Gate Set Tomography (2009.07301v2)

Abstract: Gate set tomography (GST) is a protocol for detailed, predictive characterization of logic operations (gates) on quantum computing processors. Early versions of GST emerged around 2012-13, and since then it has been refined, demonstrated, and used in a large number of experiments. This paper presents the foundations of GST in comprehensive detail. The most important feature of GST, compared to older state and process tomography protocols, is that it is calibration-free. GST does not rely on pre-calibrated state preparations and measurements. Instead, it characterizes all the operations in a gate set simultaneously and self-consistently, relative to each other. Long sequence GST can estimate gates with very high precision and efficiency, achieving Heisenberg scaling in regimes of practical interest. In this paper, we cover GST's intellectual history, the techniques and experiments used to achieve its intended purpose, data analysis, gauge freedom and fixing, error bars, and the interpretation of gauge-fixed estimates of gate sets. Our focus is fundamental mathematical aspects of GST, rather than implementation details, but we touch on some of the foundational algorithmic tricks used in the pyGSTi implementation.

Overview of Gate Set Tomography

The paper "Gate Set Tomography" presents a comprehensive exploration of Gate Set Tomography (GST), a protocol designed for precise characterization of quantum logic gates on quantum computing processors. This method has been refined since its inception around 2012-2013 and has shown robust applicability across numerous experiments, distinguishing itself from traditional state and process tomography by being calibration-free. GST characterizes all operations in a gate set simultaneously, providing a self-consistent description relative to each other rather than relying on pre-calibrated states and measurements. This feature significantly enhances its reliability in practical applications.

One of the notable achievements of GST is its ability to estimate gates with high precision and efficiency, achieving Heisenberg scaling in some practical regimes, which demonstrates a substantial improvement over traditional methods. The paper explores the intellectual development of GST, outlining the techniques and experiments that facilitate its objectives, alongside discussions on data analysis, gauge freedom, error bars, and the interpretations of gauge-fixed estimates.

Fundamental Aspects of GST

GST is presented as a self-calibrating tomography protocol, exposing the limitations inherent to state and process tomography methodologies that assume flawless reference operations. This is consequential because real-world quantum devices often suffer from inaccuracies in state preparation and measurement that traditional tomography assumes to be ideal, leading to unreliable characterizations. GST circumvents this limitation by constructing a gate set—a holistic model encompassing state preparations, gates, and measurements—and performing tomography across this set without needing a calibrated reference frame. This principle of self-consistency within GST ensures a more accurate representation of the quantum hardware's capabilities.

The introduction of gauge freedom in GST allows for transformations that do not alter observable probabilities, thereby revealing relational properties of gates rather than intrinsic attributes applicable to isolated operations. The mathematical framework introduced for GST supports this relational characterization and highlights the importance of gauge freedom in precise quantum production environments.

Long-Sequence GST and Precision

Building upon the linear gate set tomography (LGST), which resolves pre-calibration constraints, the paper describes long-sequence GST that achieves higher precision estimations. The authors demonstrate that long-sequence GST amplifies gate errors through deep, structured circuits, allowing for a scaling of estimation precision that surpasses traditional approaches. The role of "germs," repetitive sequences used to amplify specific gate errors, is outlined, offering insights into advanced error characterization. This leads to the discussion on the flexibility in incorporating various constraints like complete positivity and trace-preservation within GST's framework, thereby acknowledging GST's adaptable nature to different quantum computing scenarios.

Implications and Future Directions

The implications of GST are multifold: practically, it provides a reliable tool for debugging and modeling quantum devices, and theoretically, it extends the characterization capabilities beyond static assumptions of traditional tomography. This offers significant promise for future quantum computing developments, where understanding and correcting errors in quantum gate implementations becomes essential for scaling up quantum processors and achieving fault tolerance. Additionally, the potential extensions of GST to multi-qubit systems signal an exciting avenue for research, emphasizing the need for scalable and robust characterization protocols in quantum technology.

The paper lays foundational work for future exploration into gauge-invariant metrics, which could offer more meaningful assessments of quantum devices' performance without relying on potentially misleading gauge-variant metrics. Overall, Gate Set Tomography emerges as a pivotal protocol in quantum technology, embodying a shift towards more accurate and reliable operational analysis of quantum computing hardware.