Quantum reservoir computing provides a framework for processing complex temporal data, yet its fundamental computational and energetic limits remain unresolved. Here, we establish a non-equilibrium thermodynamic framework that links the macroscopic predictive performance of driven open quantum systems to their microscopic energetic costs. By mapping the Holevo capacities onto the Bogoliubov-Kubo-Mori geometric manifold, we analytically prove that the computational peak within the quantum critical region originates from a strict spectral resonance: the closing of the energy gap forces the reservoir's transition frequencies to align with the chaotic drive. To evaluate the associated thermodynamic costs, we introduce quantum informational dissipation to quantify the non-predictive historical data structurally retained by the reservoir, deriving a generalized Landauer bound for continuous temporal processing. This reveals a fundamental thermodynamic trade-off: the critical resonance that unlocks optimal predictive capacity inherently maximizes informational dissipation and the irreversible work required for environmental erasure. Furthermore, coherence decomposition demonstrates that dynamic quantum coherences strictly amplify predictive capacity without demanding additional mechanical work. These findings establish the ultimate energetic limits of quantum learning devices, providing theoretical principles for designing energy-efficient quantum neuromorphic hardware.
Quantum reservoir computing provides a framework for processing complex temporal data, yet its fundamental computational and energetic limits remain unresolved. Here, we establish a non-equilibrium thermodynamic framework that links the macroscopic predictive performance of driven open quantum systems to their microscopic energetic costs. By mapping the Holevo capacities onto the Bogoliubov-Kubo-Mori geometric manifold, we analytically prove that the computational peak within the quantum critical region originates from a strict spectral resonance: the closing of the energy gap forces the reservoir's transition frequencies to align with the chaotic drive. To evaluate the associated thermodynamic costs, we introduce quantum informational dissipation to quantify the non-predictive historical data structurally retained by the reservoir, deriving a generalized Landauer bound for continuous temporal processing. This reveals a fundamental thermodynamic trade-off: the critical resonance that unlocks optimal predictive capacity inherently maximizes informational dissipation and the irreversible work required for environmental erasure. Furthermore, coherence decomposition demonstrates that dynamic quantum coherences strictly amplify predictive capacity without demanding additional mechanical work. These findings establish the ultimate energetic limits of quantum learning devices, providing theoretical principles for designing energy-efficient quantum neuromorphic hardware.
Enhanced performance of quantum reservoir computing has been associated with dynamical phase transitions, but whether this connection extends to dissipative systems and which relaxation mechanisms underlie it remain insufficiently understood. We systematically compare driven-dissipative Kerr reservoirs across first-order and continuous dissipative phase transitions and find enhanced memory and nonlinear processing near both phase boundaries. Although the closure of the Liouvillian gap marks the transitions in the thermodynamic limit, computational performance does not generally follow gap suppression. An exact post-training Liouvillian-mode decomposition quantitatively attributes the trained memory to intrinsic relaxation channels. It shows that the gap mode contributes only weakly, while finite-rate modes and their cross-contributions dominate the enhanced capacity. These results go beyond phenomenological correlations by directly linking memory capacity to the intrinsic Liouvillian relaxation spectrum. Moreover, these findings provide a physical basis for designing dissipative quantum reservoirs and testing memory-enhancement mechanisms in experimentally accessible Kerr platforms.
Quantum reservoir computing (QRC) provides a hardware-efficient paradigm for temporal information processing on near-term quantum devices. Despite rapid experimental progress, a rigorous understanding of the structural conditions required for its scalable quantum-enhanced performance remains lacking. Here, we develop a theoretical framework in Pauli-Liouville space that provides a unified analytical treatment of the echo state property (ESP), nonlinear expressive power, and quantum resources. We first analyze the widely used qubit-resetting scheme and establish that quantum magic generated by reservoir dynamics is a necessary condition for effective computation, a requirement more fundamental than ESP. However, we prove that this architecture faces inherent expressivity limitations: all nonlinear processing originates exclusively from the classical encoding map, imposing an unavoidable trade-off between nonlinearity and memory capacity. To circumvent this structural bottleneck, we rigorously analyze Hamiltonian encoding, in which temporal inputs are embedded directly into the continuous dynamics generator. We show that the ESP is natively guaranteed by the Liouvillian spectral gap, decoupling it from quantum magic. Crucially, for any non-trivial drive Hamiltonian, the discrete-time update map exhibits a transcendental, infinite-order nonlinear dependence on the instantaneous input. Moreover, the intrinsic non-commutativity of the open-system generators governs the temporal coupling of these nonlinearities, producing highly non-separable processing of the input history. Our results establish a rigorous theoretical hierarchy of QRC architectures and provide prescriptive design principles for experiments targeting genuine quantum advantages in temporal processing.
