The reservoir learning power across quantum many-body localization transition

Abstract

Harnessing the quantum computation power of the present noisy-intermediate-size-quantum devices has received tremendous interest in the last few years. Here we study the learning power of a one-dimensional long-range randomly-coupled quantum spin chain, within the framework of reservoir computing. In time sequence learning tasks, we find the system in the quantum many-body localized (MBL) phase holds long-term memory, which can be attributed to the emergent local integrals of motion. On the other hand, MBL phase does not provide sufficient nonlinearity in learning highly-nonlinear time sequences, which we show in a parity check task. This is reversed in the quantum ergodic phase, which provides sufficient nonlinearity but compromises memory capacity. In a complex learning task of Mackey–Glass prediction that requires both sufficient memory capacity and nonlinearity, we find optimal learning performance near the MBL-to-ergodic transition. This leads to a guiding principle of quantum reservoir engineering at the edge of quantum ergodicity reaching optimal learning power for generic complex reservoir learning tasks. Our theoretical finding can be tested with near-term NISQ quantum devices.

Graphical abstract

Keywords

quantum reservoir computing / many-body localization / quantum ergodic / edge of quantum ergodicity,optimal learning power

1 Introduction

Recent experimental demonstration of the fascinating quantum computation advantage with superconducting qubits [1] and interfering photons [2] has triggered monumental research interests in applications of quantum computing. Although a scalable error-corrected quantum computer is still unavailable, a large degree of controllability has been available in a broad spectrum of noise-intermediate-scale quantum (NISQ) devices [3–5], including atomic [6], photonic [7, 8], and solid-state systems [9, 10], whose numerical simulations are exponentially costly in classical computation resources. This suggests compelling quantum computational power embedded in the present NISQ devices, of potential importance to applications in science discovery [11], cryptography [12], and optimization [13]. How to harness the quantum computational power and how the computational power manifests in the present NISQ devices demand theoretical investigation. This is particularly crucial for quantum systems lacking full programmability.

For classical systems, one way to characterize their computational power is from the learning capability within the framework of reservoir computing [14], where the system is interfered by manipulating the input and output only with the system itself untouched as a “black box”. In time sequence learning, it has been found that the complex dynamics in the “black box” mediates both memory and nonlinearity in mapping the input to the output. It has been demonstrated that a classical dynamical system at the phase edge of a chaotic region [15–19] develops optimal learning power as it maintains a balance between integrability and chaoticity that nurture long-term memory and complex nonlinearity.

Generalizing the concept of reservoir computing to quantum dynamical systems leads to quantum reservoir computing (QRC) [20–24], that does not require precise or full quantum control of the quantum system. This approach provides a plausible route to harness the ultimate quantum computation power of NISQ devices [25]. However, what quantum systems would exhibit optimal learning power in reservoir computing remains to be understood.

In this work, we study the learning power of a one-dimensional quantum reservoir composed of long-range randomly-coupled qubits (Fig. 1). This system has a quantum many-body localization (MBL) transition with varying the disorder strength. We investigate the learning performance on two computation tasks of Boolean function emulation including the short-term memory (STM), and the parity check (PC) tasks, which characterize the memory capacity and the amount of nonlinearity of the quantum reservoir, respectively [20]. In the MBL phase, learning the STM task outperforms the PC task significantly, which we attribute to the extensive number of emergent local integrals of motion in the MBL phase [26–29]. In the quantum ergodic phase [30], we find the opposite, which implies the quantum dynamics in the ergodic phase mediates larger amount of nonlinearity, more resembling the chaotic dynamics in classical systems. We further examine the learning power of the disordered quantum many-body system on the Mackey–Glass (MG) prediction task [31], whose performance requires both memory and nonlinearity. In this task, we find that the learning power develops a peak around the quantum MBL-to-ergodic phase transition. This work would shed light on the quantum reservoir engineering to invest most quantum learning power for complex reservoir computing tasks.

