Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Learning out-of-time-ordered correlators with classical kernel methods

John Tanner, Jason Pye, Jingbo Wang

TL;DR

The paper addresses the high cost of computing Out-of-Time-Ordered Correlators (OTOCs) in quantum many-body systems by evaluating classical kernel methods to learn OTOC values as a function of parameterised 1D Hamiltonians. Using MPO-based data for systems up to 40 qubits, the authors compare six kernels and show that Laplacian and RBF kernels achieve high predictive accuracy (testing $R^2$ up to about 0.98 on average) with relatively small training sets. The results demonstrate that, after training, kernel models can substitute expensive tensor-network computations to estimate OTOCs across parameter space, offering a practical route to extensive scrambling analyses. The work also discusses limitations (data-generation cost, extrapolation bounds) and outlines potential extensions to other correlators, higher dimensions, and quantum kernels to further enhance efficiency and reach.

Abstract

Out-of-Time Ordered Correlators (OTOCs) are widely used to investigate information scrambling in quantum systems. However, directly computing OTOCs with classical computers is an expensive procedure. This is due to the need to classically simulate the dynamics of quantum many-body systems, which entails computational costs that scale rapidly with system size. Similarly, exact simulation of the dynamics with a quantum computer (QC) will either only be possible for short times with noisy intermediate-scale quantum (NISQ) devices, or will require a fault-tolerant QC which is currently beyond technological capabilities. This motivates a search for alternative approaches to determine OTOCs and related quantities. In this study, we explore four parameterised sets of Hamiltonians describing local one-dimensional quantum systems of interest in condensed matter physics. For each set, we investigate whether classical kernel methods (KMs) can accurately learn the XZ-OTOC and a particular sum of OTOCs, as functions of the Hamiltonian parameters. We frame the problem as a regression task, generating small batches of labelled data with classical tensor network methods for quantum many-body systems with up to 40 qubits. Using this data, we train a variety of standard kernel machines and observe that the Laplacian and radial basis function (RBF) kernels perform best, achieving a coefficient of determination (\(R^2\)) on the testing sets of at least 0.7167, with averages between 0.8112 and 0.9822 for the various sets of Hamiltonians, together with small root mean squared error and mean absolute error. Hence, after training, the models can replace further uses of tensor networks for calculating an OTOC function of a system within the parameterised sets. Accordingly, the proposed method can assist with extensive evaluations of an OTOC function.

Learning out-of-time-ordered correlators with classical kernel methods

TL;DR

The paper addresses the high cost of computing Out-of-Time-Ordered Correlators (OTOCs) in quantum many-body systems by evaluating classical kernel methods to learn OTOC values as a function of parameterised 1D Hamiltonians. Using MPO-based data for systems up to 40 qubits, the authors compare six kernels and show that Laplacian and RBF kernels achieve high predictive accuracy (testing up to about 0.98 on average) with relatively small training sets. The results demonstrate that, after training, kernel models can substitute expensive tensor-network computations to estimate OTOCs across parameter space, offering a practical route to extensive scrambling analyses. The work also discusses limitations (data-generation cost, extrapolation bounds) and outlines potential extensions to other correlators, higher dimensions, and quantum kernels to further enhance efficiency and reach.

Abstract

Out-of-Time Ordered Correlators (OTOCs) are widely used to investigate information scrambling in quantum systems. However, directly computing OTOCs with classical computers is an expensive procedure. This is due to the need to classically simulate the dynamics of quantum many-body systems, which entails computational costs that scale rapidly with system size. Similarly, exact simulation of the dynamics with a quantum computer (QC) will either only be possible for short times with noisy intermediate-scale quantum (NISQ) devices, or will require a fault-tolerant QC which is currently beyond technological capabilities. This motivates a search for alternative approaches to determine OTOCs and related quantities. In this study, we explore four parameterised sets of Hamiltonians describing local one-dimensional quantum systems of interest in condensed matter physics. For each set, we investigate whether classical kernel methods (KMs) can accurately learn the XZ-OTOC and a particular sum of OTOCs, as functions of the Hamiltonian parameters. We frame the problem as a regression task, generating small batches of labelled data with classical tensor network methods for quantum many-body systems with up to 40 qubits. Using this data, we train a variety of standard kernel machines and observe that the Laplacian and radial basis function (RBF) kernels perform best, achieving a coefficient of determination () on the testing sets of at least 0.7167, with averages between 0.8112 and 0.9822 for the various sets of Hamiltonians, together with small root mean squared error and mean absolute error. Hence, after training, the models can replace further uses of tensor networks for calculating an OTOC function of a system within the parameterised sets. Accordingly, the proposed method can assist with extensive evaluations of an OTOC function.
Paper Structure (25 sections, 2 theorems, 49 equations, 6 figures, 14 tables)

This paper contains 25 sections, 2 theorems, 49 equations, 6 figures, 14 tables.

