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Learning to Classify Quantum Phases of Matter with a Few Measurements

Mehran Khosrojerdi, Jason L. Pereira, Alessandro Cuccoli, Leonardo Banchi

Abstract

We study the identification of quantum phases of matter, at zero temperature, when only part of the phase diagram is known in advance. Following a supervised learning approach, we show how to use our previous knowledge to construct an observable capable of classifying the phase even in the unknown region. By using a combination of classical and quantum techniques, such as tensor networks, kernel methods, generalization bounds, quantum algorithms, and shadow estimators, we show that, in some cases, the certification of new ground states can be obtained with a polynomial number of measurements. An important application of our findings is the classification of the phases of matter obtained in quantum simulators, e.g., cold atom experiments, capable of efficiently preparing ground states of complex many-particle systems and applying simple measurements, e.g., single qubit measurements, but unable to perform a universal set of gates.

Learning to Classify Quantum Phases of Matter with a Few Measurements

Abstract

We study the identification of quantum phases of matter, at zero temperature, when only part of the phase diagram is known in advance. Following a supervised learning approach, we show how to use our previous knowledge to construct an observable capable of classifying the phase even in the unknown region. By using a combination of classical and quantum techniques, such as tensor networks, kernel methods, generalization bounds, quantum algorithms, and shadow estimators, we show that, in some cases, the certification of new ground states can be obtained with a polynomial number of measurements. An important application of our findings is the classification of the phases of matter obtained in quantum simulators, e.g., cold atom experiments, capable of efficiently preparing ground states of complex many-particle systems and applying simple measurements, e.g., single qubit measurements, but unable to perform a universal set of gates.
Paper Structure (3 sections, 13 equations, 3 figures)

This paper contains 3 sections, 13 equations, 3 figures.

Table of Contents

  1. Quantum-enhanced support vector machines
  2. Comparison with previous studies
  3. Generalization error bounds

Figures (3)

  • Figure 1: (a) Expected phase diagram of the ANNNI model in the thermodynamic limit. (b-c) Predicted phase diagram for the ANNNI Hamiltonian \ref{['eq:annni']} with $N=50$ spins, where the ground state is simulated using tensor network methods with a small bond dimension (20). The training points are marked with a black cross and the phases and phase transition lines are the same of Fig. \ref{['fig:annni']}. In (b) the training point are uniformly scattered, while in (c) they are located on the line $h=0.25$.
  • Figure 2: (a) Scaling of the generalization error bound Eq. \ref{['eq:bound']}, for the ground state of the ANNNI model with $h\in[0,2]$, $k\in[0,1]$, as a function of the number of qubits $N$, and for the two different parity sectors (even and odd). (b) Kernel matrix entries for the training states shown in Fig. \ref{['fig:annni']}.
  • Figure 3: Predicted phase diagram for the Hamiltonian \ref{['eq:Htopo']} with $N=41$ spins, where the ground state is simulated using tensor network methods with a bond dimension 150. The training points, marked with black crosses, are located along the line $h=1$. The phase transition lines (dashed white) are estimated by interpolating the numerically observed points (white asterisks) from Ref. cong2019quantum.