Integrated Tool Set for Control, Calibration, and Characterization of Quantum Devices Applied to Superconducting Qubits

Efforts to scale-up quantum computation have reached a point where the principal limiting factor is not the number of qubits, but the entangling gate infidelity. However, the highly detailed system characterization required to understand the underlying error sources is an arduous process and impractical with increasing chip size. Open-loop optimal control techniques allow for the improvement of gates but are limited by the models they are based on. To rectify the situation, we provide an integrated open-source tool set for control, calibration, and characterization (${\mathbb{C}}^3$), capable of open-loop pulse optimization, model-free calibration, model fitting, and refinement. We present a methodology to combine these tools to find a quantitatively accurate system model, high-fidelity gates, and an approximate error budget, all based on a high-performance, feature-rich simulator. We illustrate our methods using simulated fixed-frequency superconducting qubits for which we learn model parameters with less than $1\mathrm{\%}$ error and derive a coherence-limited cross-resonance gate that achieves $99.6\mathrm{\%}$ fidelity without the need for calibration.

Figure 1

Diagram of the ${\mathbb{C}}^3$ tool set in an integrated characterization loop. Here $C_1$ is a tool for obtaining optimal pulses by finding the control parameters $\alpha$ that minimize a goal function $f_1(\alpha )$ in simulation. The gate set $\mathcal{G}$ includes all the operations that one wishes to perform on the experiment, including the information of the ideal logical operations and the optimal pulse parameters $\alpha$ that implement them. By $C_2$ we denote a model-free experimental calibration procedure that optimizes pulse shapes with a gradient-free search to minimize an infidelity function $f_2(\alpha )$ by varying all parameters at once. A data set is a collection of experiment-result pairs, including information about the pulse parameters used, $\alpha$, the sequences $S_k$ performed, and the final outcomes measured, $m_k$. By $C_3$ we denote a tool for model learning that determines the model parameters $\beta$ that best explain the data set. It minimizes a goal function $f_3(\beta |\mathcal{D})$ obtained by recreating experiments $S_k(\alpha )$ in simulation and comparing the results to those in the experiment. In ${\mathbb{C}}^3$ different parameterized models can be provided to represent various elements of the experiment to find the one that best describes it. After the learning, the resulting model can be the basis for another characterization loop, refining both the model and controls.

Figure 2

The $C_2$ calibration on the device for single-qubit gates of qubit $B$. The initial point is suggested by $C_1$ before (after) learning of the model. The light blue diamonds (light red circles) represent the values of the ORBIT goal function, Eq. (9), for varying pulse parameters $\alpha$ as chosen by the search algorithm. The larger blue diamonds (larger red circles) highlight the best of $25$ points generated and sampled at each iteration. In experimental practice, this batching helps reduce the overhead of loading pulses in AWG programming [32]. Both calibrations achieve the same final fidelity; however, the optimal gates derived from the learned model provide a better initial guess. Assuming no SPAM errors, the ORBIT value can be translated into an error per gate, indicated on the right axis. This is only meant to provide a rough estimate of the performance of the gate, noting that an ORBIT value of 0.5 represents maximum error per gate, i.e., completely depolarizing channels.

Figure 3

Progress of the $C_3$ optimization on a hierarchy of models: simple model (green, dashed), intermediate model (blue, dot-dash), and full model (red, solid), as described in the text. The model match goal function $f_{\mathrm{LL}}(\beta )$ is defined in Eq. (11). The crosses show the switch over from the CMA-ES algorithm to the L-BFGS algorithm. The CMA-ES algorithm evaluates a batch of points for each iteration (8, 9, and 12 for the simple, intermediate, and full model, respectively); only the best of each batch is shown. The L-BFGS algorithm takes on average approximately 1.2 evaluations per iteration, for all three models. The function $f_{\mathrm{LL}}$ is rescaled to express the match in terms of standard deviations of the binomial distribution that the experimental results are drawn from. The simple model is a close dispersive approximation of the intermediate model, demonstrated by their similar final match score. By including all relevant device properties, the full model reaches an almost perfect match score.

