The study of quantum memories out of thermal equilibrium is motivated both by the need for memories that withstand realistic noise in fault-tolerant quantum computers and by the fundamental question of when quantum memory can emerge in autonomous many-body systems. We establish a general equivalence between CSS-based quantum memories subject to Pauli noise and classical memories via two associated probabilistic processes. In the asynchronous setting, where the quantum dynamics is generated by a Lindbladian, we identify a broad class of syndrome-based Lindbladians for which the quantum memory problem is equivalent to memory in continuous-time classical Markov dynamics. In the synchronous setting, we introduce syndrome-based probabilistic cellular automata (SPCAs) and establish a one-to-one correspondence between their classical memories and CSS-based quantum memories under local quantum channels. This framework provides a systematic route for deriving and constructing quantum memories from classical probabilistic processes, and recovers known results, including the fact that the 4D toric code is a quantum memory in the presence of thermal noise.
As an application, we give the first rigorous proof that the 4D toric code with Toom's rule is a quantum memory in the presence of local Pauli noise, provided the error is below a constant threshold. We prove this both for the corresponding synchronous, PCA-type quantum dynamics and for a Floquet Lindbladian. More generally, we show that classical memory in an SPCA can be lifted to a quantum memory under Floquet Lindbladian dynamics; whether this can be done with a time-independent Lindbladian remains open. Our results provide a general framework for relating classical and quantum memory, shed light on the conditions for fully autonomous quantum memories, and may guide the construction of provable autonomous quantum memories in low dimensions.
The study of quantum memories out of thermal equilibrium is motivated both by the need for memories that withstand realistic noise in fault-tolerant quantum computers and by the fundamental question of when quantum memory can emerge in autonomous many-body systems. We establish a general equivalence between CSS-based quantum memories subject to Pauli noise and classical memories via two associated probabilistic processes. In the asynchronous setting, where the quantum dynamics is generated by a Lindbladian, we identify a broad class of syndrome-based Lindbladians for which the quantum memory problem is equivalent to memory in continuous-time classical Markov dynamics. In the synchronous setting, we introduce syndrome-based probabilistic cellular automata (SPCAs) and establish a one-to-one correspondence between their classical memories and CSS-based quantum memories under local quantum channels. This framework provides a systematic route for deriving and constructing quantum memories from classical probabilistic processes, and recovers known results, including the fact that the 4D toric code is a quantum memory in the presence of thermal noise.
As an application, we give the first rigorous proof that the 4D toric code with Toom's rule is a quantum memory in the presence of local Pauli noise, provided the error is below a constant threshold. We prove this both for the corresponding synchronous, PCA-type quantum dynamics and for a Floquet Lindbladian. More generally, we show that classical memory in an SPCA can be lifted to a quantum memory under Floquet Lindbladian dynamics; whether this can be done with a time-independent Lindbladian remains open. Our results provide a general framework for relating classical and quantum memory, shed light on the conditions for fully autonomous quantum memories, and may guide the construction of provable autonomous quantum memories in low dimensions.
We study fault-tolerant quantum computation using a fixed, local error-correction rule and a geometrically simple code: the cubical (d,d′)-Toric code with d′,d−d′≥2, including the 4D Toric code. Under sufficiently weak local-stochastic Pauli noise, our construction uses noiseless, bounded-depth classical processing per physical time step. Its correction cycle acts only on the current syndrome, has constant depth, and is repeated unchanged during idle logical computation; it requires neither multiscale decoder states nor scale-dependent communication delays.
Our proof treats errors and syndromes as chains in a cubical complex, combining geometric shrinking under the sweep rule with a space--time explanation-tree argument inspired by Toom's theorem. It does not rely on expansion. We establish three results:
1.Explicit higher-dimensional sweep rule. We give a local algorithm for choosing sweep corrections on cubical (d,d′)-Toric complexes, resolving the local-existence question of Kubica and Preskill.
2. Fault-tolerant computation. We prove a circuit-level threshold theorem for noisy sweep cycles interleaved with logical gates. In particular, the 4D Toric code admits a fixed, constant-depth syndrome-based correction cycle without a multiscale decoder or expansion-based encoding. We do not claim constant total-depth overhead.
3. Ideal decoding under local-stochastic noise. For an initial local-stochastic Pauli error of rate p below threshold, nd+1 noiseless sweep cycles recover the encoded state on a side-length-n(d,d′)-Toric code with probability at least 1−O(poly(n))pΩ(n), extending earlier guarantees for independent errors.
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.
Correlated noise is a critical failure mode in quantum error correction (QEC), yet a gap remains between the stochastic Pauli models used for scalable QEC analyses and the microscopic, non-Markovian descriptions of noise in physical devices. We bridge this gap by introducing Spatiotemporal Pauli Processes (SPPs): the natural multi-time generalization of Pauli channels, joint probability distributions over Pauli faults across space and time, obtained exactly from any noise process, however non-Markovian, under standard Pauli-frame randomization. SPPs thereby provide a single language in which correlated noise can be recast, compared, and extended, compatible with both microscopic open systems modeling and the stabilizer workflow of QEC design and analysis. Our central result is constructive: the multi-time Pauli twirl acts by local contractions on a process tensor network, yielding an explicit classical tensor network whose virtual bonds encode memory, bounded by the environment's Liouville-space dimension. Transfer operator diagnostics link memory spectra to correlation decay, and hidden Markov representations enable efficient sampling of correlated Pauli fault trajectories, mapping microscopically derived noise directly into circuit-level QEC simulation. We demonstrate this with surface code memory and stability benchmarks up to distance 19. A temporal "storm" model with tunable correlation time shows that memory alone, at strictly fixed marginal error rates, systematically erodes the exponential error suppression expected from code distance. A genuinely spatiotemporal quantum cellular automaton bath maps exactly, under system twirling, to a nonlinear probabilistic cellular automaton; tuning its coherent interactions drives the noise into a pseudo-critical regime with critical slowing down and macroscopic error avalanches that reverse surface code distance scaling.
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.
Probabilistic cellular automata provide a simple framework for exploring classical nonequilibrium processes. Recently, quantum cellular automata have been proposed that rely on the propagation of a one-dimensional quantum state along a fictitious discrete time dimension via the sequential application of quantum gates. The resulting (1+1)-dimensional space-time structure makes these automata special cases of recurrent quantum neural networks which can implement broad classes of classical nonequilibrium processes. Here, we present a general prescription by which these models can be extended into genuinely quantum nonequilibrium models via the systematic inclusion of asynchronism. This is illustrated for the classical contact process, where the resulting model is closely linked to the quantum contact process (QCP), developed in the framework of open quantum systems. Studying the mean-field behavior of the model, we find evidence of an "asynchronism transition," i.e., a sudden qualitative change in the phase transition behavior once a certain degree of asynchronicity is surpassed, a phenomenon we link to observations in the QCP.
Local error correction can preserve classical information against noise, but protection under synchronized updates need not survive when sites evolve at independent random times. Establishing stability is particularly challenging when protection relies on the collective action of competing local rules. Here we develop a spacetime renormalization framework that reveals how independently acting local updates can collectively stabilize memory. We identify sufficient conditions under which the geometry of stable interfaces and suppression of defect propagation produce reliable erosion at large scales. Applying this framework, we prove stability of the asynchronous \textsf{R} squeezing code and introduce robust processes in two and three dimensions. In these systems, both encoded phases persist under sufficiently weak noise, even when the noise favors one phase. Our results provide a criterion for designing autonomous memories sustained by competing local stochastic dynamics.