Abstract Local decoders, also known as cellular-automaton decoders, offer a promising path toward real-time quantum error-correction by replacing centralized classical decoding, with inherent hardware constraints, with a natively parallel and streamlined architecture from a simple local transition rule. We propose two new types of local decoders for the quantum repetition code in one dimension. The signal-rule decoders interpret odd parities between neighboring qubits as defects, attracted to each other via the exchange of binary-signals. We prove the existence of a threshold in the code-capacity model and present numerical evidence of exponential logical error suppression under a phenomenological noise model, with data and measurement errors at each error correction cycle. Compared to previously known local decoders that suffer from sub-optimal threshold and scaling, our construction significantly narrows the gap with global decoders for practical system sizes and error rates. Implementation requirements can be further reduced by eliminating the need for local classical memories, with a new rule defined on two rows of qubits. This shearing-rule works well at relevant system sizes making it an appealing short-term solution. When combined with biased-noise qubits, such as cat qubits, these decoders enable a fully local quantum memory in one dimension.
Abstract Local decoders, also known as cellular-automaton decoders, offer a promising path toward real-time quantum error-correction by replacing centralized classical decoding, with inherent hardware constraints, with a natively parallel and streamlined architecture from a simple local transition rule. We propose two new types of local decoders for the quantum repetition code in one dimension. The signal-rule decoders interpret odd parities between neighboring qubits as defects, attracted to each other via the exchange of binary-signals. We prove the existence of a threshold in the code-capacity model and present numerical evidence of exponential logical error suppression under a phenomenological noise model, with data and measurement errors at each error correction cycle. Compared to previously known local decoders that suffer from sub-optimal threshold and scaling, our construction significantly narrows the gap with global decoders for practical system sizes and error rates. Implementation requirements can be further reduced by eliminating the need for local classical memories, with a new rule defined on two rows of qubits. This shearing-rule works well at relevant system sizes making it an appealing short-term solution. When combined with biased-noise qubits, such as cat qubits, these decoders enable a fully local quantum memory in one dimension.
Execution of quantum algorithms on large-scale quantum computers will require extremely low logical error rates, which necessitates the development of scalable decoding architectures. Local decoders are promising candidates for this task, as they avoid the communication and data processing bottlenecks inherent in global decoding strategies. Cellular automaton (CA) decoders represent a distinct class of local decoders, offering a path toward the low-latency, real-time decoding required for practical applications. In this work, we present SCALA (Signaling CA with Local Attraction), a novel non-hierarchical cellular automaton decoder for quantum repetition and toric codes. By evaluating SCALA alongside the hierarchical CA decoder proposed by Harrington, we provide a direct comparison between non-hierarchical and renormalization-group-style local decoding strategies. We characterize SCALA across three key metrics: Performance, scalability, and robustness. Our results show that SCALA achieves a code-capacity threshold of approximately pc≈7.5% and provides strong sub-threshold scaling of about pL∝pd/4 on the toric code. In terms of scalability, our non-hierarchical design ensures that the local computational resources remain independent of system size, yielding a modular local architecture suitable for hardware implementation. Finally, SCALA demonstrates strong robustness to qubit measurement errors and noise within the decoder itself, a critical advantage for real-time decoding on noisy hardware. Our results establish SCALA as a high-performance, scalable, and robust local decoder for scalable quantum error correction.
Local decoders provide a promising approach to real-time quantum error-correction by replacing centralized classical decoding, with significant hardware constraints, by a fully distributed architecture based on a simple, local update rule. We propose a new local decoder for Kitaev's toric code: the 2D signal-rule, that interprets odd parity stabilizer measurements as defects, attracted to each other via the exchange of binary signals. We present numerical evidence of exponential suppression of the logical error rate with system size below a threshold, under a phenomenological noise model with data and measurement errors at each iteration. The construction achieves a significantly improved threshold and optimal finite-size scaling relative to hierarchical schemes. It also provides a lightweight alternative to windowed local decoder constructions while maintaining strong performance, thus enabling a streamlined architecture for a two-dimensional local quantum memory.
