Adaptive Tomography Achieves Optimal Scaling with N State Copies, Benchmarking Systems Accurately

Quantum tomography, the process of reconstructing an unknown quantum state or process, underpins many advances in quantum technologies and fundamental physics, yet achieving the highest possible accuracy remains a significant hurdle. Shuixin Xiao from Australian National University, Xiangyu Wang and Zhibo Hou from University of Science and Technology of China, alongside Yuanlong Wang, Jun Zhang and Ian R. Petersen, present a unified mathematical framework that streamlines the evaluation of accuracy across state, detector and process tomography. This work establishes clear criteria for optimal tomography protocols, demonstrating how to achieve the best possible scaling with the number of measurements needed, a marked improvement over existing static methods. The researchers then develop adaptive algorithms grounded in this framework, and validate them through both numerical simulations and optical experiments, achieving, for the first time, optimal performance in complex process tomography, paving the way for more efficient and reliable quantum characterisation.

Entangled Photons Characterize Phase Damping Process

This research details an investigation into adaptive quantum process tomography, specifically a technique called Adaptive Averaging Process Tomography (AAPT). Scientists meticulously characterized a phase damping process, a source of errors in quantum systems, using entangled photons. The experimental setup generates pairs of entangled photons and employs a sophisticated optical system to precisely control and measure their properties, allowing researchers to implement a wide range of measurements on the quantum system. Single-photon detectors then record the outcomes of these measurements, providing the data needed to reconstruct the quantum process.

The core of the method involves intelligently selecting measurements during the tomography process, significantly improving efficiency. Researchers discovered that prioritizing measurements in the initial stages of the algorithm, allocating 90% of resources to this step, yields higher precision while maintaining an optimal scaling of accuracy with the number of measurements, meaning accuracy improves predictably as more data is collected. Experimental results align with simulations, confirming the effectiveness of the adaptive AAPT method and demonstrating its advantage over non-adaptive approaches.

Optimal Tomography via Unified Infidelity Scaling

Researchers have developed a comprehensive framework for quantum tomography, aiming to maximize the accuracy of characterizing quantum states, detectors, and processes. Recognizing limitations in existing methods, the team unified the concept of infidelity, a measure of estimator accuracy, across all three tomography tasks. This unification extends the definition of infidelity to encompass not only the density matrices describing quantum states, but also the elements defining detectors and the matrices representing quantum processes, enabling a single analytical approach. The study establishes both a sufficient and necessary condition for any tomography protocol to achieve optimal scaling, attaining an infidelity that decreases proportionally to one over the number of state copies consumed.

This represents a significant improvement over static, non-adaptive methods. Scientists employed rigorous mathematical analysis to demonstrate that achieving this optimal scaling requires specific properties within the tomography protocol itself, providing a crucial benchmark for evaluating existing and future algorithms. To validate their theoretical findings, the team conducted both numerical simulations and quantum optical experiments, successfully demonstrating, for the first time, the attainment of this optimal scaling in ancilla-assisted process tomography.

Optimal Tomography Scales with Measurement Count

Researchers have achieved a significant breakthrough in quantum tomography, developing new methods to accurately characterize quantum states, processes, and detectors with unprecedented precision. Traditionally, determining the complete description of a quantum system requires a large number of measurements, and the accuracy of these reconstructions has been a persistent challenge. This work unifies the metrics for assessing the accuracy of state, detector, and process tomography within a single framework, establishing a clear benchmark for optimal performance. The team discovered a fundamental principle governing the scaling of accuracy with the number of measurements, demonstrating that optimal tomography protocols should achieve an error that decreases proportionally to one over the number of state copies consumed, a rate significantly faster than previously attainable with standard methods.

This optimal scaling is achieved by accurately estimating both the entirety of the quantum system being measured and its zero eigenvalues, previously overlooked in many approaches. Researchers formally proved that satisfying both conditions is both necessary and sufficient for achieving this optimal scaling, providing a substantial advance beyond existing tomography methodologies. To realize this theoretical improvement, the scientists developed adaptive algorithms for state, detector, and process tomography, employing a two-step approach that first performs a preliminary estimation of the quantum system, then refines the measurement process by focusing on the eigenbasis of the initial estimate. Experiments validate these methods, demonstrating, for the first time, the achievement of optimal infidelity scaling in ancilla-assisted process tomography.

Optimal Scaling in Quantum Tomography Algorithms

This research unifies the analysis of errors in several quantum characterisation techniques, state, detector, and process tomography, within a single framework. The team establishes a clear condition determining the optimal rate at which accuracy improves as more measurements are taken, specifically demonstrating a scaling of one over the number of samples. This contrasts with less efficient methods that scale at a slower rate. Guided by this theoretical understanding, the researchers developed adaptive algorithms for each tomography type, achieving this optimal scaling in practice. Experimental validation, including the first demonstration of optimal scaling in adaptive process tomography, confirms the effectiveness of their approach. These algorithms require fewer measurements than existing methods while still achieving improved accuracy. The authors acknowledge that their adaptive algorithms rely on an initial estimation, which can temporarily reduce performance when very few measurements are available, and future work could explore the use of mixed input states to further optimise the algorithms.