Adaptive agents realise complex reactive behaviours by using a memory of past input stimuli and output actions to guide structured future responses. Quantum adaptive agents can operate while storing less information in memory than optimal classical counterparts; yet, this does not necessarily translate into a reduced dimension of the memory that must be physically realised. We introduce a route-truncate-repair procedure that converts entropic quantum memory advantages into reductions in memory dimension. Routing a reference input process through an agent yields a temporal matrix product state representation whose canonical bond is identified with the agent's memory. Truncating this bond and locally repairing the resulting dynamics produces a smaller, physically-valid agent that remains capable of responding to arbitrary input sequences. A fidelity-divergence certificate quantifies the resulting trade-off between accuracy and memory dimension. Benchmark adaptive processes exhibit substantial dimension reduction whilst preserving the underlying behaviour with high fidelity. These results establish a route from entropic memory advantages to practical, dimension-reduced adaptive quantum agents.
Adaptive agents realise complex reactive behaviours by using a memory of past input stimuli and output actions to guide structured future responses. Quantum adaptive agents can operate while storing less information in memory than optimal classical counterparts; yet, this does not necessarily translate into a reduced dimension of the memory that must be physically realised. We introduce a route-truncate-repair procedure that converts entropic quantum memory advantages into reductions in memory dimension. Routing a reference input process through an agent yields a temporal matrix product state representation whose canonical bond is identified with the agent's memory. Truncating this bond and locally repairing the resulting dynamics produces a smaller, physically-valid agent that remains capable of responding to arbitrary input sequences. A fidelity-divergence certificate quantifies the resulting trade-off between accuracy and memory dimension. Benchmark adaptive processes exhibit substantial dimension reduction whilst preserving the underlying behaviour with high fidelity. These results establish a route from entropic memory advantages to practical, dimension-reduced adaptive quantum agents.
Recurrent quantum models (RQMs) realize sequential quantum processes through repeated application of a unitary operation on a memory system coupled with a series of output registers. However, such models often rely on unnecessarily large memory spaces, introducing redundancy and limiting scalability. Here, we introduce a \textit{variational quantum dimension reduction} framework that identifies and removes irrelevant memory degrees of freedom while preserving the recurrent dynamics of the target model. Our approach employs two parameterized quantum circuits: a decoupling unitary V(θ1) that isolates the essential memory subspace; and a compressed recurrent unitary U~(θ2) that reconstructs the dynamics in the reduced space. The optimization is guided by a unified cost function combining decoupling fidelity and dynamical accuracy, evaluated using the \textit{Quantum Fidelity Divergence Rate} (QFDR), a metric that quantifies long-term fidelity per time step. Applied to a cyclic random walk model, our framework achieves up to three orders of magnitude smaller QFDR compared to variational matrix product state truncation, while requiring only trajectory samples rather than explicit state reconstructions. This establishes a scalable, data-driven paradigm for learning minimal recurrent quantum architectures, enabling variational circuit optimization and quantum process compression for near-term quantum devices.
Quantum process tomography, the task of learning an unknown quantum channel from black-box access, is a central problem in quantum information. In this setting, protocols with quantum memory can coherently store and jointly process quantum information obtained from multiple channel uses, whereas protocols without quantum memory must measure after each use and retain only a classical transcript of the measurement outcomes. A fundamental open question is whether quantum memory provides a query-complexity advantage even when protocols without quantum memory may adapt their experiments based on all previous outcomes with unbounded classical computational power. In this work, we show that it does. We determine the optimal query complexity of quantum process tomography without quantum memory up to a constant factor to be Θ(din3dout3/ε2), where din and dout are the channel input and output dimensions, respectively, and ε is the target diamond-norm accuracy. More precisely, we prove that any incoherent protocol for this task, including adaptive protocols, requires Ω(din3dout3/ε2) queries, even when each channel use may be assisted by arbitrary fresh ancilla, and we present a non-adaptive, ancilla-free incoherent protocol achieving the matching upper bound O(din3dout3/ε2). Our results thereby generalize the optimal sample-complexity bounds for single-copy state tomography, recovered as the special case din=1. By contrast, coherent protocols with quantum memory achieve query complexity Θ(din2dout2/ε2). Hence, our results establish a rigorous learning separation between quantum process tomography with and without quantum memory.
