Implementing a strand of a scalable fault-tolerant quantum computing fabric

Methods Summary

The device is fabricated on a 720 $\mu$m thick silicon substrate. All superconducting coplanar waveguide resonators are defined via optical lithography and subtractive reactive ion etching of a sputtered niobium film (200 nm thick). The three single-junction transmon qubits are patterned using electron-beam lithography, followed by double-angle deposition of aluminum, with layer thicknesses of 35 nm and 85 nm. Liftoff process is used to form the final junction structure.

Device parameters

The three transmon qubits ($i\in$) have transition frequencies {$\omega_{i}$}$/2\pi$ = {5.0388, 5.0080, 5.2286} GHz, with readout resonators at {$\omega_{\text{R}i}$}$/2\pi$ = {6.698, 6.585, 6.695} GHz, relaxation times {$T_{1(i)}$} = {24, 29, 20} $\mu$s, {$T_{2(i)}^{\text{echo}}$} = {32, 25, 18} $\mu$s. The bus resonators are un-measured but $R_{12}$ ($R_{23}$) is designed to resonate at 8 (8.5) GHz. The dispersive cavity shifts of the readout resonators are measured to be {$\chi_{i}$}/$\pi$ = {-2.0, -2.0, -2.3} MHz and the readout resonators have line-widths {$\kappa_{\text{R}i}$}$/2\pi$ = {443, 976, 793} kHz. All qubits have measured anharmonicities of $-340$ MHz. From the above we calculate coupling strengths {$|g_{i}|$}$/2\pi$ = {70, 67, 67} MHz to the readout resonators, which is consistent with electromagnetic simulations.

Methods

The half-plaquette device is cooled to 15 mK in an Oxford Triton dilution refrigerator. A full schematic of the wiring and experimental control hardware is depicted in Extended Data Fig. 1. Each qubit has its own dedicated readout line with an associated set of isolators and Caltech HEMT (noise temp $\sim$6K) amplifiers. Q2 is unique in that its readout signal is reflected off of a UC Berkeley JPA before going onto the isolator and HEMT chain. The device is housed in a light-tight Ammuneal cryoperm-shield which is coated throughout with a layer of lossy eccosorb (Emerson & Cuming CR-124). Besides explicit cryogenic attenuators at the different stages of the cryostat, all qubits are also attenuated at the lowest temperature stage with in-house eccosorb coaxial filters.

Outside the cryostat, all microwave qubit control signals are generated via vector modulation combining off-the-shelf electronics. The microwave readout signals are pulse modulated using Arbitrary Pulse Sequencers built by Raytheon BBN Technologies. The readout signals are processed via two Alazartech ATS9870. All single-shot readout traces are processed with an optimal quadrature rotation filter, described in parallel work 28.

Calibration sequences

Complete tune-up of all microwave gates is accomplished using sets of automated repeated sequences. For single-qubit gates, the repeated calibration sequences are described in a previous publication 16.

The cross-resonance pulse amplitude is calibrated in close analogy to single qubit amplitude calibrations. An odd number $2N-1$ of $ZX_{90}$ pulses are applied and the amplitude is adjusted so that for each $N$ the expected signal is halfway between 0 and 1. Any amplitude miscalibrations lead to departures from this expected signal and are amplified for increasing $N$.

In addition to amplitude we must also calibrate the phase of the $ZX_{90}$ pulse between Q3 and Q2 (as well as Q1 and Q2). In our experiment we use a separate microwave generator to supply the cross-resonance pulse on Q3 at the frequency of Q2. The phase of this microwave signal must be calibrated to match that of the microwave generator supplying the single qubit pulses on Q2. This is done by applying the pulse sequence $IY_{90}(ZU_{180}IX)^{N}IX_{90}$. The $U$ denotes the rotation axis defined by the second generator and the goal is to calibrate for an $X$ rotation. In the case of an $X$-rotation we expect the signal to be halfway between 0 and 1 for each $N$ and miscalibrations of the phase lead to deviations that are amplified with increasing $N$. These methods provide a routine for automated calibration with high precision. In the experiments all cross-resonance pulses were calibrated on a regular basis because of phase drift between the two microwave generators.

