Gate set tomography (GST) is a self-consistent and highly accurate method for the tomographic reconstruction of a quantum information processor's quantum logic operations, including gates, state preparations, and measurements. However, GST's experimental cost grows exponentially with qubit number. For characterizing even just two qubits, a standard GST experiment may have tens of thousands of circuits, making it prohibitively expensive for platforms. We show that, because GST experiments are massively overcomplete, many circuits can be discarded. This dramatically reduces GST's experimental cost while still maintaining GST's Heisenberg-like scaling in accuracy. We show how to exploit the structure of GST circuits to determine which ones are superfluous. We confirm the efficacy of the resulting experiment designs both through numerical simulations and via the Fisher information for said designs. We also explore the impact of these techniques on the prospects of three-qubit GST.
Gate set tomography (GST) is a self-consistent and highly accurate method for the tomographic reconstruction of a quantum information processor's quantum logic operations, including gates, state preparations, and measurements. However, GST's experimental cost grows exponentially with qubit number. For characterizing even just two qubits, a standard GST experiment may have tens of thousands of circuits, making it prohibitively expensive for platforms. We show that, because GST experiments are massively overcomplete, many circuits can be discarded. This dramatically reduces GST's experimental cost while still maintaining GST's Heisenberg-like scaling in accuracy. We show how to exploit the structure of GST circuits to determine which ones are superfluous. We confirm the efficacy of the resulting experiment designs both through numerical simulations and via the Fisher information for said designs. We also explore the impact of these techniques on the prospects of three-qubit GST.
Gate set tomography (GST) is a technique for self-consistently and precisely estimating the entire gate set of a quantum processor. GST carries nontrivial experimental cost, and as such the use of GST in practice has so far been limited to characterizing subsets of systems of at most two qubits at anyone time. In this work we present a new protocol for streamlining GST experiment designs by stripping away nearly all of the redundancy present in a traditional GST experiment. We do so by determining how GST base circuits (called “germs”) are sensitive to variations in particular gate set parameters, and leverage this information to identify and subsequently strip out redundancy across a germ set as a whole. We present the application of this new protocol to GST experiment designs for two-qubit systems and show that we can reliably produce experiment designs approaching information theoretic lower bounds in size. We demonstrate in both simulation and in a theoretical analysis based on the Fisher information that these streamlined experiment designs nonetheless maintain the Heisenberg-like precision scaling properties of traditional GST. Moreover, we show that in practice it is possible using this technique to achieve a final precision in our estimates comparable to those of much larger experiment designs using many times fewer circuits overall. We also discuss the impacts of this technique for the prospects of scaling up the practical use of GST for three-qubit systems.
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.
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.
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.
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.
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.