Table of Contents

  1. Introduction
  2. Background
  3. Regularised empirical risk minimisation
  4. Kernel methods
  5. Information Spreading and Out-of-Time-Ordered Correlators
  6. Methods
  7. The parameterised sets of Hamiltonians
  8. The learning problem
  9. The kernels
  10. Generating the data
  11. Results
  12. Discussion
  13. Conclusion
  14. Supplementary results
  15. Convergence in the bond dimension
  16. ...and 10 more sections

Key Result

Theorem A.1

Let $\mathcal{X}\equiv\mathbb{R}^d$ be the input data domain and $\mathcal{D}=\{(\mathbf{x}_i,y_i)\}_{i=1}^{M}\subseteq\mathcal{X}\times\mathbb{R}$ be a training dataset. Let $\mathcal{K}:\mathcal{X}\times\mathcal{X}\to\mathbb{R}$ be a kernel and define $\widetilde{\mathcal{L}}_{\mathcal{D}}:\mathbb where $\|\cdot\|$ is the Euclidean norm, $\langle\cdot,\cdot\rangle$ is the Euclidean inner product

Figures (6)

  • Figure 1: A kernel function $\mathcal{K}$, which implicitly computes an inner-product in a high-dimensional feature space $\mathcal{F}$, can be used to simplify a regression problem if the associated feature map $\phi$ arranges the inputs in $\mathcal{F}$ in a desirable way. For example, above we see points associated with different continuous labels (indicated by the varying colours) being arranged into different parallel hyperplanes. This allows the continuous value associated with the points to be extracted via a simple projection along some axis in $\mathcal{F}$.
  • Figure 2: The Alice-Bob classical communication protocol in the case where the system is a 1D spin chain with open boundary conditions, evolving under a time-independent local Hamiltonian $H$. Alice and Bob have access to the individual qubits at opposite ends of the chain. Alice wants to send a bit $a \in \{0,1\}$ to Bob by applying $V_j$ if $a=1$, or doing nothing if $a=0$. The system then evolves for a time $t$, during which the influence of Alice's operation propagates through the system. Bob then performs a measurement of $W_k$ to try and determine the value of the bit $a$.
  • Figure 3: Tensor network representation of the systems involved in the Alice-Bob quantum communication protocol. Initially, subsystems $R_1$ and $q_A$ form a Bell state, while subsystems $\mathcal{S}\setminus q_A$ and $R_2$ form $n-1$ Bell states. The unitary operator $e^{-i H t}$ is then applied to the system $\mathcal{S}$. Bob's aim is to detect and quantify the entanglement with $R_1$ that has spread from $q_A$ to $q_B$.
  • Figure 4: Coefficient of determination ($R^2$) for the models trained on all 1000 training datapoints with the best hyperparameter values trialled in the cross-validation, making predictions on the testing sets. The top row of plots shows the results for the datasets with labels determined by $\mathscr{O}_{XZ}$ with (a) $\mathscr{H}_1$, (b) $\mathscr{H}_2$, (c) $\mathscr{H}_3$ and (d) $\mathscr{H}_4$. The bottom row of plots shows the results for the datasets with labels determined by $\mathscr{O}_{Sum}$ with (e) $\mathscr{H}_1$, (f) $\mathscr{H}_2$, (g) $\mathscr{H}_3$ and (h) $\mathscr{H}_4$.
  • Figure 5: Average coefficient of determination (Average $R^2$) for each kernel making predictions on the 40-qubit testing sets, plotted against the number of training data samples $M$ used to train the associated ML models. Each point in the plots corresponds to the numerical value of $R^2$ averaged over 20 different models using the same kernel. Each of the models is the result of using a random subset of the training data containing $M$ of the total 1000 training data samples. The hyperparameters are fixed to be the best hyperparameters found during the 10-fold cross-validation performed on all 1000 training data samples. Plots (a), (b), (c) and (d) show the average $R^2$ score on the testing datasets with labels determined by $\mathscr{O}_{XZ}$ with $\mathscr{H}_1$, $\mathscr{H}_2$, $\mathscr{H}_3$ and $\mathscr{H}_4$, respectively. Plots (e), (f), (g) and (h) show the average $R^2$ score on the testing datasets with labels determined by $\mathscr{O}_{Sum}$ with $\mathscr{H}_1$, $\mathscr{H}_2$, $\mathscr{H}_3$ and $\mathscr{H}_4$, respectively.
  • ...and 1 more figures

Theorems & Definitions (4)

  • Theorem A.1
  • proof
  • Theorem A.2
  • proof