Optimized cross-resonance gate for coupled transmon systems

The cross-resonance (CR) gate is an entangling gate for fixed-frequency superconducting qubits. While being simple and extensible, it is comparatively slow, at 160 ns, and thus of limited fidelity due to on-going incoherent processes. Using two different optimal control algorithms, we estimate the quantum speed limit for a controlled-not cnot gate in this system to be 10 ns, indicating a potential for great improvements. We show that the ability to approach this limit depends strongly on the choice of ansatz used to describe optimized control pulses and limitations placed on their complexity. Using a piecewise-constant ansatz, with a single carrier and bandwidth constraints, we identify an experimentally feasible 70-ns pulse shape. Further, an ansatz based on the two dominant frequencies involved in the optimal control problem allows for an optimal solution more than twice as fast again, at under 30 ns, with smooth features and limited complexity. This is twice as fast as gate realizations using tunable-frequency, resonantly coupled qubits. Compared to current CR-gate implementations, we project our scheme will provide a sixfold speed-up and thus a sixfold reduction in fidelity loss due to incoherent effects.

Figure 1

We observe a 5- to 10-ns quantum speed limit (QSL) for the system described by Eq. (2) and Table 1, using arbitrary amplitude piecewise-constant controls with a high sampling rate ($\le 0.1$ ns) and no filter. The unitary error (infidelity; $1-{\mathrm{\Phi }}_{\mathrm{goal}}$) of the optimized controlled-not (cnot) gates is plotted as a function of gate duration, with each point representing the average error for multiple optimizations starting at random initial pulses. See Sec. 4 for a complete discussion.

Figure 2

High-fidelity CR control pulse sequence found with a sampling rate of 5 Gs/s. (a) Initial control guess pulse conforming to the analytical ansatz (11). (b) Optimized controls giving ${\mathrm{\Phi }}_{\mathrm{goal}}=0.9999$. The pulses are relatively smooth and short (27 ns) but contain a large degree of complexity. See Sec. 5 for full details.

Figure 3

Determination of the QSL using 167 Fourier components per control. The errors ($1-{\mathrm{\Phi }}_{\mathrm{goal}}$) averaged over optimized pulses found for different random initial conditions are plotted for different gate durations. Errors are larger and decay more slowly when compared to a time-domain parametrization containing roughly 800 (piecewise-constant) control points, depicted in Fig. 1.

Figure 4

Pulse shapes and spectra of the optimal drive shape for filtered PWC ansatz with a gate time of 70 ns and a single carrier frequency. A fidelity of 0.999 is achieved using a 331-MHz-bandwidth Gaussian filter.

Figure 5

Optimization results for control pulses using two carrier frequencies and a bandwidth filter, achieving 0.9999 fidelity when incoherent processes are ignored. (a) Controls at qubit 1 frequency. (b) Controls at qubit 2 frequency ($\delta \pm g$).

Figure 6

Infidelity of the CR gate as a function of the miscalibration of qubit 1 and qubit 2 frequencies, with other model parameters specified by Table 1. The control pulses are as depicted in Fig. 5.

Figure 7

Infidelity of the CR gate as a function of equal dephasing and decoherence rates. Model parameters follow Table 1, and control pulses follow Fig. 5.

Figure 8

Maximum transient population leakage to the higher levels of the resonator (a) and the transmons (b) during the dynamics driven by the optimal controls described in Fig. 5 and the model parameters detailed in Table 1. Note the logarithmic scaling.