Quantum Next Generation Reservoir Computing: An Efficient Quantum Algorithm for Forecasting Quantum Dynamics

TL;DR

This work introduces a quantum variant of Next Generation Reservoir Computing (QNG-RC) that forecasts full many-body quantum dynamics without assuming a dynamical model. By embedding NG-RC feature construction and the Tikhonov-regularized pseudoinverse into a block-encoded quantum framework and employing QSVT, the method achieves coherent, skipped-ahead predictions with potential quantum speedups. The approach is demonstrated on both integrable and chaotic Ising-type systems, achieving near-perfect fidelities in long-horizon forecasts and high accuracy in chaotic regimes, while addressing classical computational bottlenecks associated with exponential state spaces. The paper also provides detailed methods for block-encoding, monomial feature encoding, and error analysis, outlining pathways to generalize the feature maps and extend practical quantum forecasting to larger systems.

Abstract

Next Generation Reservoir Computing (NG-RC) is a modern class of model-free machine learning that enables an accurate forecasting of time series data generated by dynamical systems. We demonstrate that NG-RC can accurately predict full many-body quantum dynamics in both integrable and chaotic systems. This is in contrast to the conventional application of reservoir computing that concentrates on the prediction of the dynamics of observables. In addition, we apply a technique which we refer to as skipping ahead to predict far future states accurately without the need to extract information about the intermediate states. However, adopting a classical NG-RC for many-body quantum dynamics prediction is computationally prohibitive due to the large Hilbert space of sample input data. In this work, we propose an end-to-end quantum algorithm for many-body quantum dynamics forecasting with a quantum computational speedup via the block-encoding technique. This proposal presents an efficient model-free quantum scheme to forecast quantum dynamics coherently, bypassing inductive biases incurred in a model-based approach.

Figures (5)

• Figure 1: (Top) The schematic of the QNG-RC algorithm. The initial part of the algorithm encodes the history of quantum dynamics collected from a time series into nonlinear feature vectors $\ket{x_k}$, which is then processed by a regularized linear layer to predict future quantum states. (Middle) In the training phase of QNG-RC, the optimally trained weight matrix in Eq. \ref{['eq: W']} is encoded in a unitary operator and processed by the quantum singular value transform (QSVT), the total complexity of which is summarized in Theorem \ref{['thm: W']}. (Bottom) In the prediction phase, the feature vector can be constructed via the same quantum circuits as in the training phase. Consequently, the predicted states will be revealed in the blue register in the last quantum circuit after applying the optimal weight matrix from the training phase.
• Figure 2: The performances of the NG-RC in predicting unseen future states of a four-qubit transverse-field Ising model in the disordered phase with $J=0.5$ and $h=5$ across $\widetilde{T} = 4 \times 10^4$ future time steps employing two different approaches: the skipping-ahead method with a time skip of $\tau=10^6$ steps (the map $f_\tau$ represents the trained NG-RC for predicting the next $\tau$ step from the current step), and the conventional approach of iteratively predicting each successive time step ($g^{(\tau)}$ is the composition of $\tau$ successive $g$ obtained from the trained NG-RC $f_{\tau=1}$. $T=2\times 10^4$ steps of the time evolution with the same step size of $\Delta t=1/(200E_{\textrm{max}})$, where $E_{\max}$ is the highest eigen-energy of the system, are used to train the NG-RC in both cases. The training is optimized with the optimal regularization parameter $\lambda = 1.0\times 10^{-3}$ in the iterative prediction and with $\lambda = 0$ in the skipping-ahead prediction. Our benchmark targets are the real and the imaginary parts of all amplitudes in the computational basis ((a) and (b)). The comparisons of the expectation values $\expval{X_0}$, $\expval{X_0X_1}$, and the fidelity $F(\tilde{s}_t,s_t):=|\braket{\tilde{s}_t}{s_t}|$ between the target and the predicted states are shown in (c) and (d).
• Figure S1: Quantum circuit for constructing $m$-time-delay, degree-$p$ feature vectors in general NG-RC. (a) The quantum circuit $U^\text{lin}$ for the linear part of feature vectors with $m$-delay \ref{['eq: k-time-delay']}, it requires $m$ oracles and $\eta = \log{m}$ additional qubits to construct $\sum_{j=0}^{m-1} \dyad{j}\otimes\mathcal{O}_{-j\Delta}$. (b) $p$ copies of $U^\text{lin}$ are required to construct the degree-$p$ feature vectors.
• Figure S2: The recursive quantum circuit for predicting $k=2$ steps into the future. In general, at the $j$th level in the recursion ($1\le j\le k$), the state $\ket{c}_{\textrm{LB}}$ of the first register LB (stands for "label") dictates whether the circuit for the $j$th level returns $\ket{\Tilde s_{j-1}}$ (when $c=1$) or $\ket{\Tilde s_{j}}$ (when $c=0$). Thus, to create a linear superposition $\ket{\Tilde{o}_j}=(\ket{0}\ket{\tilde{s}_{j-1}}+\ket{1}\ket{\tilde{s}_{j}})/\sqrt{2}$ for the feature vector, there is always a Hadamard gate applied to the wire going into the LB register (green boxes in quantum circuits). Contained in the register BE ("block encoding") are index qubits that label the upper-left block of $U_W$ in the block encoding, whereas qubits in the register RD ("residual") are required for the zero padding to make $W$ square. The EN ("encoder") register takes the data to be encoded into the feature vector. The desired results are contained in the TG ("target") register, whereas the rest are "garbage output" (GO). The GO registers do not contain any of the relevant result but they remain in the state $\ket{0}$. Thus, we can choose to reuse some of them, and these reused registers are labeled "garbage input" (GI).
• Figure S3: The performances of the NG-RC in predicting the future unseen states of a five-qubit tilted-field Ising model \ref{['eq: tilted']} in the chaotic phase with $J = 1$, $h = 1$, and $\theta = 15\pi/32$ across $\widetilde{T} = 4 \times 10^4$ future time steps employing the skipping-ahead method with a time skip of $\tau=10^6$ steps. $T=2\times 10^4$ steps of the time evolution with the same step size of $\Delta t=1/(200E_{\textrm{max}})$ are used to train the NG-RC, with the regularization parameter $\lambda = 0$. Our benchmark targets are the real and the imaginary parts of all amplitudes in the computational basis (top). The comparisons of the observables $\expval{X_0}$, $\expval{X_0X_1}$, and the fidelity $F(\tilde{s}_t,s_t):=|\braket{\tilde{s}_t}{s_t}|$ between the target and the predicted states are also shown (bottom).

Theorems & Definitions (8)

• Definition S1: Block-encoding
• Lemma S2: Multiplication of two block-encoded matrices gilyen2019
• Lemma S3: Pre-amplification of block encoding
• Lemma S4: Applying a pre-amplified block-encoded matrix on a quantum state Chakraborty2023
• Lemma S5: QSVT polynomial for matrix inversion
• Theorem S6: Matrix inversion via QSVT
• Lemma S7: Tikhonov regularized matrix pseudoinversion
• Lemma S8: Error propagation