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arXiv:2610.26228

The Fundamental Memory Ceiling of Noisy Quantum Reservoirs

Markus Baumann, Jonas Stein, Claudia Linnhoff-Popien

quant-phSubmitted Oct 19, 2026• Announced Oct 20, 2026• 0 citations
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Abstract

Fresh qubits, not size, set how long a noisy quantum system can remember. With $k$ input qubits reinitialized per time step, the longest memory any reservoir can attain under depolarizing noise is $\exp[Θ(k)]$ time steps, whatever its size, dynamics, and input encoding. A universal upper bound and an explicit reservoir that attains the exponential scaling establish this optimal resource law. The bound depends on the dynamics only through the entropy that fresh qubits remove per time step, so it holds for any Markovian noisy dynamics without feedback, including measurement-free error correction.

Comments
25 pages, 11 figures, including Supplemental Material. Reproducibility code and data included

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Fresh qubits, not size, set how long a noisy quantum system can remember. With kkk input qubits reinitialized per time step, the longest memory any reservoir can attain under depolarizing noise is exp⁡[Θ(k)]\exp[Θ(k)]exp[Θ(k)] time steps, whatever its size, dynamics, and input encoding. A universal upper bound and an explicit reservoir that attains the exponential scaling establish this optimal resource law. The bound depends on the dynamics only through the entropy that fresh qubits remove per time step, so it holds for any Markovian noisy dynamics without feedback, including measurement-free error correction.
Quantum Reservoir Computing (QRC) processes temporal data by retaining a memory of past inputs in the recurrent state of a quantum system, whereas a Quantum Extreme-Learning Machine (QELM) discards that memory, resetting the system at every step so that only the most recent input shapes the response. The two are usually treated as separate computational paradigms. We show that they are the two limits of a single architecture, connected by the input-encoding length, that is, the number of qubits overwritten with fresh data at each step. When a single qubit is re-encoded the system operates as a standard QRC, when the whole register is re-encoded it operates as a QELM, and intermediate lengths interpolate between them. The overwritten qubits hold the recent past in an explicit register, while the remaining qubits are never reset and carry older inputs forward in their evolving quantum state, so the encoding length redistributes memory between explicit and recurrent storage at fixed system size. Tuning the reservoir Hamiltonian and the evolution time with Bayesian optimization at each encoding length, we find that recurrent quantum memory is essential when a task must reach far into the past, and dispensable when the relevant history is short, where the memoryless reset limit already suffices. For every task the best reservoirs operate at the edge of chaos, where they perform as well as a densely connected reservoir with random all-to-all couplings of the same size, indicating that what temporal processing requires is the dynamical regime rather than the connectivity.
Scrambling quantum systems have attracted attention as effective substrates for temporal information processing. Here we consider a quantum reservoir processing framework that captures a broad range of physical computing models with quantum systems. We examine the scalability and memory retention of the model with scrambling reservoirs modelled by high-order unitary designs in both noiseless and noisy settings. In the former regime, we show that measurement readouts become exponentially concentrated with increasing reservoir size, yet strikingly do not worsen with the reservoir iterations. Thus, while repeatedly reusing a small scrambling reservoir with quantum data might be viable, scaling up the problem size deteriorates generalization unless one can afford an exponential shot overhead. In contrast, the memory of early inputs and initial states decays exponentially in both reservoir size and reservoir iterations. In the noisy regime, we also prove that memory decays exponentially in time for local noisy channels. These results required us to introduce new proof techniques for bounding concentration in temporal quantum models. Beyond this extreme scrambling regime, we numerically demonstrate that exponential concentration can still exist even with a physical reservoir such as an Ising model whenever the reservoir operates in a quantum-chaotic phase. In contrast, physical reservoirs in a many-body localized phase and at the edge of chaos appear to not suffer from such phenomena
Featuring memory of past inputs is a fundamental requirement for machine learning models processing time-dependent data. In quantum reservoir computing, all architectures proposed so far rely on Markovian dynamics, which, as we prove, inherently lead to an exponential decay of past information, thereby limiting long-term memory capabilities. We demonstrate that non-Markovian dynamics can overcome this limitation, enabling extended memory retention. By analytically deriving memory bounds and supporting our findings with numerical simulations, we show that non-Markovian reservoirs can outperform their Markovian counterparts, particularly in tasks that require a coexistence of short- and long-term correlations. We introduce an embedding approach that allows a controlled transition from Markovian to non-Markovian evolution, providing a path for practical implementations. Our results establish quantum non-Markovianity as a key resource for enhancing memory in quantum machine learning architectures, with broad implications in quantum neural networks.
Recent work established that, for independent erasures, storing KKK logical qubits for SSS time steps with error at most ε\varepsilonε requires a spacetime cost of Ω(S(K+log⁡(S/ε)))Ω(S(K+\log(S/\varepsilon)))Ω(S(K+log(S/ε))). A matching construction was also given, but assumes ideal error correction. Here we show that the same scaling can be achieved with an explicit noisy error-correction circuit and efficient decoding, counting all state-preparation, gate, measurement, and wait locations, provided that the hardware supports long-range qubit connectivity and fast, reliable classical processing. Thus, reliability adds only a logarithmic overhead shared by all stored qubits. The construction works for sufficiently weak but otherwise general circuit noise, including faults during syndrome extraction and correlated faults, and for independent erasures it matches the best known lower bound up to constant factors. We attain this scaling using quantum Tanner codes, with each memory step consisting of one round of syndrome extraction followed by a fixed number of parallel classical decoder steps that reduce the residual error enough to keep later faults correctable, regardless of memory size, storage time, or target accuracy. Together, this gives exponentially small failure probability per step while keeping the cost of each memory step linear in the code size. The same analysis also extends to certain constant-depth logical Clifford operations.
How complex can the responses of a quantum device become as it runs longer with a fixed internal memory? We quantify this complexity through sequential response capacity: how many adaptive testing stages, each using a fresh run, can continue to separate possible processes by a prescribed gap in response probabilities. For fixed system and memory sizes, we establish a tight law relating this capacity to run length and probability resolution. At fixed resolution, the capacity grows on the order of Klog⁡KK\log KKlogK, where KKK is the number of time steps in each run. Our construction attains this growth using time-dependent phase rotations on a single visible qubit with no additional internal memory; its tests give response probabilities exactly zero or one. Under the same tests, classical stochastic processes that measure in a fixed basis at every step have only linear capacity at fixed sizes and resolution. For phase sequences selected by a stored classical label, we then quantify how known independent Pauli noise changes this logarithmic enhancement. With ideal controls and weak residual phase noise after correction, we prove matching capacity bounds at a fixed small probability gap. These bounds identify the inverse residual phase-flip probability as the coherence timescale that limits the extra logarithmic growth.
Practical implementations of quantum technologies require preparation of states with a high degree of purity---or, in thermodynamic terms, very low temperatures. Given finite resources, the third law of thermodynamics prohibits perfect cooling; nonetheless, attainable upper bounds for the asymptotic ground-state population of a system repeatedly interacting with quantum thermal machines have been derived. These bounds apply within a memoryless (Markovian) setting, in which each refrigeration step proceeds independently of those previous. Here, we expand this framework to study the effects of memory on quantum cooling. By introducing a memory mechanism through a generalized collision model that permits a Markovian embedding, we derive achievable bounds that provide an exponential advantage over the memoryless case. For qubits, our bound coincides with that of heat-bath algorithmic cooling, which our framework generalizes to arbitrary dimensions. We lastly describe the adaptive stepwise optimal protocol that outperforms all standard procedures.