[Paper Review] Robust, self-consistent, closed-form tomography of quantum logic gates on a trapped ion qubit

Robin Blume-Kohout, John King Gamble+4 more|arXiv (Cornell University)|Oct 16, 2013

Advanced Electron Microscopy Techniques and ApplicationsBiochemistry, Genetics and Molecular Biology2 references69 citations

TL;DR

This paper introduces gate set tomography (GST), a self-consistent framework for robustly characterizing quantum logic gates on trapped ions without relying on precalibrated reference frames. It presents linear gate set tomography (LGST), a closed-form, non-iterative method that avoids likelihood maxima pitfalls, and a predictive scoring protocol to objectively evaluate estimates—demonstrating high accuracy in experimental characterization of Clifford-generating gates.

ABSTRACT

We introduce and demonstrate experimentally: (1) a framework called "gate set tomography" (GST) for self-consistently characterizing an entire set of quantum logic gates on a black-box quantum device; (2) an explicit closed-form protocol for linear-inversion gate set tomography (LGST), whose reliability is independent of pathologies such as local maxima of the likelihood; and (3) a simple protocol for objectively scoring the accuracy of a tomographic estimate without reference to target gates, based on how well it predicts a set of testing experiments. We use gate set tomography to characterize a set of Clifford-generating gates on a single trapped-ion qubit, and compare the performance of (i) standard process tomography; (ii) linear gate set tomography; and (iii) maximum likelihood gate set tomography.

Motivation & Objective

• To address the critical problem of self-referential errors in standard quantum process tomography, which rely on precalibrated states and measurements that are themselves subject to gate-induced errors.
• To develop a complete, self-consistent framework for characterizing an entire set of quantum gates without assuming known reference frames.
• To introduce a closed-form, robust estimation protocol (LGST) that avoids local maxima in likelihood functions, ensuring reliable initial estimates for further refinement.
• To propose a novel, objective scoring method that evaluates tomographic estimates based on their predictive power on independent test experiments, rather than comparison to target gates.
• To demonstrate the feasibility and superiority of GST in practice using experimental data from a trapped-ion qubit system.

Proposed method

• Gate set tomography treats the quantum device as a black box with only classical control and classical measurement outcomes, avoiding assumptions about state preparations and measurement effects.
• The framework uses sequences of gates (including preparation and measurement gates) to generate a large number of observable probabilities, enabling self-consistent estimation of all gates, states, and measurements simultaneously.
• Linear gate set tomography (LGST) is a closed-form, linear-inversion method that provides a reliable initial estimate by minimizing a weighted least-squares objective, avoiding convergence to local maxima in likelihood functions.
• The method leverages the fact that gates can be applied multiple times in sequences, generating exponentially many observable probabilities from only a few distinct gates, thus enabling over-constrained estimation.
• A predictive scoring protocol evaluates how well a tomographic estimate predicts independent test experiments by computing the average log-likelihood of observed counts, providing an objective, reference-free metric.
• Maximum likelihood estimation is then applied starting from the LGST estimate to achieve high-accuracy results, combining robustness and precision.

Experimental results

Research questions

• RQ1 Can a self-consistent quantum gate characterization framework be developed that does not rely on precalibrated reference frames for state preparations and measurements?
• RQ2 How can a closed-form, non-iterative estimation method be designed to avoid local maxima in the likelihood function during gate set tomography?
• RQ3 What objective, reference-free metric can be used to evaluate the predictive power of a tomographic estimate without assuming a target gate?
• RQ4 How does the performance of LGST compare to standard process tomography and maximum likelihood estimation in terms of predictive accuracy and robustness?
• RQ5 Can gate set tomography be experimentally demonstrated on a trapped-ion qubit system with high-fidelity gate operations?

Key findings

• The LGST protocol successfully avoids local maxima in the likelihood function due to its closed-form, linear-inversion nature, providing a reliable initial estimate for further refinement.
• The predictive scoring protocol demonstrated that the best maximum likelihood estimate achieved a score-per-count of approximately 0.02 at sequence length L=100, indicating strong predictive power.
• LGST estimates were highly effective as seeds for maximum likelihood estimation, with predictive performance degrading only beyond sequence length L=3, while maximum likelihood estimates remained accurate for longer sequences.
• The experimental results showed that most predictive failure in standard tomography was due to SPAM (state preparation and measurement) errors rather than gate errors, highlighting the importance of self-consistent characterization.
• Gate set tomography outperformed standard process tomography and linear gate set tomography in predictive accuracy, especially when trained on longer sequences.
• The study demonstrates that GST is practically feasible and provides a robust, comprehensive solution to the self-referential problem in quantum gate characterization.