Minimalistic and Scalable Quantum Reservoir Computing Enhanced with Feedback

TL;DR

The paper introduces a minimalistic quantum reservoir computing platform comprising up to five atoms in a single-mode optical cavity, integrated with continuous quantum measurement and a software-controlled feedback loop. By enabling feedback from measured readouts and enriching outputs with polynomial regression, the approach dramatically enhances memory and nonlinear processing in small quantum reservoirs, with performance scaling tied to the Hilbert space expansion as $N_c 2^{N_{ ext{atom}}}$. Demonstrations on Mackey-Glass forecasting and sine-square waveform classification show substantial gains over classical reservoirs, highlighting the practical viability of non-destructive readouts and scalable hardware. The work emphasizes experimental feasibility through realistic cooperativity values and presents a pathway to energy-efficient, hardware-light quantum machine learning using continuous measurement and external feedback.

Abstract

Quantum Reservoir Computing (QRC) leverages quantum systems to perform complex computational tasks with exceptional efficiency and reduced energy consumption. We introduce a minimalistic QRC framework utilizing as few as five atoms in a single-mode optical cavity, combined with continuous quantum measurement. The system is conveniently scalable, as newly added atoms naturally couple with existing ones via the shared cavity field. To achieve high computational expressivity with a minimal reservoir, we include two critical elements: reservoir feedback and polynomial regression. Reservoir feedback modifies the reservoir's dynamics without altering its internal quantum hardware, while polynomial regression nonlinearly enhances output resolution. We demonstrate significant QRC performance in memory retention and nonlinear data processing through two tasks: predicting chaotic time-series data via the Mackey-Glass task and classifying sine-square waveforms. This framework fulfills QRC's objectives to minimize hardware size and energy consumption, marking a significant advancement in integrating quantum physics with machine learning technology.

Figures (7)

