The study of quantum many body systems is one of the most significant applications of quantum computing and has applications in physics, materials science, and chemistry. Classical algorithms and methods such as density functional theory, Monte Carlo, and density-matrix renormalization group have assisted in solving some problems in many body systems but are proven inefficient when it comes to a general class of problems. The reason classical computers fall short in simulating quantum many-body physics is that describing an 𝑛-qubit quantum system accurately may require an amount of classical data that is exponential in 𝑛.

Recently, classical machine learning (ML) techniques have shown promising results in investigating problems in quantum many-body physics. Although these methods are experimentally effective with present size systems, they cannot provide the theoretical proof for efficiently solving larger system sizes, which cannot be solved by these classical algorithms yet. In this work, the authors attempt to establish classical machine learning algorithms based on the concept of a classical shadow derived from randomized Pauli measurements. The objective is to examine two potential applications of classical ML. The first application involves learning to predict classical representations of quantum many-body ground states while the second attempts to classify quantum states of matter into phases in a supervised learning scenario.

In the first application, a family of Hamiltonians is considered, where the Hamiltonian 𝐻(𝑥) depends on m real parameters. The ML algorithm is trained on a set of training data consisting of sampled values of 𝑥, each accompanied by the corresponding classical shadow for the ground state of the Hamiltonians. The ML algorithm then predicts a classical representation of the ground states for new values of 𝑥, hence estimating ground state properties using the predicted classical representation. This method is shown to be efficient for the case of predicting ground state properties that do not vary too rapidly as a function of 𝑥, with provable bounds.

In the second application, in order to predict the phase label for new quantum states that were not encountered during training, it was assumed that any two phases can be distinguished by a nonlinear function of marginal density operators of subsystems of constant size. It is shown that if such a function exists, a classical ML algorithm can learn to distinguish the phases using an amount of training data and classical processing which are polynomial in the system size. The authors also performed numerical experiments on quantum systems such as the Rydberg atom chain and 2D antiferromagnetic Heisenberg model, in order to test the efficiency of the learning and prediction phases. The numerics verify the claim that classical ML algorithms can efficiently classify a wide range of quantum phases of matter, in particular when informed by data collected in physical experiments and/or simulations on quantum computers.

Overall, the work focuses on the classical shadow formalism which assumes that each quantum state is represented by its classical shadow, that could be obtained either from a classical computation or from an experiment on a quantum device. The classical ML algorithm is then trained on labeled classical shadows, andshadows and learns to predict labels for new classical shadows. This is an interesting concept that can be potentially used for other classical representations of quantum states which could be exploited by classical ML. As an added bonus, existing highly programmable quantum simulators, which lack local controls in implementing single qubit Pauli measurements, can be benefitted from such a formalism.

An interesting outlook for quantum computing can then be predicted as follows; typically, the computational scaling of quantum computaitonal algorithms can be shown to be significantly better than classical algorithms for the same task in quantum many-body problems. However, the requirements of fault-tolerant quantum computers and other overhead implies a pre-factor that allows only a limited number of (very accurate) simulations to be performed in a reasonable timeframe. In this way, quantum computing may not help in generating huge datasets or explore exhaustively a large configurational space to design new materials, by themselves. However, when combined with Machine Learning approaches such as those presented in the work discussed today, the accurate quantum-computer generated data may replace the physical experiments required for generating the data in this strategy. That in-turn allows for a pure in-silico and more cost-effective strategy for discovery and design of new materials and understanding of (potentially new) physics.