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Boosting computational power through spatial multiplexing in quantum reservoir computing
Kohei Nakajima, Keisuke Fujii, Makoto Negoro, Kosuke Mitarai, Masahiro Kitagawa
TL;DR
Quantum reservoir computing needs scalable computational resources, but increasing qubits in physical NMR implementations is difficult. The paper introduces spatial multiplexing, which combines multiple small quantum reservoirs driven by common inputs, and reports improved performance across benchmark tasks while analyzing its validity and limitations.
Problem
Increasing qubit counts to enlarge QRC computational resources requires difficult and time-consuming redesign of NMR sample molecules.
Method
The paper drives multiple disjoint small quantum systems with common input streams and combines their signals into one reservoir.
Results
Performance improved across numerical benchmark tasks, and memory capacity improved as spatial multiplexing increased.
Takeaways & Limitations
Spatial multiplexing offers an operationally easy way to increase QRC computational power using multiple small systems rather than one larger system.
Abstract
from arXiv · showhide
Quantum reservoir computing provides a framework for exploiting the natural dynamics of quantum systems as a computational resource. It can implement real-time signal processing and solve temporal machine learning problems in general, which requires memory and nonlinear mapping of the recent input stream using the quantum dynamics in computational supremacy region, where the classical simulation of the system is intractable. A nuclear magnetic resonance spin-ensemble system is one of the realistic candidates for such physical implementations, which is currently available in laboratories. In this paper, considering these realistic experimental constraints for implementing the framework, we introduce a scheme, which we call a spatial multiplexing technique, to effectively boost the computational power of the platform. This technique exploits disjoint dynamics, which originate from multiple different quantum systems driven by common input streams in parallel. Accordingly, unlike designing a single large quantum system to increase the number of qubits for computational nodes, it is possible to prepare a huge number of qubits from multiple but small quantum systems, which are operationally easy to handle in laboratory experiments. We numerically demonstrate the effectiveness of the technique using several benchmark tasks and quantitatively investigate its specifications, range of validity, and limitations in detail.
I. INTRODUCTION
Quantum reservoir computing uses natural quantum dynamics to process complex temporal data, while physical implementations face practical limits when increasing qubit counts. The paper introduces spatial multiplexing as an experimentally convenient way to enlarge computational resources using multiple small quantum systems.
- Reservoir computing uses high-dimensional dynamics with fading memory and nonlinear input processing for real-time temporal information processing.
- Quantum reservoir computing exploits quantum dynamics whose exponentially large degrees of freedom are difficult to simulate classically.
- Increasing qubit counts in NMR implementations requires redesigning sample molecules, which is operationally difficult and energy- and time-consuming.
- Spatial multiplexing drives multiple disjoint small quantum systems with common inputs and combines their signals as one reservoir.
- Physical spatial multiplexing differs from software equivalence because preparing multiple small molecules can be easier than preparing one large molecule with the same node count.
- Quantum reservoir signals include measured true nodes, while exponentially many hidden nodes arise from the Hilbert-space dimensions.
B. Temporal multiplexing
Temporal multiplexing samples quantum-reservoir dynamics at subdivided intervals to create virtual nodes. This increases the number of computational nodes used for learning without changing the underlying reservoir dynamics.
- Temporal multiplexing samples signals during each subdivided interval of unitary evolution to construct V virtual nodes.
- A reservoir with N true nodes produces NV computational nodes at each input timestep when temporal multiplexing is applied.
- Temporal multiplexing uses input-driven transient dynamics that can include hidden-node influence and increase the total computational nodes used in learning.
- The parameter τ directly modulates reservoir dynamics, whereas V determines how those dynamics are observed.
- Increasing V generally contributes to improved computational performance, while the choice of τ changes which computations are performed well.
C. Spatial multiplexing
Spatial multiplexing increases QRC resources by running multiple disjoint quantum reservoirs in parallel under a common input stream. Their signals are combined into one reservoir, offering a practical alternative to redesigning larger molecules.
- Increasing computational nodes in NMR QRC would naturally require more qubits, creating physical implementation difficulties.
- Spatial multiplexing prepares multiple spatially distant or uncoupled quantum reservoirs and drives them in parallel with a common input stream.
- NMR implementations can use readily available sample molecules to increase computational nodes instead of redesigning sample molecules.