• Figure 1: Setup of quantum optical reservoir computing with feedback. The input function is coherently integrated into the driving field of an optical cavity. The reservoir consists of atoms inside the cavity, exhibiting diverse detunings and coupling strengths across various spatial positions. Readouts are obtained via continuous measurements of photonic quadratures and atomic spin channels, denoted by $x_{kn}$ with $n$ the index of the readout channel. The feedback mechanism is implemented by modifying the input using several readouts, $x_{kn_{f}}$, where $n_{f}$ indexes the readouts used for feedback. The parameter $V_{n_{f}}$ controls the strength of the $n_{f}$-th feedback channel. A polynomial regression is then applied to map the readouts to the output with weights $W_{n}$. The parameters $V_{n_{f}}$ and $W_{n}$ are trained through a machine learning process to enhance the performance of QRC.
• Figure 2: Input and readouts for the Mackey-Glass task utilizing a single-atom reservoir with $\omega_{1}=20$ and $g_{1}=30$. (a) The input function to be processed, $f_{k}$, divided into memory fading, training, and testing regions. (b) The actual input applied to the reservoir, $\widetilde{f}_{k}$, modified by the feedback mechanism using all the 4 available readouts as feedback channels. (c)-(f) The corresponding 4 readouts from photonic quadratures and atomic spin channels. The green shaded regions highlight the distinct responses of the reservoir to different waveforms in the input signal. Parameters: $Delay=20$, $\kappa=10$, $\omega_{c}=40$, and $\epsilon=20$.
• Figure 3: Performance testing for the Mackey-Glass task. (a)(b) The tested NRMSE plotted against the number of atoms (or neurons), using all available readouts from the cavity field and all atoms, with no feedback (black), $2$ feedbacks via $x_{k1}$ and $x_{k2}$ channels (blue), or $4$ feedbacks via $x_{k1}$ to $x_{k4}$ channels (green), where regular linear regression (solid lines) or polynomial regression (dashed lines) is applied. In panel (a), the coupling strength $g_i = 30$ is fixed for all atoms, and the detunings $\omega_i$ vary: $\omega_i = 20$ for one atom, $\omega_i = [0, 40]$ for two atoms, $\omega_i = [0, 20, 40]$ for three atoms, $\omega_i = [0, 10, 30, 40]$ for four atoms, and $\omega_i = [0, 10, 20, 30, 40]$ for five atoms. In panel (b), $\omega_i = 20$ is fixed for all atoms, and $g_i$ varies: $g_i = 30$ for one atom, $g_i = [10, 50]$ for two atoms, $g_i = [10, 30, 50]$ for three atoms, $g_i = [10, 20, 40, 50]$ for four atoms, and $g_i = [10, 20, 30, 40, 50]$ for five atoms. In panels (a)(b), classical reservoir computing (CRC) (red solid line) uses all available readouts from all neurons in classical echo state networks (ESN) (detailed in the "Supplementary Information"). (c) The tested NRMSE as a function of the number of unmeasured atoms (for QRC) or unmeasured neurons (for CRC), using only the $4$ readout channels (taking 4 measurements) from the cavity field and a particular atom with $g=30$ and $\omega=20$, where the total number of atoms is increased by fixing either $g_{i}$ (solid lines) or $\omega_{i}$ (dashed-dotted lines). (d)(e) Actual (red) and target (blue) outputs from QRC utilizing one atom with no feedback with linear regression, and three atoms with $4$ feedbacks with polynomial regression, corresponding to the points marked by letters "d" and "e" in panel (b), respectively. (f)(g) The influence of different selections for the $2$ feedback channels out of the $6$ readout channels in two-atom QRC (black lines), where ($n$,$m$) denotes the feedback channels $x_{kn}$ and $x_{km}$ with $n<m$. The results from one-atom and three-atom QRC using feedback channels $x_{k1}$ and $x_{k2}$ (orange lines) are plotted for reference. Panel (f) fixes $g_i$ for all atoms, while panel (g) fixes $\omega_i$. (h) Comparison of three methods for training feedback parameters $V_{n_{f}}$: differential evolution, brute force, and brute force plus Nelder Mead, under varying numbers of feedbacks for one-atom QRC with $g_{1}=30$ and $\omega_{1}=20$ using linear regression. Parameters: $Delay=20$, $\kappa=10$, $\omega_{c}=40$, $\epsilon=20$.
• Figure 4: Performance testing for the Mackey-Glass task using one-atom QRC with various $Delay$ and decay rates $\kappa$. (a) NRMSE as a function of $Delay$, with no feedback or 4 feedbacks, for $\kappa=10$. (b)(c) The actual (red) and target (blue) outputs for $Delay=200$ and $Delay=2$ with $\kappa=10$, corresponding to the points marked by letters "b" and "c" in panel (a), respectively. (d) NRMSE as a function of $\kappa$, with no feedback or 4 feedbacks, for $Delay=20$. (e)(f) The actual (red) and target (blue) outputs for $\kappa=10^{5}$ and $\kappa=10$ with $Delay=20$, corresponding to the points marked by letters "e" and "f" in panel (d), respectively. Parameters: $\omega_{1}=20$, $g_{1}=30$, $\omega_{c}=40$, $\epsilon=20$.
• Figure 5: Input and readouts for the sine-square waveform classification task. A total of $110$ random waveforms are sent as input, with $10$ waveforms allocated for memory fading, $50$ waveforms for training, and $50$ waveforms for testing. (a) A segment of the input function, $f_{k}$, showing the first $10$ waveforms during the testing phase. (b) The corresponding segment of the actual input applied to the reservoir, $\widetilde{f}_{k}$, modified by the use of $4$ feedback channels. (c)-(f) The corresponding readouts from a single-atom reservoir with $\omega_{1}=20$ and $g_{1}=30$. Parameters: $\omega_{c}=40$, $\kappa=10$, $\epsilon=20$, $\omega_{ss}=10$, $N_{ss}=16$.