirable. We have shown that our decoder maintains a good performance with the train-test-ratio as low as 40:60.
Quantum error correction is a crucial technology for realizing quantum computers. These computers achieve fault-tolerant quantum computing by detecting and correcting errors using decoding algorithms. Quantum error correction using neural network-based machine learning methods is a promising approach that is adapted to physical systems without the need to build noise models. In this paper, we use a distributed decoding strategy, which effectively alleviates the problem of exponential growth of the training set required for neural networks as the code distance of quantum error-correcting codes increases. Our decoding algorithm is based on renormalization group decoding and recurrent neural network decoder. The recurrent neural network is trained through the ResNet architecture to improve its decoding accuracy. Then we test the decoding performance of our distributed strategy decoder, recurrent neural network decoder, and the classic minimum weight perfect matching (MWPM) decoder for rotated surface codes with different code distances under the circuit noise model, the thresholds of these three decoders are about 0.0052, 0.0051, and 0.0049, respectively. Our results demonstrate that the distributed strategy decoder outperforms the other two decoders, achieving approximately a 5% improvement in decoding efficiency compared to the MWPM decoder and approximately a 2% improvement compared to the recurrent neural network decoder.
Abstract The decoding of error syndromes of surface codes with classical algorithms may slow down quantum computation. To overcome this problem it is possible to implement decoding algorithms based on artificial neural networks. This work reports a study of decoders based on convolutional neural networks, tested on different code distances and noise models. The results show that decoders based on convolutional neural networks have good performance and can adapt to different noise models. Moreover, explainable machine learning techniques have been applied to the neural network of the decoder to better understand the behaviour and errors of the algorithm, in order to produce a more robust and performing algorithm.
A fault-tolerant quantum computation requires an efficient means to detect and correct errors that accumulate in encoded quantum information. In the context of machine learning, neural networks are a promising new approach to quantum error correction. Here we show that a recurrent neural network can be trained, using only experimentally accessible data, to detect errors in a widely used topological code, the surface code, with a performance above that of the established minimum-weight perfect matching (or blossom) decoder. The performance gain is achieved because the neural network decoder can detect correlations between bit-flip (X) and phase-flip (Z) errors. The machine learning algorithm adapts to the physical system, hence no noise model is needed. The long short-term memory layers of the recurrent neural network maintain their performance over a large number of quantum error correction cycles, making it a practical decoder for forthcoming experimental realizations of the surface code.