Approximate Message Passing Reduces Quantum State Tomography Infidelity by over One Order of Magnitude

Quantum state tomography, the process of fully characterizing quantum systems, faces significant challenges as complexity increases with system size. Noah Siekierski, Kausthubh Chandramouli, and colleagues at North Carolina State University and the University of Central Florida demonstrate a new approach to overcome these limitations, leveraging a technique called approximate message passing. Their work introduces a compressed sensing method that achieves substantially improved performance in low-rank quantum state tomography, reducing reconstruction errors by over an order of magnitude compared to existing methods. By carefully designing the algorithm, the team not only advances the field of quantum state characterisation, but also opens possibilities for applying similar techniques to other complex tomography protocols, and they validated their approach with experiments on real quantum hardware, considering the impact of device noise.

Approximate Message Passing Reconstructs Quantum States

Quantum state tomography aims to reconstruct an unknown quantum state from a set of measurements, a task complicated by the high dimensionality of the state space. This research introduces a new approach to quantum state tomography based on Approximate Message Passing (AMP), an iterative algorithm originally developed for signal processing and machine learning. The proposed AMP-based algorithm efficiently estimates the density matrix representing the quantum state, offering significant advantages in computational complexity and scalability compared to conventional techniques. The method formulates quantum state estimation as a Bayesian inference problem and then applies the AMP algorithm to approximate the distribution of the density matrix elements.

The algorithm iteratively refines estimates by exchanging messages between measurement data and a prior distribution, effectively leveraging all available information. Theoretical analysis demonstrates the algorithm’s ability to accurately reconstruct states under certain conditions, and numerical simulations validate its performance with various quantum states and measurement scenarios. Results show that this AMP algorithm outperforms existing methods in both accuracy and efficiency, particularly for high-dimensional quantum systems. A key achievement of this work is the development of a novel AMP variant tailored to the specific characteristics of quantum state tomography.

The algorithm addresses challenges such as ensuring the estimated density matrix remains physically valid throughout the iterative process, enhancing its robustness and reliability. Furthermore, the research provides insights into the fundamental limits of quantum state tomography, establishing a connection between the algorithm’s performance and the information-theoretic properties of the quantum state. This allows for a better understanding of the trade-offs between measurement resources, estimation accuracy, and computational complexity in quantum state reconstruction.

Quantum state tomography is an indispensable tool for characterizing many-body quantum systems. However, the exponential scaling of computational cost with system size necessitates the development of specialized approaches for certain quantum states, such as those with low rank. This research demonstrates how approximate message passing (AMP), a compressed sensing technique, can be used to perform low-rank quantum state tomography. AMP provides asymptotically optimal performance for large systems, making it well-suited for this application.

Rank Control and Observable Estimation Methods

This appendix provides mathematical justification and detailed explanations for the methods used in the main research. Specifically, it focuses on bounding the rank of the estimated quantum state and providing a rigorous way to extract information about the quantum state from a limited set of measurement outcomes. A lower rank simplifies the problem, and accurate extraction of information is crucial for state reconstruction. The appendix demonstrates that even with noisy measurements, the rank of the estimated state remains reasonably controlled. It uses the properties of density matrices and error channels to show how the rank is preserved, or at least bounded, under the combined effect of errors.

This is a crucial result because it means the problem of reconstructing the state does not become exponentially hard as the number of qubits increases. The appendix also provides a detailed explanation of how to extract information about the quantum state from a set of measurement outcomes. It introduces projectors for the Pauli matrices and explains how to calculate the expectation value of a Pauli operator by summing over the measurement outcomes. This method allows for efficient analysis of measurement data, reducing the computational cost of state reconstruction. These results are essential for performing quantum state tomography and for developing error mitigation techniques, crucial for building practical quantum computers.

Low-Rank Tomography via Approximate Message Passing

This research presents a significant advancement in quantum state tomography, specifically for systems described by low-rank density matrices. Scientists developed a novel approach using approximate message passing, a technique originally from compressed sensing, to reconstruct quantum states with improved accuracy and efficiency. The method demonstrably outperforms existing low-rank tomography techniques across a range of tested states, reducing reconstruction errors by a substantial margin. The team successfully applied this new algorithm to experimental data obtained from a quantum device, IBM Kingston, demonstrating its feasibility and practical application.

By carefully considering the characteristics of real quantum hardware, they were able to inform the tomography process and achieve reliable state preparation fidelity estimates. This work not only improves the accuracy of quantum state reconstruction but also provides a valuable tool for characterizing and validating quantum devices. The authors acknowledge that the performance of the algorithm is influenced by device noise, a common challenge in quantum experiments, and suggest that future research could explore the application of this method to other tomography protocols and investigate its performance with more complex quantum systems. While the current work focuses on low-rank states, the underlying principles could potentially be extended to address broader challenges in quantum state characterization.