2 The quantum reservoir computing setup

In QRC [20], the system is initialized as an infinite temperature state at time t = 0. The quantum reservoir dynamics corresponds to a unitary time evolution of N qubits described by a time-dependent density operator, ρ(t), sequentially interrupted by an input signal {\left\{s_k\right\}}_{k=1}^M, where s_k takes binary numbers 0 and 1 for a discrete signal, or values in between in a continuous setting. The time evolution is chopped into multiple τ -durations, i.e., t = kτ , with each duration further split into V subintervals. At time t = kτ , the time evolution is interrupted by the input as \begin{array}{l}\rho (t)\mapsto \rho (t+0^{+})={\rho }_{s_k}^{[1]}\otimes {\text{tr}}_1[\rho (t)]\hfill \end{array}, where tr₁ is a partial trace with respect to the first qubit, and the state of the first qubit is modified to {\rho }_{s_k}^{[1]}\to |{\psi }_{s_k}\rangle \langle {\psi }_{s_k}|, with |{\psi }_{s_k}\rangle =\sqrt{1-s_k}|+\rangle +\sqrt{s_k}|-\rangle, and |±i, the eigenstates of the Pauli-X operator. Note that here the information is encoded with the σ_x basis rather than the σ_z basis encoding. The σ_z basis encoding which has applied to the NARMA task [20, 32], does not perform well for the complicated learning tasks presented in this work (see Appendix A). Measurements are performed in the Pauli-X basis at the end of each subinterval. The resulting output signal is denoted as S, with elements S_k,v,i = (1 + tr [ ρ (kτ + τ v/V ) X_i ]) /2, v and i indexing subintervals and qubits, respectively.

In learning a time-sequence defined by a nonlinear function, y_{k^{\prime }}^{\star }=f(\{s_{k\le k^{\prime }}\};k^{\prime }), we cast the time steps into three groups {K₁, K₂, K₃}, with the casted time sequence and the reservoir output signal represented by a set of vectors \left\{{\mathit{y}}_1^{\star }\mathrm{,}{\mathit{y}}_2^{\star }\mathrm{,}{\mathit{y}}_3^{\star }\right\}, and matrices {S₁, S₂, S₃} accordingly. The starting time steps K₁ are discarded to remove the effect of initialization. The following time steps K₂ are used for training a linear regression model, y_k={\displaystyle {\sum }_{v\mathrm{,}i}}{S}_{k\mathrm{,}v\mathrm{,}i}{W}_{v\mathrm{,}i}+B_k with W s and Bs fitting parameters introduced to minimize the Frobenius norm ||y−y^⋆||. The final time steps K₃ are for prediction, using the reservoir output S₃ and the trained linear regression model, by which a time sequence y₃ is predicted. The performance of the reservoir computing is quantified by a normalized covariance, C=\text{cov}\left({\mathit{y}}_3^{\star }\mathrm{,}{\mathit{y}}_3\right)/\left[\sigma \left({\mathit{y}}_3^{\star }\right)\sigma \left({\mathit{y}}_3\right)\right], with σ the standard deviation. Its average \overline{C} is obtained by sampling distinctive input-signals (s_k ).

3 Disordered quantum spin chain reservoir

We consider a one-dimensional long-ranged coupled transverse-field Ising model as the quantum reservoir (Fig. 1), whose dynamics is governed by the Hamiltonian,

Here, X and Z are the Pauli operators, J_ij = J₀|i−j|^−α represent the long-range power-law decaying Ising couplings, and the transverse field contains a background constant part B, and a random part, ϕ_i drawn from \left(\begin{array}{c}-\frac{W}{2}\mathrm{,}\frac{W}{2}\end{array}\right) according to a uniform distribution. Hereafter, the coupling strength J₀ is set as energy unit. In this work, we choose B = 4, and the main findings are largely independent of this parameter choice. This disorder spin model has a MBL to quantum ergodic transition, at a critical disorder strength W_C [33]. Long-range coupling is considered here to support sufficiently complex dynamics for QRC. To characterize the learning power, we average over the disorder samples and input signals.

