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Improved machine learning algorithm for predicting ground state properties

Laura Lewis, Hsin-Yuan Huang, Viet T. Tran, Sebastian Lehner, Richard Kueng, John Preskill

TL;DR

A classical machine learning algorithm for predicting ground state properties with an inductive bias encoding geometric locality that outperforms any non-learning classical algorithm but requires extensive training data is given.

Abstract

Finding the ground state of a quantum many-body system is a fundamental problem in quantum physics. In this work, we give a classical machine learning (ML) algorithm for predicting ground state properties with an inductive bias encoding geometric locality. The proposed ML model can efficiently predict ground state properties of an $n$-qubit gapped local Hamiltonian after learning from only $\mathcal{O}(\log(n))$ data about other Hamiltonians in the same quantum phase of matter. This improves substantially upon previous results that require $\mathcal{O}(n^c)$ data for a large constant $c$. Furthermore, the training and prediction time of the proposed ML model scale as $\mathcal{O}(n \log n)$ in the number of qubits $n$. Numerical experiments on physical systems with up to 45 qubits confirm the favorable scaling in predicting ground state properties using a small training dataset.

Improved machine learning algorithm for predicting ground state properties

TL;DR

A classical machine learning algorithm for predicting ground state properties with an inductive bias encoding geometric locality that outperforms any non-learning classical algorithm but requires extensive training data is given.

Abstract

Finding the ground state of a quantum many-body system is a fundamental problem in quantum physics. In this work, we give a classical machine learning (ML) algorithm for predicting ground state properties with an inductive bias encoding geometric locality. The proposed ML model can efficiently predict ground state properties of an -qubit gapped local Hamiltonian after learning from only data about other Hamiltonians in the same quantum phase of matter. This improves substantially upon previous results that require data for a large constant . Furthermore, the training and prediction time of the proposed ML model scale as in the number of qubits . Numerical experiments on physical systems with up to 45 qubits confirm the favorable scaling in predicting ground state properties using a small training dataset.
Paper Structure (27 sections, 27 theorems, 237 equations, 5 figures)

This paper contains 27 sections, 27 theorems, 237 equations, 5 figures.

Table of Contents

  1. Introduction
  2. ML algorithm and rigorous guarantee
  3. Definitions
  4. Feature mapping and ML model
  5. Rigorous guarantee
  6. Proof ideas
  7. Simple form for ground state property
  8. Norm inequality for observables
  9. Prediction error bound for the ML algorithm
  10. Numerical experiments
  11. Outlook
  12. Simple form for ground state property
  13. Approximation by a sum of smooth functions
  14. Simplification using discretization
  15. Simple form for ground state property
  16. ...and 12 more sections

Key Result

Theorem 1

Given $n, \delta > 0$, $\tfrac{1}{e} > \epsilon > 0$ and a training data set $\{x_\ell, y_\ell\}_{\ell = 1}^N$ of size where $x_\ell$ is sampled from an unknown distribution $\mathcal{D}$ and $|y_\ell - \Tr(O \rho(x_\ell))| \leq \epsilon$ for any observable $O$ with eigenvalues between $-1$ and $1$ that can be written as a sum of geometrically local observables. With a proper choice of the effici

Figures (5)

  • Figure 1: Overview of the proposed machine learning algorithm. Given a vector $x \in [-1, 1]^m$ that parameterizes a quantum many-body Hamiltonian $H(x)$. The algorithm uses a geometric structure to create a high-dimensional vector $\phi(x) \in \mathbb{R}^{m_\phi}$. The ML algorithm then predicts properties or a representation of the ground state $\rho(x)$ of Hamiltonian $H(x)$ using the $m_\phi$-dimensional vector $\phi(x)$.
  • Figure 2: Predicting ground state properties in 2D antiferromagnetic random Heisenberg models. (A) Prediction error. Each point indicates the root-mean-square error for predicting the correlation function in the ground state (averaged over Heisenberg model instances and each pair of neighboring spins). Left figure fixes the training set size $N$ to be $50$ and system size $n$ to be $9 \times 5 = 45$. Center figure fixes the shadow size $T$ to be $500$ and $n = 45$. Right figure fixes $N = 50$ and $T = 500$. The shaded regions show the standard deviation over different spin pairs. (B) Visualization. We plot how much each coupling $J_{ij}$ contributes to the prediction of the correlation function over different pairs of qubits in the trained ML model. Thicker and darker edges correspond to higher contributions. We see that the ML model learns to utilize the local geometric structure.
  • Figure 3: Intuition behind Lemma \ref{['lemma:approxlocal']}. The qubits (blue circles) are arranged in a two-dimensional lattice with local Hamiltonian terms (light gray shading) acting between all pairs of neighboring qubits. A Pauli term $P$ acts on a subset of these qubits indicated by the light blue region. The dark blue circle represents a neighborhood around the region on which $P$ acts. The idea of Lemma \ref{['lemma:approxlocal']} is that when changing the parameters $x$, only $\vec{x}_j$ such that $h_j(\vec{x}_j)$ within the neighborhood around the region that $P$ acts on should significantly change $\Tr(P\rho(x))$. Hence, $f_P$ depends only on those parameters. It is implicit in the figure that $h_j$ depends on $\vec{x}_j$ for all $j$. Hence, $f_P$ depends only on the vectors $\vec{x}_{14}, \vec{x}_{19}, \vec{x}_{20}, \vec{x}_{25}$.
  • Figure 4: Example of Definition \ref{['def:discretization']}. Illustration of the set $T_{x', P}$ (light blue shading) for specific $x' \in X_P$ (blue circle), fixing $\delta_2 = 1/2$ for simplicity. (a)Example for $m=1$.$I_P$ is fixed to $\{1\}$ so that $X_P = \{0, \pm 1/2, \pm 1\}$ according to Def. \ref{['def:discretization']}. $T_{x', P}$ is depicted for the chosen $x' = 1/2$. (b)Example for $m = 2$.$I_P$ is fixed to $\{2\}$, and $T_{x', P}$ is depicted for the chosen $x' = (0, -1/2)$.
  • Figure 5: Intuition behind proof construction of Theorem \ref{['thm:normineq']} for the cases of $d = 1$ (a) and $d = 2$ (b). In both cases, the idea is to divide our qubits (blue circles) in $d$-dimensional space into blocks (light blue boxes), and consider the quantity we wish to bound in these blocks. Note that all qubits not highlighted are in the buffer region. The first column in the figure depicts the unshifted blocks, i.e., $\vec{j} = 0$. The second column displays an example of shifted blocks (dashed boxes). Finally, the last column considers Pauli terms (dark blue circles) acting on the qubits circled and indicates if they are contained in $U_{0}$, defined in Eq. \ref{['eq:uj']}.

Theorems & Definitions (52)

  • Theorem 1: Sample and computational complexity
  • Corollary 1
  • Proposition 1: A variant of Proposition 1 in huang2021provably
  • Lemma 1: Training error bound
  • Theorem 2: Pauli $1$-norm bound
  • Definition 1
  • Definition 2
  • Lemma 2: Approximation using smooth local functions; simple case
  • Corollary 2: Approximation using smooth local functions; general case
  • Lemma 3: Change one coordinate; directional derivative
  • ...and 42 more