Randomized benchmarking

All single-qubit gates are 40 ns Gaussian-shaped microwave pulses (Gaussian width $\sigma=10$ ns) resonant with the transition frequencies of the qubits, with scaled derivative-of-Gaussian shapes applied on the quadrature channel to minimize leakage effects 29. The gates are all autonomously calibrated with a set of repeated pulse experiments, correcting for: amplitude of $X_{90}$ and $X$ gates, amplitude imbalance between $X$- and $Y$- rotations, mixer skew, and derivative of Gaussian shape parameter. Single-qubit gates are all independently characterized via Clifford 30 randomized benchmarking (RB), and summarized in Table 1. To characterize the addressability error of the system, we perform simultaneous 25 RB, applying different sets of randomized single-qubit Clifford gates to all three qubits at the same time. These results are also summarized in Table 1 and essentially indicate that addressability errors are at the 0.1% error level.

The two-qubit $ZX_{90}$ gates for both pairs of qubits are shaped with Gaussian turn-on (3$\sigma$, $\sigma=24$ ns), a flat section, and then a Gaussian turn-off, for a total gate time of 350 ns. The $ZX_{90}$ gates are tuned-up also using repeated pulse experiments (described in previous section). It is also important to note that the pair of two-qubit gates can be applied simultaneously, as they commute with one another. To characterize the gates, we generate two-qubit Clifford operations 17 and perform RB. The results for the two cases are shown in Extended Data Fig. 2, where we show the average fidelity decay over 35 different randomized two-qubit Clifford sequences. Analyzing the decay curves gives us error per two-qubit Clifford gate of $0.058\pm 0.003$ for the Q1 and Q2 and $0.065\pm 0.002$ for Q3 and Q2. We find the reduced chi-square for these fits are 0.583 and 0.385 respectively. This demonstrates that the model is a faithful representation of the data. As each two-qubit Clifford gate is composed of 1.5 $ZX_{90}$ generators, we estimate the two-qubit $ZX_{90}$ gate errors to be 3.8% and 4.3%.

Readout characterization

For this experiment each qubit has its own measurement resonator. On Q1 and Q3 high-power readout was used and for Q2 a dispersive linear readout with a JPA was used. The readout was performed by using an integrating kernel that takes into account the response of the cavity (see Ref. 28 for more details). This is important when most of the information is in the initial transients of the signal. The integration time for the experiment was 4 $\mu$s for the high power readout and 2 $\mu$s for the dispersive readout with the JPA.

Shown in Extended Data Fig. 3 are typical histograms for the three readout channels averaged over all computational basis for the qubits not measured. Here we see that the assignment fidelity, defined by

for the three channels is $0.84$, $0.91$ and $0.89$ respectively. These are typical values and we see about a $2-3\%$ fluctuation over the course of a typical experiment. By fitting a double Gaussian model to the data we find that the ratio of the undesired state to the desired state for Q1 prepared in the ground (excited) is $9.9\%$($22\%$) for Q2 $5.7\%$($8.3\%$) and for Q3 $13.7\%$($6.0\%$). We believe most of the error is due to the high-power non-linear readout of Q1 and Q3 and is not due to the qubits being initialized in the wrong state. With no power applied to the Q1 and Q3 resonator the assignment fidelity is $0.95$ and the ratios of the two Gaussians are $5.6\%$ when Q2 is prepared in the excited state and negligible when Q2 is prepared in the ground state.

State tomography

We find that in all cases the fluctuations in the state fidelity from statistics is much smaller than the difference between the linear reconstruction and the semi-definite program. Furthermore, we find typically the sum of all the negative eigenvalues in the three-qubit space to be less than 0.03.

Measurement tomography

An ideal Z-parity check can be described by the quantum operation

and the extra system is used to label the outcome of measurement of the syndrome qubit. In the noisy case this is represented by the operation

By binning the results of the measurement on the syndrome qubit, tomography on the two-qubit subspace is performed by preparing a complete set of different input states and measurement bases via pre and post-rotations, and reconstructing the operations from the measurement results. The complete set of rotations that we use are the same as those used in state tomography. We use both a linear reconstruction and a minimization to make the maps physical. For more details on how quantum process tomography can be performed see Ref. 16.

We use the Pauli transfer matrix 16 defined by

to represent the measurement operations where $P_{j}$ are the standard Pauli operators $\{I,X,Y,Z\}^{\otimes 2}$.

Since the nullspace of a projection operation has measure zero and the noisy realization typically will also have a nullspace of zero measure this integral is well defined. To compute this we draw 150,000 different random states from the Fubini-Study measure and compute the average.

One could also define a process fidelity by computing the state fidelity between normalized Choi matrices of the ideal and noisy operations

however for non-unitary processes there is no simple relationship between them.

The unconditional map can be defined by tracing Eq. (5) over the syndrome qubit giving

Since this is a quantum operation the standard fidelity between quantum operations can be used.

Note

During the completion of this manuscript, we became aware of similar work by O. P. Saira et al. 31.

References