Resonant Coupling Parameter Estimation with Superconducting Qubits

Today’s quantum computers are composed of tens of qubits interacting with each other and the environment in increasingly complex networks. To achieve the best possible performance when operating such systems, it is necessary to have accurate knowledge of all parameters in the quantum computer Hamiltonian. In this paper, we demonstrate theoretically and experimentally a method to efficiently learn the parameters of resonant interactions for quantum computers consisting of frequency-tunable superconducting qubits. Such interactions include, for example, those with other qubits, resonators, two-level systems, or other wanted or unwanted modes. Our method is based on a significantly improved swap spectroscopy calibration and consists of an offline data collection algorithm, followed by an online Bayesian learning algorithm. The purpose of the offline algorithm is to detect and coarsely estimate resonant interactions from a state of zero knowledge. It produces a quadratic speedup in the scaling of the number of measurements. The online algorithm subsequently refines the estimate of the parameters to accuracy comparable with that of traditional swap spectroscopy calibration but in constant time. We perform an experiment implementing our technique with a superconducting qubit. By combining both algorithms, we observe a reduction of the calibration time by 1 order of magnitude. Our method will improve present medium-scale superconducting quantum computers and will also scale up to larger systems. Finally, the two algorithms presented here can be readily adopted by communities working on different physical implementations of quantum computing architectures.

Popular Summary

Large quantum computers, those with many physical quantum bits (qubits), require extensive calibration before they can execute algorithms. In addition, because the physical parameters of the quantum computer change in time, the calibration must be repeated regularly. This ensures that the qubits will always perform optimally. It is therefore necessary to execute this task as quickly and accurately as possible. We explain and experimentally demonstrate two algorithms that give us the information needed to optimize two-qubit gates and avoid errors caused by defects in the quantum device. Together, these algorithms reduce the time needed by 1 order of magnitude. The algorithms are tested on a superconducting qubit, with which we characterize performance and robustness.

The goal of the algorithms is to detect the interaction between qubits and other systems. These other systems can be other qubits, quantum harmonic oscillators (which are used for gate and measurements), or unwanted physical defects called “two-level systems.” Detecting and estimating the parameters of these interactions, the resonance frequency and the coupling strength, let us tune each qubit. For example, knowing the parameters of the interaction between the two qubits is essential to two-qubit gates. Moreover, knowing if there are two-level systems lets us avoid them, reducing errors. The algorithms are based on the dynamics of resonant interactions and reduce the number of measurements needed by a square-root factor.

We believe that integrating these algorithms in the regular calibration of quantum processors will help improve performance. In particular, the information they provide can greatly benefit the problem of tuning the operating frequency of qubits.

Figure 1

Pulse sequence for a swap spectroscopy experiment. The initial $\pi$-pulse (red) excites the qubit, which is initialized at the so-called idle frequency. The flux pulse (blue) changes the qubit transition frequency $f_q$ to the probe frequency $f_p(A)$ for duration $t$. The qubit is then measured (green) after being set back to its idle frequency.

Figure 2

Swap spectra for two frequency ranges. The $x$ axis shows the probe frequency of the qubit, which is set by the amplitude of the flux pulse applied to the SQUID. The $y$ axis indicates the length of the flux pulse before measurement and, therefore, corresponds to the interaction time with potential resonance modes. (a) Distinct features are visible in the full spectrum, including two chevron patterns around 4.8 GHz and one at 5.1 GHz. The resonance at 5.1 GHz looks aliased because of the low-resolution sampling of the time axis. A low-$T_1$ streak is also visible at 4.35 GHz, likely caused by an incoherent TLS. (b) Enlargement of the region with the slower coherent chevron patterns caused by synthesized resonance modes. For this experiment, the synthesizers are set at 4.8100 and 4.8314 GHz. A wide frequency scan is needed to see if and where there are resonance modes, as in (a), but a more detailed experiment, as in (b), is needed to properly estimate the resonance parameters.

Figure 3

Offline octave sampling. (a) Contour plot of the probability of finding the qubit in $|e⟩$ [see Eq. (2)] as a function of time $t$ and frequency detuning $\delta f$; both axes are normalized by the coupling strength $g$. The highlighted box in the center indicates which portion of the chevron pattern is meant to be detected by the algorithm. (b) Swap spectroscopy experiment with octave sampling. Starting from a bin spanning the full measurement range, at each subsequent octave the bin width is halved and the bin length is doubled. The color of each bin represents the average value of the measured $P_e$ over $n_s=5$ samples. The red boxes indicate the resonances reported by the analysis explained in Sec. 4b.

Figure 4

Illustration of a simulated iteration of the online particle filter algorithm with 40 000 particles. (a) Given a prior distribution, we heuristically generate measurement settings $f_p$ and $t$ meant to increase information. (b) Following the measurement, the likelihood $\mathcal{L}$ of the result is computed for each particle. The values shown on the scale are normalized. (c) We apply Bayes’ theorem to determine the posterior distribution. This task is achieved by resampling the particles according to their likelihoods. The distribution is split into two “clouds.” After resampling, the posterior distribution can be used as the next iteration’s prior.

Figure 5

Performance of the online estimation algorithm over $1000$ runs with RM2. (a),(b) Histograms over the means of the initial and final particle distributions. (c),(d) Mean of the posterior particle distribution computed after each iteration’s measurement. As more iterations are made, the particle distribution converges toward the true value of the parameters. The zeroth iteration corresponds to the initial distribution. The true values of the parameters are identified by a red line in the histograms. The total runtime of the experiment is 6.4 h. Each individual run executes $35$ iterations and thus $35$ measurements, taking approximately $23\mathrm{s}$. Most of the time (60%) is spent acquiring data. Each measurement comprises $786$ shots at a repetition rate of 2 kHz. The remaining time is attributable to data transfer and processing. The total computation time for the estimation algorithm is approximately $1\mathrm{s}$.

Figure 6

Images of a sample identically fabricated to the one used in this work. (a) Optical image of the Xmon transmon qubit, with the drive line on the left, the measurement resonator coupler above, and the flux bias line below. The cross-shaped island constitutes a capacitor to ground, while the SQUID acts as a nonlinear inductor. (b) Scanning electron microscope image of the SQUID, which is located at the end of the bottom arm of the cross. (c) Scanning electron microscope image of a Josephson junction made with a Dolan bridge.

Figure 7

Dilution refrigerator wiring schematics. The sample is mounted in an aluminum package at the mixing chamber stage of a dilution refrigerator, at a temperature of approximately 10 mK. We use a dedicated microwave line to drive the qubit. Two coaxial lines are joined by a custom Eccosorb tee at the mixing chamber for the flux biasing of the SQUID. We use a homodyne readout setup, with the in-phase quadrature mixers configured for image rejection. ADC, analog-to-digital converter; CNT, carbon nanotube; DAC, digital-to-analog converter; CP, cold plate; MC, mixing chamber.

Figure 8

Peaks found in the highest octave in Fig. 3 with the prominence value set to 0.09. Many peaks are found for RM3. They are merged when the detection analysis proceeds to lower octaves.

Figure 9

Initial particle distribution generated from the octave data in Fig. 3 after splitting of the spectrum according to the detections. The distribution corresponding to RM1 is plotted in blue, the distribution corresponding RM2 is plotted in orange, and the distribution corresponding to RM3 is plotted in green.

Figure 10

Particle resampling.