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 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 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 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.
Abstract To progress in the characterization of noise for quantum computers, gate set tomography (GST) has emerged as a self-consistent protocol that accurately estimates noisy gates, state preparations, and measurements. In its original incarnation, GST improves estimation precision by applying gates sequentially, assuming the noise yields fixed completely-positive and trace-preserving (CPTP) maps independent of prior gate history. This ‘Markovian’ assumption can conflict with experiments, where time-correlated noise may induce non-Markovian dynamics, or slow drifts and cumulative calibration errors introduce context dependence, causing CP-divisible maps to vary with circuit depth. In this work, we address this issue for trapped-ion devices with phonon-mediated two-qubit gates. Using detailed microscopic modeling of high-fidelity light-shift gates, we tailor GST to capture the main source of context dependence: motional degrees of freedom. Rather than invalidating GST, context dependence can be incorporated into the gate-set parametrization, reducing sampling cost. Our results identify a promising research avenue that might be applicable to other platforms where microscopic modeling can be incorporated: the development of a context-aware GST.
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.
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.
Abstract Gate set tomography (GST) allows for a self-consistent characterization of noisy quantum information processors (QIPs). The standard device-agnostic approach treats the QIPs as black boxes that are only constrained by the laws of physics, attaining full generality at aconsiderable resource cost: numerous circuits built from the gate set must be run in order to amplify each of the gate set parameters. In this work, we show that a microscopic parametrization of quantum gates under time-correlated noise on the driving phase, motivated by recent experiments with trapped-ion gates, reduces the required resources enabling a more efficient version of GST. By making use of the formalism of filter functions over the noise spectral densities, we discuss the minimal parametrizations of the gate set that include the effect of finite correlation times and non-Markovian quantum evolutions during the individual gates. We compare the estimated gate sets obtained by our method and the standard long-sequence GST, discussing their accuracies in terms of established metrics, as well as showcasing the advantages of the parametrized approach in terms of the sampling complexity for specific examples.