- The scheme permits each reservoir to have its own qubit count, input-injection interval, unitary evolution, and number of virtual nodes.
- Spatial multiplexing produces a total of Σc=1^C NcVc computational nodes that are exploited as one reservoir.
D. Output settings and learning procedure
The system combines all temporally and spatially multiplexed signals into reservoir states and learns a linear static readout. Training uses least squares after washing out initial transients, followed by evaluation with the trained weights.
- Temporal and spatial multiplexing produce Ntotal = Σc=1^C NcVc computational nodes, whose states are collected for readout training.
- The output is a weighted sum of reservoir states using linear and static weights attached to each node.
- A constant bias node is included alongside the computational nodes in the system output.
- The optimal readout weights are obtained by solving a least-squares problem from the target sequence and training-data matrix.
- Each trial includes washout, training, and evaluation phases, with washout removing initial transients before trained weights generate outputs.
E. Theoretical insights into the effect of spatial multiplexing
Theoretical analysis shows that spatial multiplexing cannot worsen training performance and can improve it by combining reservoir representations, while its practical benefits depend on rank and overfitting conditions.
- Spatial multiplexing always improves computational performance, or at worst leaves it unchanged, according to the theoretical analysis.
- Combining reservoirs A and B forms a larger reservoir whose least-squares residual is determined by the combined projector P_A+B.
- The difference Q_A,A+B = P_A+B − P_A is positive semidefinite, so adding reservoir B cannot increase the residual for reservoir A.
- The largest eigenvalues of the projector differences bound residual reductions and can predict the improvement without evaluating the combined reservoir directly.
- Overfitting can reverse evaluation-phase gains because spatial multiplexing increases computational nodes and may exceed the available training-data scale.
- The theoretical result assumes full-rank reservoir matrices; common-input synchronization can violate this assumption and reduce the benefit of multiplexing.
III. PERFORMANCE ANALYSES
The performance analyses numerically vary spatial multiplexing in quantum reservoirs using memory capacity and NARMA tasks, with controlled quantum-system and input settings.
- The numerical experiments assess spatial multiplexing through memory capacity and NARMA emulation of nonlinear dynamical systems.
- Each quantum reservoir uses a fully connected transverse-field Ising model with randomly assigned coupling strengths.
- The experiments use identical qubit counts, input intervals, and virtual-node counts across reservoirs, while varying their random coupling strengths.
- The order of spatial multiplexing ranges from 1 to 5, with five-qubit systems, virtual-node counts of 1, 5, or 25, and inputs injected into the first qubit.
A. Memory Capacity
Memory capacity measures how well a quantum reservoir reconstructs delayed inputs. Across tested virtual-node settings, increasing the spatial-multiplexing order improves memory capacity, with stronger values for selected input-interval parameters.
- Memory capacity evaluates short-term memory by testing reconstruction of an input from d timesteps earlier using the current reservoir state.
- The memory function MF_d measures delayed-input reconstruction quality, with larger values indicating better recovery of u_k−d.
- Increasing spatial-multiplexing order improves memory capacity across virtual-node settings 1, 5, and 25.
- The improvement appears as larger MF_d values at longer delays when the spatial-multiplexing order increases.
- Memory capacity is relatively larger when τ∆ = 0.5 and 1 than under the other tested τ∆ settings.
B. NARMA tasks
The NARMA benchmarks evaluate nonlinear processing with long time dependence, and spatial multiplexing improves quantum-reservoir traceability and averaged NMSE across task orders.
- NARMA tasks evaluate whether a learning system can emulate nonlinear dynamical systems with long time dependence.
- The benchmark includes NARMA2, NARMA5, NARMA10, NARMA15, and NARMA20, with multitasking requiring simultaneous emulation of all systems.
- The evaluation compares target and system outputs using NMSE after washout, training, and evaluation phases, averaged over 100 trials with different QR systems.
- As NARMA order increases, overall task performance worsens, reflecting greater task difficulty.
- Increasing spatial multiplexing improves traceability and task performance for every NARMA task, with especially strong effects for NARMA2, NARMA5, and NARMA10.Figure 4 illustrates outputs through five quantum systems with five qubits, 25 virtual nodes, and τ∆=2; Figure 5 quantitatively confirms the improvement.
