Abstract A central problem in open quantum systems is the characterization of non-Markovian processes, where an environment retains the memory of its interaction with the system. A key distinction is whether or not this memory can be simulated classically, as this can lead to efficient modelling and noise mitigation. Powerful tools have been developed recently within the process matrix formalism, a framework that conveniently characterizes all multi-time correlations through a sequence of measurements. This leads to a detailed classification of classical and quantum-memory processes and provides operational procedures to distinguish between them. However, these results leave open the question of what type of system-environment interactions lead to classical memory. More generally, process-matrix methods lack a direct connection to joint system-environment evolution, a cornerstone of open-system modelling. In this work, we characterize Hamiltonian and circuit-based models of system-environment interactions leading to classical memory. We show that general time-dependent Hamiltonians with product eigenstates, and where the environment’s eigenstates form a time-independent, orthonormal basis, always produce a particular type of classical memory: probabilistic mixtures of unitary processes. Equivalently, these Hamiltonians are characterized as commuting with a complete set of observables on the environment. Additionally, we show that the most general type of classical memory processes can be generated by a quantum circuit in which the system and environment interact through a specific class of controlled unitaries. Our results establish the first strong link between process-matrix methods and traditional Hamiltonian-based approaches to open quantum systems.
Abstract A central problem in open quantum systems is the characterization of non-Markovian processes, where an environment retains the memory of its interaction with the system. A key distinction is whether or not this memory can be simulated classically, as this can lead to efficient modelling and noise mitigation. Powerful tools have been developed recently within the process matrix formalism, a framework that conveniently characterizes all multi-time correlations through a sequence of measurements. This leads to a detailed classification of classical and quantum-memory processes and provides operational procedures to distinguish between them. However, these results leave open the question of what type of system-environment interactions lead to classical memory. More generally, process-matrix methods lack a direct connection to joint system-environment evolution, a cornerstone of open-system modelling. In this work, we characterize Hamiltonian and circuit-based models of system-environment interactions leading to classical memory. We show that general time-dependent Hamiltonians with product eigenstates, and where the environment’s eigenstates form a time-independent, orthonormal basis, always produce a particular type of classical memory: probabilistic mixtures of unitary processes. Equivalently, these Hamiltonians are characterized as commuting with a complete set of observables on the environment. Additionally, we show that the most general type of classical memory processes can be generated by a quantum circuit in which the system and environment interact through a specific class of controlled unitaries. Our results establish the first strong link between process-matrix methods and traditional Hamiltonian-based approaches to open quantum systems.
We present a method to detect quantum memory in a non-Markovian process. We call a process Markovian when the environment does not provide a memory that retains correlations across different system-environment interactions. We define two types of non-Markovian processes, depending on the required memory being classical or quantum. We formalise this distinction using the process matrix formalism, through which a process is represented as a multipartite state. Within this formalism, a test for entanglement in a state can be mapped to a test for quantum memory in the corresponding process. This allows us to apply separability criteria and entanglement witnesses to the detection of quantum memory. We demonstrate the method in a simple model where both system and environment are single interacting qubits and map the parameters that lead to quantum memory. As with entanglement witnesses, our method of witnessing quantum memory provides a versatile experimental tool for open quantum systems.
Quantum systems are rarely isolated from their surroundings. These hidden environments can retain information and return it later, producing correlated errors in quantum computers and complicating the interpretation of experiments. On a quantum processor, for example, one job can therefore leave an environmental trace that affects the next, even after the qubits themselves have been reset. Learning how this hidden influence persists and affects later behavior is therefore important, but the environment is often impossible to measure directly, and fully reconstructing its effects through tomography rapidly becomes impractical. Here, we introduce a response matrix that uses controlled interventions and observations of the system alone to reveal the temporal structure of environmental memory. The matrix records how future observations change across different controlled histories. Its spectrum characterizes how memory is structured and how it persists under specified system resets, while suitable linear combinations of its entries can rule out specified classical models of that memory. Because its history-future queries can be evaluated independently, numerical response matrices can be assembled in parallel. We establish the framework theoretically, characterize response spectra in an interacting spin process, and certify quantum memory in a two-qubit benchmark. This approach turns system observations into a probe of inaccessible environments and could support the diagnosis of correlated noise, tests of open system models, and memory-aware control design.
Currently, there is no systematic way to describe a quantum process with memory solely in terms of experimentally accessible quantities. However, recent technological advances mean we have control over systems at scales where memory effects are non-negligible. The lack of such an operational description has hindered advances in understanding physical, chemical, and biological processes, where often unjustified theoretical assumptions are made to render a dynamical description tractable. This has led to theories plagued with unphysical results and no consensus on what a quantum Markov (memoryless) process is. Here, we develop a universal framework to characterize arbitrary non-Markovian quantum processes. We show how a multitime non-Markovian process can be reconstructed experimentally, and that it has a natural representation as a many-body quantum state, where temporal correlations are mapped to spatial ones. Moreover, this state is expected to have an efficient matrix-product-operator form in many cases. Our framework constitutes a systematic tool for the effective description of memory-bearing open-system evolutions.
Memory is the fundamental form of temporal complexity: when present but uncontrollable, it manifests as non-Markovian noise; conversely, if controllable, memory can be a powerful resource for information processing. Memory effects arise from/are transmitted via interactions between a system and its environment; as such, they can be either classical or quantum. From a practical standpoint, quantum processes with classical memory promise near-term applicability: they are more powerful than their memoryless counterpart, yet at the same time can be controlled over significant timeframes without being spoiled by decoherence. However, despite practical and foundational value, apart from simple two-time scenarios, the distinction between quantum and classical memory remains unexplored. Here, we analyse multi-time quantum processes with memory mechanisms that transmit only classical information forward in time. Complementing this analysis, we also study two related – but simpler to characterise – sets of processes that could also be considered to have classical memory from a structural perspective, and demonstrate that these lead to remarkably distinct phenomena in the multi-time setting. Subsequently, we systematically stratify the full hierarchy of memory effects in quantum mechanics, many levels of which collapse in the two-time setting, making our results genuinely multi-time phenomena.
Abstract Established methods for characterizing quantum information processes do not capture non-Markovian (history-dependent) behaviors that occur in real systems. These methods model a quantum process as a fixed map on the state space of a predefined system of interest. Such a map averages over the system’s environment, which may retain some effect of its past interactions with the system and thus have a history-dependent influence on the system. Although the theory of non-Markovian quantum dynamics is currently an active area of research, a systematic characterization method based on a general representation of non-Markovian dynamics has been lacking. In this article we present a systematic method for experimentally characterizing the dynamics of open quantum systems. Our method, which we call quantum process identification (QPI), is based on a general theoretical framework which relates the (non-Markovian) evolution of a system over an extended period of time to a time-local (Markovian) process involving the system and an effective environment. In practical terms, QPI uses time-resolved tomographic measurements of a quantum system to construct a dynamical model with as many dynamical variables as are necessary to reproduce the evolution of the system. Through numerical simulations, we demonstrate that QPI can be used to characterize qubit operations with non-Markovian errors arising from realistic dynamics including control drift, coherent leakage, and coherent interaction with material impurities.
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.