Learning quantum many-body systems from a few copies (2107.03333v4)

Abstract: Estimating physical properties of quantum states from measurements is one of the most fundamental tasks in quantum science. In this work, we identify conditions on states under which it is possible to infer the expectation values of all quasi-local observables of a state from a number of copies that scales polylogarithmically with the system's size and polynomially on the locality of the target observables. We show that this constitutes a provable exponential improvement in the number of copies over state-of-the-art tomography protocols. We achieve our results by combining the maximum entropy method with tools from the emerging fields of classical shadows and quantum optimal transport. The latter allows us to fine-tune the error made in estimating the expectation value of an observable in terms of how local it is and how well we approximate the expectation value of a fixed set of few-body observables. We conjecture that our condition holds for all states exhibiting some form of decay of correlations and establish it for several subsets thereof. These include widely studied classes of states such as one-dimensional thermal and high-temperature Gibbs states of local commuting Hamiltonians on arbitrary hypergraphs or outputs of shallow circuits. Moreover, we show improvements of the maximum entropy method beyond the sample complexity that are of independent interest. These include identifying regimes in which it is possible to perform the postprocessing efficiently as well as novel bounds on the condition number of covariance matrices of many-body states.

• The paper introduces an efficient quantum tomography method that requires significantly fewer samples than traditional approaches.
• It employs maximum entropy, classical shadows, and quantum optimal transport to estimate local observable expectations with polylogarithmic scaling.
• The method demonstrates potential for practical quantum state characterization in complex systems while reducing computational complexity.

Evaluating Efficient Quantum State Inference via Few Samples and Transportation Cost Inequalities

The paper "Learning quantum many-body systems from a few copies" by Cambyse Rouzé and Daniel Stilck França presents a significant advancement in the field of quantum state tomography. The authors focus on creating efficient tomographic protocols for quantum many-body systems by leveraging the characteristics of quantum states and incorporating modern computational techniques. Their approach potentially circumvents the traditional challenges of resource-intensive tomographic methods which tend to have sample complexities that grow exponentially with the system size.

Central Concepts and Methodologies

Quantum tomography typically requires exponentially large resources when dealing with general quantum states due to the vast state space. This research, however, is centered around deducing the properties and expectation values of quasi-local observables in a manner that scales polylogarithmically with the system size, provided specific locality conditions are met. The key innovation lies in employing the maximum entropy method in conjunction with techniques from classical shadows and quantum optimal transport.

Maximum Entropy Method

This method seeks to identify the most likely state matching a set of known expectation values while maximizing the von Neumann entropy. By solving a convex optimization problem, the authors ensure that the tomographic process accommodates statistical noise and finite data access, leading to reliable state reconstruction.

Classical Shadows

Classical shadows play a pivotal role in efficiently estimating the expectation values of local observables. This technique allows deducing numerous properties of a quantum state from relatively few measurements, lending itself well to the problem of many-body quantum state tomography.

Quantum Optimal Transport and Lipschitz Observables

An intriguing aspect of the paper is the application of concepts in optimal transport, quantifying state differences through transportation cost inequalities and the Lipschitz continuity of observables. These mathematical tools provide a framework for approximating the physical properties of quantum states, focusing on robustness against statistical fluctuations and finite-sample effects.

Results and Findings

Rouzé and França exhibit substantial improvements over traditional tomography protocols, particularly highlighting exponential sample complexity reductions. Their method is shown to scale polynomially with the locality of observables and logarithmically with system size. The work applies broadly to a class of physically relevant quantum states, such as high-temperature Gibbs states and outputs of shallow quantum circuits.

Numerical and Theoretical Implications

Numerically, the results demonstrate that even for moderate locality and system size, the method achieves a considerable reduction in the complexity required to infer system properties compared to prior methods. Theoretically, the research underscores the potential to extend learning methods to new quantum states like ground states of gapped Hamiltonians.

Conclusion and Future Directions

The paper positions itself as a cornerstone contribution in reducing the barrier to practical quantum state tomography. It opens pathways for future research in several directions, including the adaptation of these techniques to other quantum information processes like quantum machine learning and process tomography.

As quantum devices continue to grow in size and complexity, the findings in this paper may prove stratagem in characterizing and verifying quantum systems efficiently. The blend of convex optimization, classical shadow techniques, and optimal transport inequalities introduces a paradigm shift in addressing the computational challenges inherent in many-body quantum physics, showcasing a promising direction toward practical quantum information processing. This method may catalyze further developments in simulating and understanding complex quantum environments and their intrinsic properties in a scalable and resource-efficient manner.