Long-Distance Transmon Coupler with cz-Gate Fidelity above $99.8\mathrm{\%}$

Tunable coupling of superconducting qubits has been widely studied due to its importance for isolated gate operations in scalable quantum processor architectures. Here, we demonstrate a tunable qubit-qubit coupler based on a floating transmon device, which allows us to place qubits at least 2 mm apart from each other while maintaining over 50-MHz coupling between the coupler and the qubits. In the introduced tunable-coupler design, both the qubit-qubit and the qubit-coupler couplings are mediated by two waveguides instead of relying on direct capacitive couplings between the components, reducing the impact of the qubit-qubit distance on the couplings. This leaves space for each qubit to have an individual readout resonator and a Purcell filter, which is needed for fast high-fidelity readout. In addition, simulations show that the large qubit-qubit distance significantly lowers unwanted non-nearest-neighbor coupling and allows multiple control lines to cross over the structure with minimal crosstalk. Using the proposed flexible and scalable architecture, we demonstrate a controlled-$Z$ gate with ($99.81\pm 0.02$)% fidelity.

Popular Summary

Most superconducting-based quantum processors aiming for fault tolerance are designed as a quantum-bit (qubit) lattice, where each qubit has three or more neighboring qubits. Two-qubit gates between two neighboring qubits can be implemented by utilizing an intermediate coupler, which can turn on and off the interaction between the qubits. Even though these couplers can be implemented in various ways, the use of a qubit as a coupler has been demonstrated to be one of the most promising candidates for high-fidelity two-qubit gates.

Besides the most essential components consisting of qubits, couplers, and readout resonators, the quantum processor would benefit from additional components, such as band-pass filters for readout resonators, which require physical space on the chip. However, a qubit coupler yields the undesirable requirement that the two qubits need to be physically close to each other to ensure the ability to turn the interaction between the qubits completely off. This requirement leads to a chip that is densely packed with qubits and has no available space for additional components, ultimately limiting the performance of the quantum processor.

Here, we present a coupling structure that allows the neighboring qubits to be separated 4 times further apart than in previous demonstrations while utilizing all the advantages of a qubit as a coupler. This can be achieved by combining the qubit coupler with specially designed waveguides in such a way that the qubits are still interacting with each other, even when placed far apart. With this long-distance coupler, we repeatably implement two-qubit gates with an average fidelity of 99.81%.

The additional space in the qubit lattice provided by the introduced coupler opens up many possibilities to improve the performance of a quantum processor. For example, high-fidelity readout of the qubit state—a key requirement for quantum error correction—can be improved by placing individual band-pass filters for the readout line of each qubit in between the lattice. Furthermore, by combining short and long coupling structures, sophisticated designs beyond a square lattice are possible, which may help solve specific real-world problems during the noisy intermediate-scale quantum era.

Figure 1

(a) A quasi-lumped-element circuit diagram of qubits (blue and orange) coupled by a tunable coupling structure consisting of waveguide extenders (turquoise) and a floating coupler qubit (red). The gray lines show the effective lumped-element model for the coupling capacitances arising from the black waveguide extenders C and F. (b) A false-colored micrograph of the qubit-coupler-qubit system. The qubits, coupler, and waveguide extenders follow the color code from (a). The air bridges are colored nontransparently to protect unpublished intellectual property. (c) The simulation of the effective coupling strengths between qubit 1 and the coupler, $g_{1\mathrm{c}}$, qubit 2 and the coupler, $g_{2\mathrm{c}}$, and qubits 1 and 2, $g_{12}$, for different qubit-qubit distances $d_{\mathrm{qq}}$ (solid lines). The measured coupling values (dots with 68% confidence intervals) are shown for our device with $d_{\mathrm{qq}}=1960\text{μ}\mathrm{m}$ (dashed line). (d) The simulated spurious next-nearest-neighbor (NNN) coupling $g_{13}$ between ${\mathrm{Q}}_1$ and ${\mathrm{Q}}_3$ in a square-lattice configuration (schematic) and the nearest-neighbor (NN) coupling $g_{12}$ as a function of the distances $d_{\mathrm{qq}}$ between ${\mathrm{Q}}_1$ and ${\mathrm{Q}}_2$. The gray shaded area indicates the region where the coupling values fall below the simulation accuracy. (e) A schematic of a simplified flip-chip architecture, where the qubits and the coupler are connected via waveguide extenders (light gray) on the bottom chip and a long perpendicular transmission line (black) is crossing above the qubit-coupler structure on the top chip. (f) The simulated capacitance coupling ratios (see the main text) from each component to the transmission line [color code same as in (e)] at various crossing positions $x_{\mathrm{cross}}$ of the transmission line, the total coupling (black dashed line), and the region for low-crosstalk crossings (blue shading).