Figure 4

Sensitivity analysis of selected parameters for different models and data sets. In (a) we sweep the qubit frequency ${\omega }_B$ and evaluate the goal function $f_{\mathrm{LL}}$ on the learning data at each value. The star represents the optimal value returned by $C_3$. Intermediate (blue, dot-dash) and full (red, solid) models show the same frequency value, while the value of the simple model (green, dashed) is dispersively shifted, as expected. In (b),(c) we perform the same sweep for the chip temperature and $T_2^{\ast }$ of qubit $B$, respectively, evaluated on the full model for different learning data sets. The ORBIT data are the same as used in the full model in (a). Introducing the quantum process tomography (QPT) data (purple, dotted) allows a more precise definition of the temperature. To determine $T_2^{\ast }$, we use relaxation and dephasing data (orange, dot-dash). Match values below 0 can occur because of noisy data, finite sampling, and deviation from the assumption of a Gaussian distribution of the data. More sensitivity plots are shown in the Supplemental Material [35].

Figure 5

The $C_3$ learning of the two-qubit model parameters. Blue, left (red, right) triangles indicate qubit $A$ ($B$) parameters, while shared properties are shown with green upward triangles. The true values of the “real” model are indicated as dashed lines. Learning begins using just ORBIT data (left white section) that fixes qubit frequencies, anharmonicities, coupling, and the line transfer function to their true values. Then tomography data from a two-qubit experiment are added (right gray section), which allows better identification of the chip temperature $T$ and the misclassification constants $p_{0\to 1}$ and $p_{1\to 0}$.

Figure 6

(a)–(c) Correlation between simulation and experiment expressed as a density of points $({\tilde{m}}_k,m_k)$ for the simple, intermediate, and full models, respectively. The data set is the so-called validation set: the data points $k$ that are not used in optimization of the model parameters. The simple and intermediate models show poor correlation as the simulation predicts a wide distribution of measurement outcomes for each recorded value. They also exhibit a tilt that can be attributed to SPAM errors not considered in the models. Only the full model produces a consistently high density distribution centered around the diagonal, with minimal spread due to the noisy data.

Figure 7

The $C_3$ learning of the relaxation ($T_1$) and dephasing ($T_2^{\ast }$) parameters. Blue, left (red, right) triangles indicate qubit $A$ ($B$) parameters. The true values of the “real” model are indicated as dashed lines. Background sections represent different learning data sets: just ORBIT data (left white section), a mix of ORBIT and QPT data (center gray section), decay and Ramsey data (right white section). The decay times are correctly identified only when specific data, sensitive to the decoherence effects, are used for learning, at which point they quickly converge to the real values.

Figure 8

The process of simulating experimental procedure for signal processing and readout. The $k\mathrm{th}$ control function is specified by some function ${\epsilon }_k(t)$ and specifies the line voltage $u_k(t)$ by an AWG with limited bandwidth. Electrical properties of the setup, such as impedances, are expressed as a line transfer function $\phi$, resulting in a control field $c_k(t)=\phi [u_k(t)]$, as in Eq. (19). After solving the equation of motion for the system, readout and misclassification are modeled by applying rescaling and transformations to the simulated populations $p_i=|{\rho }_{ii}{|}^2$, according to Eq. (24).

Figure 9

A 2D cut through the landscape of a calibration goal function. The system chosen is a multilevel qubit, with the control sequence being a single length RB sequence. Control parameters ${\alpha }_j$ are the Gaussian pulse amplitude, DRAG coefficient, and frequency offset of a DRAG-corrected pulse. The RB sequence is chosen to implement the identity operation, ideally leaving the qubit in the ground state. The infidelity (seen as the color ranging from 0 to 1) is defined as the overlap of the final state with the system’s ground state. (a) The landscape of the simulated system, as a cut through the plane of pulse amplitude $A$ and DRAG coefficient $\eta$, for a fixed RB sequence. Multiple local minima can be observed. (b) A higher-resolution plot of the same landscape. Further local minima can be observed in the neighborhood of the global minimum.