The QRC model in Eq. (1) can be realized in experiments with the present NISQ devices including NMR [34], trapped ions [35, 36], and Rydberg atoms [37]. With NMR, the required Hamiltonian is realizable by pulse shaping [38, 39]. With trapped ions, the model has already been used to investigate the MBL transition [40] and discrete time crystals [41]. With Rydberg atoms, the Ising couplings take specific forms with exponents α = 3, 6 [37, 42, 43], and generic long-range interacting models can be engineered with the quantum wiring scheme [44]. In our numerical results to present below, we neglect the backaction effects in the measurements to comply with the NMR system [34]. We also confirm the main findings still hold when the backaction is incorporated.

4 Short-term memory task

We first evaluate the memory capacity of our quantum spin-chain reservoir through STM task [20], where the targeting time-sequence function is y_k^{\star }=s_{k-k_{\text{Δ}}} with k_∆ representing the time delay. The learning performance on this linear task reflects the reservoir memory capacity. Figure 2 shows the QRC performance across the MBL transition. We find that the learning performance has a non-monotonic dependence on the time duration τ , reaching an optimum at a certain intermediate value τ_c. The increase of \overline{C} at small τ is because the reservoir dynamics is essentially frozen with a too small choice of τ , and that the information injected through the first qubit is washed away by the subsequent inputs without capability of sharing and storing the information collectively in the reservoir. At τ > τ_c, the decrease of \overline{C} with τ happens because the information would become more hidden in many-body correlations [45, 46], which cannot be extracted by the local measurements. An intermediate value of τ should be used in order to maximize the learning power of the reservoir. As we increase the disorder strength, the learning performance exhibits a monotonic increase, demonstrating an apparent advantage of the MBL phase (W > W_C) in memory holding, compared with the quantum ergodic phase (W < W_C). We attribute this to the emergence of local integrals of motion in the MBL phase [26–29]. With these conserved quantities, the information stored in the Pauli-X basis scrambles very slowly through a logarithmic dephasing process in quantum dynamics [47–49].

5 Parity check task

The second learning task we examine with our QRC is PC, for which the learning function is y_k^{\star }= \left({\displaystyle {\sum }_{m=0}^{k_{\text{Δ}}}{s}_{k-m}}\right) mod 2. This function is highly nonlinear, and the QRC learning performance then reflects the amount of nonlinearity in the reservoir. Figure 3 shows our numerical results. As the time duration τ is increased, we find that \overline{C} monotonically increases, and then saturates to 1 at large τ . With increasing disorder strength for J₀τ≤1, the learning performance becomes slightly worse within the quantum ergodic phase, and bends downward dramatically across the MBL transition — the learning power of the quantum ergodic phase thus outperforms the MBL phase for the PC task. In the quantum ergodic phase, the Heisenberg time evolution of the Pauli-X operators is strongly coupled with an exponential number (4^N−1) of all Pauli operators, which then resembles the classical chaotic dynamics, providing a larger degree of nonlinearity than the integrable MBL phase.

To gain more insight about the nonlinearity of the quantum reservoir, we look into the intrinsic chaotic properties of the quantum reservoir and calculate the out-of-time-order-correlator (OTOC) [50, 51],

\begin{array}{}\text{(2)}& O(\tau )=1-\frac{1}{N-1}{\displaystyle \sum _{i=2}^N\langle X_1^{†}(\tau )X_i^{†}{X}_1(\tau )X_i\rangle }\mathrm{,}\end{array}

with X₁(τ ) = e^iHτ X₁e^−iHτ and 〈·〉 an average over a thermal ensemble at infinite temperature. The OTOC has been introduced in the literature to diagnose quantum chaos by generalizing Poisson brackets and Lyapunov exponent from classical dynamical systems [52]. Here it captures the information spreading from the first qubit to the rest. From Fig. 4(a), we see that OTOC increases monotonically with the time duration τ and then saturates, which closely correlates with the learning performance in Fig. 3. Quantum chaotic dynamics as illustrated by OTOC is beneficial for producing nonlinearity. Since all the present NISQ devices have limited quantum coherence time [3, 4], it is crucial to know the required time scale for the QRC to reach satisfactory learning performance. We thus introduce a time scale, τ_th, where OTOC exceeds a certain threshold value (set to be 0.2 here). As shown in Fig. 4(b), we find in general the information spreads much more rapidly in the quantum ergodic phase than the MBL phase. We also find the information spreads more rapidly at a smaller α, which implies larger amount of nonlinearity with longer ranged interacting models for a given amount of evolution time. More analysis about the mechanisms of the learning power for QRC in MBL and quantum ergodic phase can be found in Appendix B. Quantum reservior learning performances in the deep MBL regions are presented in Appendix C. These findings would shed light on quantum reservoir engineering for harnessing the learning power of NISQ devices on nonlinear reservoir computing tasks.