Quantum error correction (QEC) normally requires active maintenance using measurements, classical processing and externally controlled correction schedules. Passive QEC, by contrast, was previously established only in unphysical spatial dimensions. Here, we give an explicit scheme for autonomous quantum error correction and computation in two dimensions, formulated as a dissipative quantum cellular automaton with a fixed, local and translation-invariant update rule. The construction uses hierarchical, self-simulating control elements based on ideas from the seminal classical results of Gács (1986, 1989) together with a measurement-free concatenated quantum code. We prove the existence of a nonzero noise threshold under a local noise model. Below this threshold, logical errors on encoded initial states are suppressed exponentially with increasing system size and the memory lifetime diverges in the thermodynamic limit. As an extension, we outline how this mechanism can also be implemented in continuous time by a time-independent, translation-invariant local Lindbladian. Quantum circuits can moreover be encoded in the automaton's initial state and executed fault-tolerantly with polylogarithmic overhead under the same circuit-independent rule. Our scheme thus realizes a self-correcting quantum computer capable of universal computation.
We introduce a new framework for constructing topological quantum memories, by recasting error recovery as a dynamical process on a field generating cellular automaton. We envisage quantum systems controlled by a classical hardware composed of small local memories, communicating with neighbours and repeatedly performing identical simple update rules. This approach does not require any global operations or complex decoding algorithms. Our cellular automata draw inspiration from classical field theories, with a Coulomb-like potential naturally emerging from the local dynamics. For a 3D automaton coupled to a 2D toric code, we present evidence of an error correction threshold above 6.1% for uncorrelated noise. A 2D automaton equipped with a more complex update rule yields a threshold above 8.2%. Our framework provides decisive new tools in the quest for realising a passive dissipative quantum memory. A new error correction method for quantum computing memories is based on local computing elements. Michael Herold from the Freie Universität Berlin in Germany, with colleagues in Germany, Denmark and the UK, sought to address the challenge of maintaining information stored in topological quantum memories. Without a stable memory, delicate quantum states can decay quickly, introducing errors in stored information. Error correction is an important process in stabilizing topological memories, but was previously conceived as a system-wide process. The proposed practical error correction mechanism relies on parallel cellular operations within the topological quantum memory, so that the local operations replace the need for a complex system-wide scheme. The concept has the further benefit of being compatible with classical hardware, and it is easily scalable.
A quantum computation can be described by the evolution of a collection of quantum bits, or qubits, according to the laws of quantum mechanics, from an initial state encoding the input of a problem, to its solution represented by the final state. A machine capable of performing this task, a quantum computer, must be robust against errors caused by the inherent imperfections in the fabrication and control of such complex systems. This thesis examines the theory of fault‑tolerant quantum computation, tailored to the physical constraints of today’s leading platforms, most notably superconducting circuits and neutral‑atom arrays.A fault-tolerant quantum computation requires to correct errors in real time, at the cost of a dedicated architectural layer whose integration may introduce additional noise. A possible simplification uses a quantum memory model where information is encoded in a pair of configurations of a multi-qubit system, stabilized by the uniform application of a simple local rule fitting within the theory of cellular automata. We propose a short-term solution, the shearing-rule which protects a one-dimensional repetition code against errors with good performance but lacks a threshold. This shortcoming is addressed by adding small local classical memories, used to define the signal-rules decoders, where error correction relies on exchanging binary signals. Combined with biased-noise qubits, such as cat qubits, these local decoders define a fully local one-dimensional quantum memory.The final part of the manuscript addresses the advantage of quantum machines over their classical counterparts. We emphasize that any such demonstration must account for the presence of noise to be meaningful — for example, while exact sampling from the output distribution of circuits drawn from certain families is believed to be hard for classical computers, the task may become tractable if one allows for an approximate solution that reflects the presence of noise. In this context, we propose a constant-depth quantum circuit that solves a sampling problem in a fault-tolerant manner. Because our approach is particularly well-suited to the atom rearrangement capabilities of neutral-atom array platforms, we hope it will contribute to a future experimental demonstration of a fault-tolerant quantum advantage.
Self-correcting quantum memories demonstrate robust properties that can be exploited to improve active quantum error-correction protocols. Here we propose a cellular automaton decoder for a variation of the color code where the bases of the physical qubits are locally rotated, which we call the XYZ color code. The local transformation means our decoder demonstrates key properties of a two-dimensional fractal code if the noise acting on the system is infinitely biased towards dephasing, namely, no string-like logical operators. As such, in the high-bias limit, our local decoder reproduces the behavior of a partially self-correcting memory. At low error rates, our simulations show that the memory time diverges polynomially with system size without intervention from a global decoder, up to some critical system size that grows as the error rate is lowered. Furthermore, although we find that we cannot reproduce partially self-correcting behavior at finite bias, our numerics demonstrate improved memory times at realistic noise biases. Our results therefore motivate the design of tailored cellular automaton decoders that help to reduce the bandwidth demands of global decoding for realistic noise models.