Figure 2

(a) The effective $ZZ$ interaction strength $\zeta$ between the qubits is shown as a function of the coupler frequency ${\omega }_{\mathrm{c}}/2\pi$ and the frequency detuning between the two qubits, $\mathrm{\Delta }/2\pi$. If ${\alpha }_2<\mathrm{\Delta }<-{\alpha }_1$, there exists a contour (black dotted line) along which the $ZZ$ interaction vanishes. The qubit frequencies are shown with orange dashed lines and the detuning values corresponding to their anharmonicities are marked with white dashed lines. The solid black line indicates the lowest coupler frequency at which a conditional phase of ${\varphi }_{11}=\pi$ can be accumulated in 25 ns. The solid red line illustrates the path along which the coupler is operated in the experiments. (b) The experimentally measured interaction strength (dots) as a function of the coupler frequency ${\omega }_{\mathrm{c}}/2\pi$ in units of flux quanta ${\mathrm{\Phi }}_0$ for the specific qubit-qubit detuning chosen to implement the cz gate. The numerical model (purple line) is fitted to the data to extract the coupling strengths between the two qubits and between each qubit and the coupler. (c) The experimentally measured eigenfrequencies (white to blue spectrum) and fitted eigenfrequencies (lines) of the qubits and the coupler as a function of the external flux ${\mathrm{\Phi }}_{\mathrm{ext}}^{\mathrm{c}}$ threading through the coupler superconducting quantum interference device (SQUID) loop. The coupler spectrum below 3.7 GHz is measured using ${\mathrm{Q}}_2$ as an ancilla (see Appendix pp6). The red circles show the idling and the operation configuration of the coupler.

Figure 3

The interleaved randomized benchmarking experiment for two qubits. (a) The experimental data of the sequence fidelity for randomized benchmarking with interleaved cz gates (purple dots) and the reference Clifford sequence (blue dots) as a function of the number of Clifford gates. The purple and blue shaded regions indicate the standard deviation of 30 random sequences for the interleaved randomized benchmarking experiment and the reference Clifford sequence, respectively. The histograms in the inset show the error per Clifford gate (blue) and the error per cz gate (purple) for the 14 repeated experiments, from which the average error per Clifford gate of ${ϵ}_{\mathrm{Clifford}}^{\mathrm{avg}}=(1.39\pm 0.03)\times {10}^{-2}$ and the average error per cz gate of ${ϵ}_{\mathrm{CZ}}^{\mathrm{avg}}=(1.9\pm 0.2)\times {10}^{-3}$ are estimated. (b) The experimentally obtained error per cz gate as a function of the number of interleaved cz gates $n$ in the randomized benchmarking sequence. The error bars indicate the standard deviation extracted from the residuals of the best model fit of the sequence decay for $n$ interleaved cz gates. The gate sequence is depicted in the inset: a random Clifford gate $C$ followed by $n$ cz gates, repeated $m$ times, after which a reversing Clifford gate $C_{\mathrm{r}}$ is applied. Extraction of the fitting error for the sequence fidelity for $n$ interleaved cz gates yields a reduction in the uncertainty of the error per cz gate by a factor of $1/n$.

