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arXiv:2411.03285

ShadowGPT: Learning to Solve Quantum Many-Body Problems from Randomized Measurements

Jian Yao, Yi‐Zhuang You

Published Nov 5, 2024• 2 citations
arXivHTMLPDF

Abstract

We propose ShadowGPT, a novel approach for solving quantum many-body problems by learning from randomized measurement data collected from quantum experiments. The model is a generative pretrained transformer (GPT) trained on simulated classical shadow data of ground states of quantum Hamiltonians, obtained through randomized Pauli measurements. Once trained, the model can predict a range of ground state properties across the Hamiltonian parameter space. We demonstrate its effectiveness on the transverse-field Ising model and the $\mathbb{Z}_2 \times \mathbb{Z}_2$ cluster-Ising model, accurately predicting ground state energy, correlation functions, and entanglement entropy. This approach highlights the potential of combining quantum data with classical machine learning to address complex quantum many-body challenges.

DOI
10.48550/arxiv.2411.03285

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We propose ShadowGPT, a novel approach for solving quantum many-body problems by learning from randomized measurement data collected from quantum experiments. The model is a generative pretrained transformer (GPT) trained on simulated classical shadow data of ground states of quantum Hamiltonians, obtained through randomized Pauli measurements. Once trained, the model can predict a range of ground state properties across the Hamiltonian parameter space. We demonstrate its effectiveness on the transverse-field Ising model and the Z2×Z2\mathbb{Z}_2 \times \mathbb{Z}_2Z2​×Z2​ cluster-Ising model, accurately predicting ground state energy, correlation functions, and entanglement entropy. This approach highlights the potential of combining quantum data with classical machine learning to address complex quantum many-body challenges.
Machine learning has emerged recently as a powerful tool for predicting properties of quantum many-body systems. For many ground states of gapped Hamiltonians, generative models can learn from measurements of a single quantum state to reconstruct the state accurately enough to predict local observables. Alternatively, classification and regression models can predict local observables by learning from measurements on different but related states. In this work, we combine the benefits of both approaches and propose the use of conditional generative models to simultaneously represent a family of states, learning shared structures of different quantum states from measurements. The trained model enables us to predict arbitrary local properties of ground states, even for states not included in the training data, without necessitating further training for new observables. We first numerically validate our approach on 2D random Heisenberg models using simulations of up to 45 qubits. Furthermore, we conduct quantum simulations on a neutral-atom quantum computer and demonstrate that our method can accurately predict the quantum phases of square lattices of 13$\times$13 Rydberg atoms.
Recent theoretical progress has established conditions under which machine learning models can efficiently predict ground-state properties of gapped local Hamiltonians when trained on quantum-generated data. Previous experimental demonstrations in this paradigm, however, have largely been limited to small systems or highly structured states, due to the difficulty of preparing many-body ground states on quantum processors. In this work, we demonstrate learning from experimental quantum data generated from approximate ground states of the two-dimensional Heisenberg XXZ model with system sizes up to 115 qubits. We construct a dataset of single-site expectation values, two-point correlations, and 12-body loop correlations across the antiferromagnetic phase. We then train neural networks on this data and show that they can accurately predict spatially resolved observables for previously unseen Hamiltonian parameters, both within the training distribution and in an out-of-distribution regime approaching the phase boundary. Our results demonstrate the practical realization of learning from quantum data for an interacting two-dimensional many-body system at scale, motivating a path toward regimes where quantum processors could provide training data beyond the reach of classical approximation methods.
Abstract Advancements in the implementation of quantum hardware have enabled the acquisition of data that are intractable for emulation with classical computers. The integration of classical machine learning (ML) algorithms with these data holds potential for unveiling obscure patterns. Although this hybrid approach extends the class of efficiently solvable problems compared to using only classical computers, this approach has been realized for solving restricted problems because of the prevalence of noise in current quantum computers. Here, we extend the applicability of the hybrid approach to problems of interest in many-body physics, such as predicting the properties of the ground state of a given Hamiltonian and classifying quantum phases. By performing experiments with various error-reducing procedures on superconducting quantum hardware with 127 qubits, we managed to acquire refined data from the quantum computer. This enabled us to demonstrate the successful implementation of classical ML algorithms for systems with up to 44 qubits. Our results verify the scalability and effectiveness of the classical ML algorithms for processing quantum experimental data.
We introduce the generative quantum eigensolver (GQE), a new quantum computational framework that operates outside the variational quantum algorithm paradigm by applying classical generative models to quantum simulation. The GQE algorithm optimizes a classical generative model to produce quantum circuits with desired properties. Here, we develop a transformer-based implementation, which we name the generative pre-trained transformer-based (GPT) quantum eigensolver (GPT-QE). We show a proof-of-concept of training and pretraining of GPT-QE applied to electronic structure Hamiltonians, and demonstrate its ability illustrated by surpassing coupled cluster singles and doubles (CCSD) for the strong bond dissociation of the nitrogen molecule and approaching chemical accuracy. We also demonstrate the method on real quantum hardware.
Estimating quantum state properties is essential across a wide range of applications, from studying quantum many-body physics to benchmarking quantum hardware. We present Qurrium, a Python package built on Qiskit that implements randomized measurement protocols for estimating purity, second-order Rényi entropy, expectation values of Pauli observables, and state overlap. In this paper, we focus on the classical shadow protocol and demonstrate two workflows in Qurrium. One is the end-to-end workflow that integrates quantum circuit preparation, simulation, measurement, and analysis. The other is the standalone estimation workflow that accepts pre-collected measurement data from any hardware platform in Qurrium's data format. We demonstrate both workflows through examples of a cluster state and an Ising time-evolved state. Furthermore, we report results obtained from a superconducting quantum processor developed by Academia Sinica and analyzed using the standalone estimation workflow. Qurrium is built on Qiskit, the dominant framework in quantum computing software, making it directly accessible to the large community of researchers already working with Qiskit. The source code is openly available at https://github.com/qurrium/qurrium and the example code is provided at https://github.com/qurrium/classical-shadow-examples.
There is no doubt that quantum computing has opened up new horizons and perspectives in many fields, including scientific research. With the advancement of technology and quantum computers, we can now conduct new kinds of scientific experiments: observing quantum properties of individual systems, atoms, electrons and photons as well as influencing and controlling quantum systems. Quantum mechanics describes the properties of quantum systems by their quantum states. Unlike in classical mechanics, quantum states themselves cannot be directly observed in experiments, so many-body dynamics mathematical simulations are useful. Even though many-body dynamics problems can be solved for limited cases, but when we consider real-world pieces of matter this leads to computation-intensive calculations that are beyond classic digital computers. The memory and time needed to describe the quantum state of a many-body system scales exponentially with the size of the system. This has led physics communities to focus their attention on the algorithms underlying modern machine learning with the goal of making progress in quantum matter research. As a consequence, the successful application of machine learning requires effective and informative data representation. This paper discusses some novel aspects of applying classical machine learning algorithms to quantum data derived from measurements associated with quantum systems in many-body simulation problems. In this paper, we investigate and experimentally demonstrate one of the effective methods of the quantum data representation suited for the classical machine learning applications based on quantum classical shadows.