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arXiv:2605.11158

Scalable linearized gate set tomography

Ashe Miller, Corey Ostrove, Jordan Hines, Noah Siekierski, Kevin Young, Robin Blume-Kohout, Timothy Proctor

quant-phSubmitted May 11, 2026• Announced May 13, 2026• 0 citations
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Abstract

Characterizing errors on many-qubit quantum computers remains a key challenge to understanding and improving the performance of these devices. Current characterization methods either don't scale beyond a few qubits, or make simplifying assumptions (such as assuming stochastic Pauli error) that obscure the underlying physical error mechanisms. In this work, we present a scalable extension to gate set tomography-linearized gate set tomography-that enables characterization of many-qubit systems. Linearized gate set tomography relies on sparse error models, a linear approximation to enable efficient data fitting, and data from shallow circuits-so that the systematic error in the linear approximation is small. We demonstrate the accuracy of our technique using simulations of a ten-qubit system with coherent and stochastic errors, including coherent crosstalk, and we demonstrate that it is robust in presence of additional errors that are not included within the sparse error model ansatz.

Comments
10 pages, 6 figures

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Characterizing errors on many-qubit quantum computers remains a key challenge to understanding and improving the performance of these devices. Current characterization methods either don't scale beyond a few qubits, or make simplifying assumptions (such as assuming stochastic Pauli error) that obscure the underlying physical error mechanisms. In this work, we present a scalable extension to gate set tomography-linearized gate set tomography-that enables characterization of many-qubit systems. Linearized gate set tomography relies on sparse error models, a linear approximation to enable efficient data fitting, and data from shallow circuits-so that the systematic error in the linear approximation is small. We demonstrate the accuracy of our technique using simulations of a ten-qubit system with coherent and stochastic errors, including coherent crosstalk, and we demonstrate that it is robust in presence of additional errors that are not included within the sparse error model ansatz.
Flexible characterization techniques that provide a detailed picture of the experimental imperfections under realistic assumptions are crucial to gain actionable advice in the development of quantum computers.Gate set tomography self-consistently extracts a complete tomographic description of the implementation of an entire set of quantum gates, as well as the initial state and measurement, from experimental data.It has become a standard tool for this task but comes with high requirements on the number of sequences and their design, making it already experimentally challenging for only two qubits.In this work, we show that low-rank approximations of gate sets can be obtained from significantly fewer gate sequences and that it is sufficient to draw them at random.This coherent noise characterization however still contains the crucial information for improving the implementation.To this end, we formulate the data processing problem of gate set tomography as a rank-constrained tensor completion problem.We provide an algorithm to solve this problem while respecting the usual positivity and normalization constraints of quantum mechanics.For this purpose, we combine methods from Riemannian optimization and machine learning and develop a saddle-free second-order geometrical optimization method on the complex Stiefel manifold.Besides the reduction in sequences, we numerically demonstrate that the algorithm does not rely on structured gate sets or an elaborate circuit design to robustly perform gate set tomography.Therefore, it is more flexible than traditional approaches.We also demonstrate how coherent errors in shadow estimation protocols can be mitigated using estimates from gate set tomography.
The efficiency of Quantum Characterisation, Verification, and Validation (QCVV) protocols highly hinges on the agreement between the assumed noise model and the underlying error mechanisms. As a matter of fact, errors in Quantum Processing Units (QPUs) incorporate various aspects of context-dependability which are overlooked by the majority of the commonly used QCVV protocols. As QCVV protocols are indispensable when it comes to characterizing and evaluating quantum operations, there is a serious need for a detailed characterization taking into account such aspects. In this work, we address these shortcomings by designing a context-aware version of the gate set tomography (GST) protocol. Our experiment selection approach is based on a polynomial quantification of the accumulation of errors within the designed circuits. Using simulated QPUs, we show that this technique enables a characterization with an inaccuracy reaching 10−510^{-5}10−5. Furthermore, we use our proposed protocol to experimentally infer context-dependent errors, namely crosstalk and memory effects, in a publicly accessible cloud-based superconducting qubits platform. Our results show that when the GST is upgraded to include such features of context-awareness, a large coherence in the errors is observed. These findings open up possibilities of drastically reducing the errors within the currently demonstrated QPUs.
Crosstalk is a leading source of failure in multiqubit quantum information processors.It can arise from a wide range of disparate physical phenomena, and can introduce subtle correlations in the errors experienced by a device.Several hardware characterization protocols are able to detect the presence of crosstalk, but few provide sufficient information to distinguish various crosstalk errors from one another.In this article we describe how gate set tomography, a protocol for detailed characterization of quantum operations, can be used to identify and characterize crosstalk errors in quantum information processors.We demonstrate our methods on a two-qubit trapped-ion processor and a two-qubit subsystem of a superconducting transmon processor.
Quantum error correction (QEC) can strongly suppress noise, but predicting whether a computation succeeds requires understanding not only how much effective noise persists after QEC, but what form that noise takes. To thoroughly characterize this noise, we introduce logical gate set tomography (``logical GST''). Logical GST produces detailed representations of noisy logical operations that can be used to derive higher-level metrics, such as gate fidelity and diamond distance, and to test for non-Markovian and coherent logical errors. We use logical GST to obtain the first full, self-consistent characterization of a universal single-qubit logical gate set, implemented in the seven-qubit Steane code using Quantinuum System Model H1, a 20-qubit trapped-ion quantum processor. In the experimental data from H1, logical GST identified predominantly logical Pauli-stochastic errors, found no evidence of logical non-Markovianity, and detected weak potential signatures of coherent logical error. By contrast, higher-shot datasets generated with Quantinuum's detailed system emulator, identify mechanisms for weak but statistically significant logical non-Markovianity and strong evidence of coherent logical error. Finally, we use logical GST to perform error attribution with the system emulator, revealing how specific physical error sources shape logical error mechanisms.
Gate set tomography (GST) characterizes the process matrix of quantum logic gates, along with measurement and state preparation errors in quantum processors. GST typically requires extensive data collection and significant computational resources for model estimation. We propose a more efficient GST approach for qudits, utilizing the qudit Hadamard and virtual Z gates to construct fiducials while assuming virtual Z gates are error-free. Our method reduces the computational costs of estimating characterization results, making GST more practical at scale. We experimentally demonstrate the applicability of this approach on a superconducting transmon qutrit.
Quantum gate set tomography (GST) has emerged as a promising method for the full characterization of quantum logic gates. In contrast to quantum process tomography (QPT), GST self-consistently and correctly accounts for state preparation and measurement (SPAM) errors. It therefore provides significantly more accurate estimates than QPT as gate fidelities increase into the fault-tolerant regime. We give a detailed review of GST and provide a self-contained guide to its implementation. The method is presented in a step-by-step fashion and relevant mathematical background material is included. Our goal is to demonstrate the utility of GST as both an accurate characterization technique and a simple and effective diagnostic tool. As an illustration, we compare the output of GST and QPT using simulated example data for a single qubit. In agreement with the original literature, we find that coherent errors are poorly estimated by QPT near quantum error correction thresholds, while GST is accurate in this regime.