Universal Gate for Fixed-Frequency Qubits via a Tunable Bus

A challenge for constructing large circuits of superconducting qubits is to balance addressability, coherence, and coupling strength. High coherence can be attained by building circuits from fixed-frequency qubits; however, leading techniques cannot couple qubits that are far detuned. Here, we introduce a method based on a tunable bus which allows for the coupling of two fixed-frequency qubits even at large detunings. By parametrically oscillating the bus at the qubit-qubit detuning we enable a resonant exchange ($XX+YY$) interaction. We use this interaction to implement a 183-ns two-qubit iswap gate between qubits separated in frequency by 854 MHz, with a measured average fidelity of 0.9823(4) from interleaved randomized benchmarking. This gate may be an enabling technology for surface-code circuits and for analog quantum simulation.

Figure 1

(a) Schematic of an $N$ qubit, one-bus device. (b) Size of the expansion terms for $\omega$ and $J$ versus the dc flux tuning the bus. Calculations are for the device parameters given in Sec. 2b.

Figure 2

(a) Optical image and (b) schematic of our circuit consisting of two fixed-frequency transmon qubits ($Q1$, $Q2$) coupled via a third tunable bus qubit (“tunable bus,” TB). $Q1$ and $Q2$ have individual readout resonators (RR1, RR2). The TB is tuned by a dc-bias coil and a high-speed flux line (HSFL). Spectroscopy of (c) TB and (d) $Q1$, $Q2$ frequency versus dc flux; $Q1$ tunes more strongly with flux because it is closer in frequency to the TB. We fit these tuning curves (the solid lines) using Eq. (4) to extract the Hamiltonian parameters for Eq. (1).

Figure 3

(a) Flux-modulation pulse of strength $\delta$. The pulse envelope (the gray dashed line) is a square pulse with Gaussian turn on and off with $\sigma =8.3\mathrm{ns}$ and the turn on or off time is $3\sigma$. (b) Exchange oscillations between $|10⟩$ and $|01⟩$ as a function of the flux-pulse length and the drive frequency ${\omega }_{\varphi }$ (with respect to the detuning between the qubits when $\delta =0$, ${\mathrm{\Delta }}_{12,\delta =0}\approx 854\mathrm{MHz}$). These data are taken at a dc flux bias of $\mathrm{\Theta }=-0.108{\mathrm{\Phi }}_0$ with a constant flux-pulse height $\delta =0.153{\mathrm{\Phi }}_0$. The flux modulation induces a dc shift of the tunable bus, so the resonance frequency (the dotted line) of the exchange oscillation is shifted down from ${\mathrm{\Delta }}_{12,\delta =0}$ by approximately 3 MHz. (c) Qubit shifts during the flux-modulation pulse as a function of the drive strength (the procedure is described in the main text). The drive strength is plotted in terms of the measured exchange rate (the bottom axis) and the flux-modulation strength $\delta$ in units of ${\mathrm{\Phi }}_0$ (the top axis) calculated theoretically from this exchange rate. The solid lines are no-free-parameter numerical calculations of the exchange rate and shift given a certain flux modulation by solving Eq. (1).

Figure 4

(a) On-resonance swap oscillations between the qubits. We first apply a $\pi$ pulse to $Q1$, then perform a variable-length flux-modulation pulse on the tunable bus with ${\omega }_{\varphi }/2\pi =850.6\mathrm{MHz}$ and $\delta =0.155{\mathrm{\Phi }}_0$. The excitation oscillates between the two qubits and, at special evolution times (indicated by arrows), entangled states are generated. (b) State tomography of the first entangled state in the Pauli representation, with a fidelity of 0.974. Single-qubit terms are illustrated in blue and two-qubit terms in red. The outlined bars show the ideal state, $|\mathrm{\Psi }⟩=(|10⟩-i|01⟩)/\sqrt{2}$.

Figure 5

Optimization and simulation of the iswap gate. (a) Gate error versus pulse width measured experimentally (the triangles) versus the numerical calculation (the circles), including levels outside the computational basis and decoherence. (b) Calculated leakage versus pulse width for different levels outside the computational basis. (c) Leakage measured experimentally by proxy by tracing over the computational states (leakage RB). These data are for the standard gate length 183 ns. (Inset) The leakage metric (the asymptote of the RB data) versus the pulse width. Because the leakage metric is flat versus the pulse width, we are not in the high-leakage regime seen in the numerics (b).

Figure 6

(a) Standard, interleaved, and purity randomized benchmarking (RB) of the two-qubit iswap gate, with the state transformation shown in the inset. For standard and interleaved RB, the ground-state population of qubit 2 is plotted as a function of the number of Clifford gates for a sample RB run consisting of 20 random seeds (14 seeds for the purity RB). The error numbers quoted in the main text are averaged over eight such independent runs. For purity RB, we plot the trace of ${\rho }^2$ versus the number of Clifford gates. (b) Pauli transfer matrix of the iswap gate measured from quantum-process tomography.