Figure 4

The impact of the coupler decoherence on the qubits. (a) The energy-relaxation times of the hybridized qubit-coupler modes measured directly (dots), measured via an ancilla qubit (diamonds), and evaluated numerically based on the hybridization with the coupler (solid lines) as a function of the coupler frequency ${\omega }_{\mathrm{c}}/2\pi$. The shaded areas indicate the coupler frequencies swept during the cz gate (gray) and regions with unknown sources of decoherence, partly attributed to parasitic two-level systems (orange and blue shading for ${\mathrm{Q}}_1$ and ${\mathrm{Q}}_2$, respectively). (b) The measured frequencies of the hybridized modes during coherence measurements (dense set of dots) and the frequencies of the uncoupled system (black dashed lines). (c) The same as in (a) but showing the Gaussian component of the pure dephasing time for the hybridized qubit and coupler modes. (d) The coherence limit of the cz gate (solid black line) as a function of the interaction time $\tau$, calculated using Eq. (3), with contributions from individual coherence times (dashed lines), and the coherence limit for no decoherence contributions from the coupler (green stars). When evaluating the gate error, an idling time of 11 ns is added on top of the interaction time. The black dot shows the experimental gate error with the 68% confidence interval. The data for $T_{\varphi ,1,\mathrm{eff}}^{{\mathrm{Q}}_2}$ are below the curve corresponding to $T_{1,\mathrm{eff}}^{{\mathrm{Q}}_2}$ and not clearly visible in the plot.

Figure 5

The lumped-element circuit diagram of the floating-coupler scheme. The drawn sizes of the capacitors roughly indicate the relative magnitudes between the designed values of the capacitances. The crosses indicate the Josephson junctions with the Josephson energy $E_{\mathrm{J}}$.

Figure 6

The derivation of the effective model using star-mesh transformations. The capacitances are defined in Eqs. (A3) and (A4).

Figure 7

The effective-circuit schematic of the floating-coupler setup. The capacitances are defined in Eq. (A3).

Figure 8

A schematic of the reduced effective circuit of the floating-coupler setup. The capacitances are defined in Eq. (A16).

Figure 9

The experimental setup. The background colors depict the different temperature stages of the dilution refrigerator. The coaxial microwave cables (straight black lines) and twisted-pair cables (braided lines) are used to control and read out the qubit and coupler states. The schematic of the sample in the white box consists of two flux-tunable qubits (blue and orange), a readout structure for each qubit (light blue and light orange), and a flux-tunable coupler (red).

Figure 10

The results from a three-tone spectroscopy experiment. (a) Qubit spectroscopy with an additional frequency sweep of the coupler drive. The horizontal black dashed line indicates the one-dimensional sweep for a three-tone experiment shown in (b). The coupler frequency ${\omega }_{\mathrm{c}}/2\pi$ is extracted by a minimum search of the one-dimensional sweep (vertical black dashed line).

Figure 11

(a) The experimentally obtained relaxation and coherence times of the coupler as functions of its frequency. (b) The pure dephasing rate from echo measurements as a function of the flux-dispersion slope. A line fit yields the flux-noise amplitude $\sqrt{A}$.

Figure 12

The experimentally obtained sequence fidelity as a function of the number of Clifford gates in the randomized benchmarking sequence with $n$ interleaved cz gates, where $n$ ranges from 1 to 20. Each data point (symbols) represents the average result of 30 random sequences. Each sequence fidelity trace is fitted using an exponential model (solid lines) to extract the error ${ϵ}_{n\mathrm{CZ}}$ per $n$ cz gates.

Figure 13

The exponential part of the dephasing times as a function of the coupler frequency for the hybridized modes. For the coupler frequencies relevant for the cz gate, the exponential part of dephasing mostly stays constant, except for a sharp drop at 3.8 GHz for qubit 1. We observe the feature to move to different frequencies over time, indicating that it originates from a spectrally active TLS. The solid lines show the median of the measured exponential dephasing rates, evaluated at the frequencies relevant for the cz gate and used for calculating $T_{\varphi ,1,\mathrm{eff}}^k$.