C. Temporal versus spatial multiplexing
The paper quantifies spatial multiplexing against its single-system baseline and temporal multiplexing, finding substantial memory-capacity gains and lower NMSE in broad experimental settings.
- Improvement ratios compare averaged memory capacity or NMSE at each multiplexing order with the corresponding order-1 baseline.
- Memory-capacity improvement exceeded twice the baseline when multiplexing increased from 1 to 5 for every virtual-node setting.The averaged improvement ratio across parameter settings was 2.11, reaching a maximum of 3.23 at τ∆=0.5, V=1, and N=5.
- For NARMA5 with V=25, increasing multiplexing from 1 to 5 reduced the improvement-ratio value by a factor of 10.
IV. TOWARD ENGINEERING QUANTUM RESERVOIR THROUGH SPATIAL MULTIPLEXING
The engineering analysis examines how to choose and combine reservoirs while recognizing overfitting and synchronization constraints. It also notes implementation alternatives for increasing computational power.
- Increasing spatial multiplexing can improve performance theoretically, but overfitting may make limiting the number of computational nodes preferable in experiments.
- Combining reservoirs is analyzed under assumptions about regression settings and least-squares solutions, with total computational nodes held fixed at N+N′.
- Choosing the individually better-performing reservoir is not always the best combination decision.
- A reservoir partner can be selected without performing the task only when the relevant performance ranges do not overlap.
- Spatial multiplexing may combine any reservoirs that are not synchronized, and the combination should depend on experimental efficiency.
V. DISCUSSION
The paper presents spatial multiplexing as an experimentally practical way to increase QRC computational power and examines its effectiveness, theoretical implications, range of validity, and limitations. The approach is especially relevant when increasing addressable qubits is operationally difficult, while larger computational-node counts can introduce overfitting concerns.
- Limitations: Overfitting is a practical limitation when spatial multiplexing creates many computational nodes, particularly in some higher-τ∆ NARMA results.Ridge or Lasso regression is proposed to selectively exploit effective degrees of freedom from the enlarged node set.
- Evaluation scope: The analyses average improvement ratios across qubit counts, virtual-node counts, and τ∆ settings, comparing spatial and temporal multiplexing at matched computational-node increases.The figure uses 100 trials with different QR systems for each condition.
- Implementation: Spatial multiplexing combines multiple small quantum systems driven by common inputs, offering an easier implementation than increasing the number of addressable qubits in NMR QRC.The paper also considers implementations using different molecules or pulse techniques with the same molecule.
- Broader applicability: Spatial multiplexing can extend beyond QRC to interacting systems whose components are experimentally difficult to manipulate or enlarge.The paper discusses composing reservoirs based on different physical systems as a possible broader application.
- Theoretical implications: The scheme increases true nodes proportionally with multiplexing order, while increasing qubits directly produces an exponential increase in hidden nodes.This distinguishes the scaling behavior of spatial multiplexing from enlarging the interacting quantum system.
Appendix A: Extended numerical experiments and analyses
The appendix expands the numerical evaluation across QR and ESN settings, documenting the parameter ranges and comparison procedures used to assess memory capacity and task performance. It varies qubit number, virtual nodes, τ∆, and ESN configuration while preserving matched input/output evaluation procedures.
- Extended QR experiments: The extended QR analyses vary N across 3, 4, and 5 and τ∆ across 0.5, 1, 2, 3, 4, 8, 16, and 32.These settings are summarized in the appendix figures for systematic parameter analysis.
- Comparison procedure: The ESN update uses tanh activation, while its input/output arrangement, washout, training, evaluation lengths, and evaluation procedures match those of the QR system.Internal weights are randomized and rescaled according to each task setting.
- ESN comparisons: The ESN comparisons use networks with 5 to 300 internal nodes and vary input-weight scaling and spectral radius across broad ranges.For NARMA tasks, multiple ESNs, parameter settings, and random trials are prepared for each configuration.
- ESN comparisons: ESN memory-capacity evaluation uses 100 networks with spectral radius 0.9 and input-weight scaling σ = 0.01 across five nonlinear dynamical-system emulation tasks.Different random input sequences are used, with multitasking applied within each trial.
- Figure 8: Figure 8 plots averaged memory capacity against total computational nodes, encoding qubit number by point shape, virtual-node number by color, and multiplexing order by connected lines.Conventional ESN references such as ESN20 are included for comparison.