Spin-Network Quantum Reservoir Computing with Distributed Inputs: The Role of Entanglement

TL;DR

The paper tackles how entanglement structure in a distributed-input quantum reservoir affects short-term memory for bilinear tasks. It uses a four-qubit Ising spin network with Lindblad dissipation, injecting two independent input streams into two qubits and training a linear readout on reservoir observables to predict delayed bilinear targets. The key findings show that the short-term memory capacity $C_{STM}$ is maximized at moderate coupling $J_s$ when input–input entanglement dominates, while the overall entanglement peaks at larger $J_s$; a finite propagation time is required for recall, and a zero-delay dip appears in the memory profile, with a long memory tail emerging in the moderate-coupling regime. This indicates that not just the amount but the localization of entanglement—especially between input qubits—plays a crucial role in optimizing memory in distributed quantum reservoirs, offering guidance for designing quantum neuromorphic systems with efficient temporal processing.

Abstract

Reservoir computing is a promising neuromorphic paradigm, and its quantum implementation using spin networks has shown some advantage when entanglement is present. Here, we consider a distributed scenario in which two distinct input time series are injected into separate qubits of a spin-network reservoir. We investigate how the overall entanglement, as well as its localization in the system, influence the performance of the reservoir. Focusing on bilinear memory tasks that require computing the product of the two inputs, we evaluate the short-term memory capacity and correlate it with logarithmic negativity as a measure of bipartite entanglement. We find that short-term memory capacity reaches its maximum at relatively small coupling strengths. In contrast, average entanglement peaks at larger couplings. Analyzing entanglement across all bipartitions, we find that the entanglement between the two input qubits is consistently the strongest and most relevant for task performance. In the small coupling strength regime where the short-term memory capacity is maximized, the reservoir exhibits an extended memory tail: performance remains high for a long time. Finally, a pronounced dip in performance at zero time delay, observed across frequencies, indicates that information requires a finite propagation time through the reservoir before it can be effectively recalled. In summary, our results show that moderate entanglement, particularly between the two input qubits, plays a key role in enhancing short-term memory performance.

Figures (4)

• Figure 1: Distributed quantum reservoir computing pipeline and task performance.(a) Along the horizontal axis, the reservoir---modeled as a system of four spin qubits---evolves in time under the Ising Hamiltonian [box (i)] together with environmental interactions described by the Lindblad master equation [box (i)]. At discrete injection steps $t_k$, the two input sequences $s_1(t_k)$ and $s_2(t_k)$ are injected simultaneously but into separate qubits, reflecting the distributed nature of the input encoding described in box (ii). The reservoir then evolves from state $\rho(t_1)$ to $\rho(t)$ until the next injection, at which point the cycle repeats. The vertical line separates the physical evolution (lower part) from the learning stage (upper part). In the measurement-and-training step [box (iii)], the expectation values of all four qubits are concatenated at each time step to form a reservoir state vector $\mathbf{r}(t)$. Stacking these vectors across all times yields the training matrix $R_{\mathrm{train}}$. The target sequence, defined as $g(t)$, is collected across all time steps to form $G_{\mathrm{train}}$. This target is mapped to $R_{\mathrm{train}}$, and the readout weights $W(\lambda)$ are obtained via Tikhonov regularization. (b) Example at the optimal delay $\tau^{\ast}$ (selected by maximizing $C_{\mathrm{STM}}(\tau)$) for coupling $J_s=0.1$ and input frequency $f=2$. Top left: input $s_1(t)$. Bottom left: $s_2(t)$. Top right: target $y^{\ast}(t)=s_1(t-\tau^{\ast})\,s_2(t-\tau^{\ast})$. Bottom right: test prediction $\hat{y}(t)$ overlaid with target $y^{\ast}(t)$.
• Figure 2: Memory capacity, average entanglement, and input–input entanglement in different coupling regimes.(a) Memory capacity vs.$J_s$ peaks at intermediate couplings ($J_s \approx 0.3$) before decaying. (b) Curves show the logarithmic negativity $E$, averaged over ten random connectivity matrices. Entanglements correspond to different partitions of the system: $E_1:{\{1\}\,|\,\{2,3,4\}}, \,E_2:{\{2\}\,|\,\{1,3,4\}}, \,E_3:{\{3\}\,|\,\{1,2,4\}}, \,E_4:{\{4\}\,|\,\{1,2,3\}}$. ($E_1$–$E_4$) show similar non-monotonic behavior with a maximum near $J_s \approx 6$. ((c) Pair–vs.–rest partitions are $E_{12}:{\{1,2\}\,|\,\{3,4\}}, \,E_{13}:{\{1,3\}\,|\,\{2,4\}},\, E_{14}:{\{1,4\}\,|\,\{2,3\}}$, following the same trend with modest differences. At $f=2$, larger $E_1$ and $E_2$ together with $E_{12}<E_{13},E_{14}$ indicate that entanglement is concentrated within the input pair $\{1,2\}$. ((d) Gray curve: difference between $E_1+E_2$ and $E_3+E_4$; green curve: difference between $E_{13}+E_{14}$ and $2E_{12}$. Both exhibit maxima around $J_s \approx 0.3$--$0.7$, following a trend similar to panel (a). ((e) Balance ratios: $(E_1+E_2)/(E_3+E_4)$ and $(E_{13}+E_{14})/(2E_{12})$, both peaking around $J_s \approx 0.05$. Overall, the optimum performance occurs along the initial rise before the entanglement peak, where the entanglement between qubits 1 and 2 dominates relative to other partitions—suggesting that this form of entanglement is the most beneficial for the task.
• Figure 3: Short-term memory capacity $C_{\mathrm{STM}}(\tau)$ versus delay $\tau$ at three coupling strengths. At very small coupling ($J_s=0.005$), the memory profile resembles that of the moderate and high-coupling case ($J_s=6$ and $J_s=75$), with a fast decaying memory. By contrast, the weak-coupling case ($J_s=0.325$) exhibits a long, low-amplitude tail (slow fading memory). These distinct decay behaviors correlates with Fig. \ref{['fig:c_ent_J_s']}: in figure Fig. \ref{['fig:c_ent_J_s']}, the performance peaks at $J_s \approx0.3$, here we can see that at this regime the memory profile is not similar to other regimes, exhibiting a slower decay. The difference can be understood as information becoming trapped between the two input qubits at $J_s=0.325$, consistent with Fig. \ref{['fig:c_ent_J_s']}, which shows that the input qubits are more strongly entangled with each other than with the rest of the network.
• Figure 4: Short-term memory capacity $C_{\mathrm{STM}}(\tau)$ versus delay $\tau$ at fixed coupling $J_s=1.5$ for three driving frequencies ($f\in\{1,2,3\}$). All curves exhibit a dip at $\tau = 0$, a near-term peak, and a subsequent decay. The depth of the zero-delay dip increases with $f$, consistent with an increasingly challenging task at higher driving frequencies. This initial dip reflects the fact that, in our distributed setting, information requires additional time to propagate throughout the reservoir before it can be effectively recalled.