The Pauli frame mechanism allows Pauli gates to be tracked in classical electronics and can relax the timing constraints for error syndrome measurement and error decoding. When building a quantum computer, such a mechanism may be beneficial, and the goal of this paper is not only to study the working principles of a Pauli frame but also to quantify its potential effect on the logical error rate. To this purpose, we implemented and simulated the Pauli frame module which, in principle, can be directly mapped into a hardware implementation. Simulation of a surface code 17 logical qubit has shown that a Pauli frame can reduce the error rate of a logical qubit up to 70% compared to the same logical qubit without Pauli frame when the decoding time equals the error correction time, and maximum parallelism can be obtained.
The Pauli frame mechanism allows Pauli gates to be tracked in classical electronics and can relax the timing constraints for error syndrome measurement and error decoding. When building a quantum computer, such a mechanism may be beneficial, and the goal of this paper is not only to study the working principles of a Pauli frame but also to quantify its potential effect on the logical error rate. To this purpose, we implemented and simulated the Pauli frame module which, in principle, can be directly mapped into a hardware implementation. Simulation of a surface code 17 logical qubit has shown that a Pauli frame can reduce the error rate of a logical qubit up to 70% compared to the same logical qubit without Pauli frame when the decoding time equals the error correction time, and maximum parallelism can be obtained.
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.
Quantum error correction (QEC), the lynchpin of fault-tolerant quantum computing (FTQC), is designed and validated against well-behaved Pauli stochastic error models. But in real-world deployment, QEC protocols encounter a vast array of other errors -- coherent and non-Pauli errors -- whose impacts on quantum circuits are vastly different than those of stochastic Pauli errors. The impacts of these errors on QEC and FTQC protocols have been largely unpredictable to date due to exponential classical simulation cost. Here, we show how to accurately and efficiently model the effects of coherent and non-Pauli errors on FTQC, and we study the effects of such errors on syndrome extraction for surface and bivariate bicycle codes, and on magic state cultivation. Our analysis suggests that coherent error can shift fault-tolerance thresholds, increase the space-time cost of magic state cultivation, and can increase logical error rates by an order of magnitude compared to equivalent stochastic errors. These analyses are enabled by a new technique for mapping any Markovian circuit-level error model with sufficiently small error rates onto a detector error model (DEM) for an FTQC circuit. The resulting DEM enables Monte Carlo estimation of logical error rates and noise-adapted decoding, and its parameters can be analytically related to the underlying physical noise parameters to enable approximate strong simulation.
Correcting errors in real time is essential for reliable large-scale quantum computations. Realizing this high-level function requires a system capable of several low-level primitives, including single-qubit and two-qubit operations, mid-circuit measurements of subsets of qubits, real-time processing of measurement outcomes, and the ability to condition subsequent gate operations on those measurements. In this work, we use a ten qubit QCCD trapped-ion quantum computer to encode a single logical qubit using the [[7,1,3]] color code, first proposed by Steane~\cite{steane1996error}. The logical qubit is initialized into the eigenstates of three mutually unbiased bases using an encoding circuit, and we measure an average logical SPAM error of 1.7(6)×10−3, compared to the average physical SPAM error 2.4(8)×10−3 of our qubits. We then perform multiple syndrome measurements on the encoded qubit, using a real-time decoder to determine any necessary corrections that are done either as software updates to the Pauli frame or as physically applied gates. Moreover, these procedures are done repeatedly while maintaining coherence, demonstrating a dynamically protected logical qubit memory. Additionally, we demonstrate non-Clifford qubit operations by encoding a logical magic state with an error rate below the threshold required for magic state distillation. Finally, we present system-level simulations that allow us to identify key hardware upgrades that may enable the system to reach the pseudo-threshold.
Building fault-tolerant quantum computers with large numbers of logical qubits requires both scalable hardware architectures and error-correcting codes that make efficient use of physical qubits. Photonic interconnects address both of these challenges by allowing the physical qubits to be distributed across multiple processors while providing the non-local connectivity necessary to implement resource-efficient codes such as high-rate quantum low-density parity-check (qLDPC) codes. A key requirement for realising this architecture is the ability to perform stabiliser measurements between remote processors, which has not previously been demonstrated experimentally. Here we report the first experimental demonstration of distributed quantum error detection and correction. We generate entanglement between network qubits in two separate trapped-ion processors and use it to perform remote syndrome measurements on data qubits. We first realise a distributed [[2, 1, 1]] repetition code, detecting phase-flip errors on a logical qubit encoded across the two modules in real time and suppressing logical errors. We then combine these mid-circuit syndrome measurements with real-time feedforward to actively correct arbitrary single-qubit Pauli errors on a distributed Bell state. These results provide an experimental foundation for quantum error correction (QEC) across modular quantum architectures.
Correcting errors in real time is essential for reliable large-scale quantum computations. Realizing this high-level function requires a system capable of several low-level primitives, including single-qubit and two-qubit operations, midcircuit measurements of subsets of qubits, real-time processing of measurement outcomes, and the ability to condition subsequent gate operations on those measurements. In this work, we use a 10-qubit quantum charge-coupled device trapped-ion quantum computer to encode a single logical qubit using the [[7,1,3]] color code, first proposed by Steane [Phys. Rev. Lett. 77, 793 (1996)PRLTAO0031-900710.1103/PhysRevLett.77.793]. The logical qubit is initialized into the eigenstates of three mutually unbiased bases using an encoding circuit, and we measure an average logical state preparation and measurement (SPAM) error of 1.7(2)×10^{-3}, compared to the average physical SPAM error 2.4(4)×10^{-3} of our qubits. We then perform multiple syndrome measurements on the encoded qubit, using a real-time decoder to determine any necessary corrections that are done either as software updates to the Pauli frame or as physically applied gates. Moreover, these procedures are done repeatedly while maintaining coherence, demonstrating a dynamically protected logical qubit memory. Additionally, we demonstrate non-Clifford qubit operations by encoding a T[over ¯]|+⟩_{L} magic state with an error rate below the threshold required for magic state distillation. Finally, we present system-level simulations that allow us to identify key hardware upgrades that may enable the system to reach the pseudothreshold.
Quantum error correction (QEC) is essential for achieving low error rates required for fault-tolerant quantum computation. In stabilizer-based codes such as the surface code, errors are inferred from repeated syndrome measurements and corrected by a classical decoder. To prevent error accumulation, decoding must be performed with both high throughput and low latency to keep pace with the QEC cycle and enable real-time feedback for universal logical operations. Here we report a hardware-integrated control architecture featuring an FPGA-based neural-network (NN) decoder and experimentally demonstrate real-time surface-code (distance-3) QEC on a superconducting quantum processor. The system achieves a deterministic closed-loop latency of 550 ns, including 124 ns for NN decoding, enabling feedback corrections within a 1.25 us QEC cycle. We show that real-time decoding and feedback correction achieve logical performance comparable to offline decoding while maintaining robustness against varying error conditions. We further demonstrate mid-circuit feedback correction in non-Clifford logical circuits, where Pauli-frame updating alone becomes insufficient. Our results establish a low-latency hardware architecture for embedded QEC control and provide a pathway towards scalable fault-tolerant quantum computing systems.