6 Mackey–Glass task

With the STM and PC tasks, we have demonstrated maximizing the memory capacity and nonlinearity favors the quantum ergodic, and the MBL phases, respectively. This suggests the QRC has optimal learning power in between for learning tasks that requires both large memory capacity and sufficient nonlinearity. We thus analyze the MG task [31, 53] as such an example. This task is defined by [31]

\begin{array}{}\text{(3)}& F_{k+1}=\gamma F_k+\frac{\lambda F_{k-k_{\text{Δ}}}}{1+F_{k-k_{\text{Δ}}}^{\beta }}\mathrm{,}\end{array}

with k_∆ a time delay. The dynamical system has a chaotic attractor with k_∆ > 16.8, and here we choose γ = 0.9, β = 10, λ = 0.2, and k_∆ = 17. We set s_k = (F_k−F_min)/(F_max−F_min), and y_k^{\star }=s_{k+1} — the time sequences are shifted and rescaled to fit into the regime of [0, 1] by introducing F_max and F_min in our numerical simulation. This learning task is to predict the chaotic time sequence, which is deterministic unlike the two tasks studied above. For the MG task, the output signals as measured from the reservoir (Fig. 1) are added with a slight amount of white noise in the range of (−σ, σ) to improve robustness [20]. We choose σ = 10⁻⁶ for the noise strength. In predicting the chaotic time sequence, i.e., for k∈K₃, s_k and y_k are decided through an iteration — y_k is obtained by giving the input s_k to the reservoir, and then y_k+1 obtained by setting s_k+1 = y_k . A piecewise linear threshold function is introduced here to enforce y_k∈[0, 1] [54].

The QRC performance on this MG task strongly depends on the number of forward steps (l = |K₃|) to predict. The dependence of \overline{C} on l is shown in Fig. 5(a). The QRC prediction for the chaotic time sequence works perfectly if the number of forward steps is small, producing \overline{C}\approx 1 for l < 10. It fails if l at request is too large. We remark here that the piecewise linear function is introduced only for numerical stability, and that the corresponding truncation is activated only when the reservoir computing starts to fail at large l. We then define a critical time step l_c, above which \overline{C} drops down below a threshold (chosen to be 0.5 here). More discussions on the choice of the threshold value are provided in Appendix D. This critical time step quantifies the learning power of the quantum reservoir in predicting the chaotic time sequence. As shown in Fig. 5(b), as we vary the disorder strength, l_c develops a peak generically around the MBL localization transition point, as demonstrated for a broad range of α∈{0.4, 0.8, 1.2}. This feature becomes more prominent for larger number of qubits. This finding implies the quantum reservoir has an optimal learning power near the edge of quantum ergodicity for complex learning tasks that require both memory and nonlinearity being sufficient.

7 Conclusion and outlook

We have investigated the learning power of a disordered quantum spin chain in both quantum ergodic and MBL phases. The MBL quantum reservoir is advantageous in holding long-term memory, for the presence of emergent local integrals of motion. The quantum ergodic phase provides a larger degree of nonlinearity, with a compromise on the memory holding. In dealing with the MG task that requires both memory capacity and nonlinearity, we find the learning performance develops a peak near the edge of the quantum ergodic phase, as analogous to the optimal computational power established for the classical reservoir computing at the edge of chaos. This leads to an important guiding principle of quantum reservoir engineering for complex reservoir computing tasks, that is to prepare the quantum system at the edge of quantum ergodicity. We expect our finding to hold in using other measures of quantum reservoir computing performance as well, for example with information processing capacity [55–57], for which the confirmation is worth future investigation.

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