Predicting Many Properties of a Quantum System from Very Few Measurements (2002.08953v2)

Abstract: Predicting properties of complex, large-scale quantum systems is essential for developing quantum technologies. We present an efficient method for constructing an approximate classical description of a quantum state using very few measurements of the state. This description, called a classical shadow, can be used to predict many different properties: order $\log M$ measurements suffice to accurately predict $M$ different functions of the state with high success probability. The number of measurements is independent of the system size, and saturates information-theoretic lower bounds. Moreover, target properties to predict can be selected after the measurements are completed. We support our theoretical findings with extensive numerical experiments. We apply classical shadows to predict quantum fidelities, entanglement entropies, two-point correlation functions, expectation values of local observables, and the energy variance of many-body local Hamiltonians. The numerical results highlight the advantages of classical shadows relative to previously known methods.

• The paper introduces a classical shadows framework that compresses quantum state information from random measurements to predict multiple properties.
• It achieves measurement efficiency with logarithmic scaling in predicted properties, reaching the information-theoretic lower bound.
• Practical experiments validate superior predictions for quantum fidelities, entanglement entropies, and correlation functions across various models.

Predicting Many Properties of a Quantum System from Very Few Measurements

Overview of the Paper

The paper "Predicting Many Properties of a Quantum System from Very Few Measurements" by Hsin-Yuan Huang, Richard Kueng, and John Preskill introduces a novel framework for efficiently predicting numerous properties of quantum systems using minimal measurements. The authors extend the concept of shadow tomography, originally proposed by Aaronson, and employ classical shadows—a compressed classical description of quantum states—to achieve their objectives.

Abstract

Accurate prediction of properties in complex quantum systems is fundamental for advancing quantum technologies. The authors present a technique for constructing approximate classical descriptions of quantum states, termed classical shadows, using a logarithmic number of measurements. This approach saturates information-theoretic lower bounds and is highly efficient in predicting diverse properties such as quantum fidelities, entanglement entropies, and various correlation functions. Their results are supported by extensive numerical experiments demonstrating the advantages over traditional methods.

Main Contributions

1. Classical Shadows Framework:
   • Classical Shadow Construction: The framework involves rotating the quantum state using a randomly selected unitary and then performing computational basis measurements. This is repeated to gather a collection of classical shadows representing the state, which can be processed to predict various quantum properties.
   • Measurement Efficiency: The number of measurements required scales logarithmically with the number of properties to be predicted, independent of the system size, making the procedure highly scalable.
2. Theoretical Guarantees:
   • Performance Bounds: The paper provides rigorous performance guarantees, showing that classical shadows enable the prediction of $M$ properties with a number of measurements that scales as $\mathcal{O}(\log(M))$. Specifically, the shadow norm of the observables determines the required number of measurements.
   • Optimality: The approach reaches the information-theoretic lower bounds, proving that no alternative method using single-copy measurements can universally require fewer samples while retaining accuracy.
3. Practical Relevance:
   • Applications: Classical shadows are shown to be effective in practical scenarios such as estimating quantum fidelities with target states, verifying entanglement, and measuring local observables in quantum simulations.
   • Software Implementation: The authors provide an open-source tool to facilitate adoption in quantum computing experiments and research.

Numerical Experiments

The paper validates the theoretical framework through numerical experiments across different quantum systems and properties:

• Quantum Fidelity: Classical shadows enable efficient fidelity estimation for GHZ states and toric code ground states, requiring significantly fewer measurements compared to neural network tomography (NNQST) methods.
• Correlation Functions: Two-point correlation functions in models like 1D transverse field Ising model and 2D anti-ferromagnetic Heisenberg model are accurately predicted using Pauli measurements, outperforming NNQST in terms of prediction accuracy and computational efficiency.
• Entanglement Entropy: The framework accurately predicts second-order Rényi entanglement entropy for subsystems, highlighting its power in capturing nonlinear properties.

Implications and Future Developments

The implications of this research are both theoretical and practical. Theoretically, the framework sets a benchmark for optimal measurement efficiency in quantum state characterization. Practically, it offers a toolbox for efficiently extracting relevant quantum properties, directly applicable to current quantum computing platforms.

Future Directions:

• Time-evolution of Classical Shadows: Investigating methods to update classical shadows as quantum states evolve, especially under local Hamiltonians, would be valuable.
• Generalization to Quantum Channels: Extending the formalism to predict properties of quantum channels rather than static states can broaden the applicability.
• Shallow Circuit Ensembles: Exploring classical shadows generated from shallow random quantum circuits could offer new insights into simulation efficiency and hardware implementation.

Conclusion

The authors have introduced a potent method for efficiently estimating multiple properties of quantum states, substantially advancing quantum state tomography. By leveraging classical shadows, the framework not only meets theoretical lower bounds but also demonstrates practical superiority in a range of quantum systems and tasks. This work paves the way for more efficient quantum state characterization, crucial for the development of quantum technologies.

For more details and practical implementation, the authors provide resources and code at GitHub Repository.