Quantum reservoir computing offers a promising route toward energy-efficient machine learning by processing temporal information with minimal training overhead. Yet, the physical principles linking its energetic cost to computational performance remain largely unexplored. Here we show that, in an interacting spin reservoir, information encoding and information processing are governed by distinct physical mechanisms. In the weak interacting regime, we derive an analytical expression for the average (switching) work, showing that the energetic cost of encoding new inputs is determined by the local response of the reservoir units. In contrast, interactions primarily redistribute the encoded information, generating memory and nonlinear features while only weakly affecting the work. This separation produces opposite correlations between energetic cost and performance for representative linear and nonlinear benchmark tasks. Our results identify the switching work as the energetic signature of information encoding and clarify when energetic efficiency and computational performance are compatible.
Autonomous quantum thermal machines have recently been proposed as physics-based computing substrates where logical inputs and outputs are encoded in temperature gradients. While such ``thermodynamic neurons'' exhibit a clear trade-off between computational fidelity and heat dissipation, the fundamental information-theoretic limits of temperature-encoded computation remain uncharacterized. Here, we derive rigorous bounds linking average error probability, channel capacity, and entropy production for finite-capacity thermal reservoirs operating far from equilibrium. We prove that the minimal dissipation required to achieve a target average error probability ⟨ξ⟩ diverges as ⟨ξ⟩ approaches a fundamental minimum error floor $\epsMin$ imposed by finite-reservoir thermal fluctuations. We further establish a thermodynamic channel capacity that saturates at high dissipation, and quantify the minimal dissipation required for cascaded networks to maintain target fidelity. We demonstrate that the required dissipation grows linearly O(L) with network depth for shallow networks, but diverges as the network depth L approaches a fundamental maximum limit $L_{\max} = \epsNet/\epsMin$ imposed by the minimum error floor. Furthermore, under strong noise amplification conditions, the required dissipation can grow up to O(L3). Our framework bridges stochastic thermodynamics, finite-time information theory, and autonomous computation, providing rigorous design principles for energy-efficient analog thermodynamic hardware.
Quantum learning provides a versatile paradigm for information processing by exploiting the intrinsic representational capacity of high-dimensional Hilbert spaces. Here, we investigate a Hamiltonian-encoding framework for quantum reservoir computing that simultaneously addresses three key challenges in quantum learning: trainability, hardware efficiency, and information stability. In this framework, input data are directly mapped onto a fixed Hamiltonian and transformed into expressive nonlinear features through quantum dynamical evolution. By employing the reservoir-computing paradigm, the approach naturally circumvents the barren plateau problem in quantum learning landscapes. We validate the framework across two complementary platforms: an analog superconducting array processor and a digital gate-based quantum circuit implementation. Despite their fundamentally different realizations, both platforms exhibit comparable representational power and achieve competitive learning performance, establishing a unified framework for cross-platform quantum learning. While both implementations achieve comparable performance, the analog processor may offer a more hardware-efficient realization by bypassing the temporal overhead of gate-based decomposition and thereby making more effective use of finite coherence times, albeit at the expense of universality. Furthermore, we find that finite dissipation suppresses quantum-scrambling-induced instabilities at long evolution times and can enhance learning performance, revealing a constructive role for environmental coupling in stabilizing quantum learning dynamics. Collectively, these results establish Hamiltonian-encoded reservoir computing as a compact, expressive, and hardware-efficient paradigm for quantum learning on current-generation quantum platforms.
The performance of a quantum reservoir computer in temporal processing tasks depends on how its driven quantum substrate retains information about past inputs, distributes it across physical degrees of freedom, and loses it through environmental dissipation and measurement feedback. We formulate these processes using a classical-quantum state obtained by restricting a reservoir process tensor to classical input encoding and single-time readout. Conditional subsystem Holevo quantities describe information about selected input histories and bound its accessibility to measurements on subsystems of the reservoir. In a six-qubit all-to-all transverse-field Ising reservoir, we find that the total stored information changes relatively little across Hamiltonian parameters, while its spatial distribution and temporal decay vary strongly. Effective diagnostics of these two behaviours identify different regions of high information-processing capacity for linear and higher-degree temporal tasks. Measurement-induced dephasing can improve noiseless task performance when it increases forgetting rates without strongly reducing information delocalisation. The framework separates storage from subsystem accessibility and provides a common description of information flow in driven quantum learning systems.