Central to the success of adaptive systems is their ability to interpret signals from their environment and respond accordingly-they act as agents interacting with their surroundings.Such agents typically perform better when able to execute increasingly complex strategies.This comes with a cost: the more information the agent must recall from its past experiences, the more memory it will need.Here we investigate the power of agents capable of quantum information processing.We uncover the most general form a quantum agent need adopt to maximize memory compression advantages and provide a systematic means of encoding their memory states.We show these encodings can exhibit extremely favorable scaling advantages relative to memoryminimal classical agents, particularly when information must be retained about events increasingly far into the past.
What makes one quantum process more complex than another? We adopt an operational approach inspired by classical complexity science, which characterises structure through the memory required for causal generation. We develop a fully-quantum extension of computational mechanics for quantum space series: Correlated sequences of quantum outputs that do not return to interact with their generator. For stationary, memoryful quantum space series admitting finite-dimensional, primitive causal models, we show that any memory-minimal causal generator can be chosen to produce outputs isometrically, without a discarded environment. The resulting canonical matrix product state representation uniquely determines the irreducible memory required for generation. In particular, the minimum memory entropy is exactly the entanglement entropy across the past-future bipartition, giving entanglement a direct operational interpretation as irreducible memory for causal generation. The same structure determines the minimum memory dimension through the corresponding Schmidt rank. Within this class, we further show that the fundamental entropy rate vanishes, that causal generation is symmetric under time reversal, and that all memoryful processes possess infinite quantum Markov order with respect to arbitrary sharp measurements. These results establish an operational framework for quantifying intrinsic quantum structure, revealing deep connections between causal generation, complexity, memory, entropy, and entanglement.
In reinforcement learning, an agent interacts sequentially with an environment to maximize a reward, receiving only partial, probabilistic feedback. This creates a fundamental exploration-exploitation trade-off: the agent must explore to learn the hidden dynamics while exploiting this knowledge to maximize its target objective. While extensively studied classically, applying this framework to quantum systems requires dealing with hidden quantum states that evolve via unknown dynamics. We formalize this problem via a framework where the environment maintains a hidden quantum memory evolving via unknown quantum channels, and the agent intervenes sequentially using quantum instruments. For this setting, we adapt an optimistic maximum-likelihood estimation algorithm. We extend the analysis to continuous action spaces, allowing us to model general positive operator-valued measures (POVMs). By controlling the propagation of estimation errors through quantum channels and instruments, we prove that the cumulative regret of our strategy scales as O(K) over K episodes. Furthermore, via a reduction to the multi-armed quantum bandit problem, we establish information-theoretic lower bounds demonstrating that this sublinear scaling is strictly optimal up to polylogarithmic factors. As a physical application, we consider state-agnostic work extraction. When extracting free energy from a sequence of non-i.i.d. quantum states correlated by a hidden memory, any lack of knowledge about the source leads to thermodynamic dissipation. In our setting, the mathematical regret exactly quantifies this cumulative dissipation. Using our adaptive algorithm, the agent uses past energy outcomes to improve its extraction protocol on the fly, achieving sublinear cumulative dissipation, and, consequently, an asymptotically zero dissipation rate.
Stochastic modeling of complex systems plays an essential, yet often computationally intensive, role across the quantitative sciences.Recent advances in quantum information processing have elucidated the potential for quantum simulators to exhibit memory advantages for such tasks.Heretofore, the focus has been on lossless memory compression, wherein the advantage is typically in terms of lessening the amount of information tracked by the model, while-arguably more practical-reductions in memory dimension are not always possible.Here, we address the case of lossy compression for quantum stochastic modeling of continuous-time processes, introducing a method for coarse graining in quantum state space that drastically reduces the requisite memory dimension for modeling temporal dynamics while retaining near-exact statistics.In contrast to classical coarse graining, this compression is not based on sacrificing temporal resolution and brings memory-efficient high-fidelity stochastic modeling within reach of present quantum technologies.