Suppression of Static $ZZ$ Interaction in an All-Transmon Quantum Processor

The superconducting transmon qubit is currently a leading qubit modality for quantum computing, but the gate performance in a quantum processor with transmons is often insufficient to support the running of complex algorithms for practical applications. It is thus highly desirable to further improve gate performance. Due to the weak anharmonicity of a transmon, a static $ZZ$ interaction between coupled transmons commonly exists, undermining the gate performance and, in the long term, it can become performance limiting. Here, we theoretically explore a promising parameter region in an all-transmon system to address this issue. We show that a feasible parameter region, where the $ZZ$ interaction is heavily suppressed, while leaving the $XY$ interaction with an adequate strength to implement two-qubit gates, can be found for all-transmon systems. Thus, two-qubit gates, such as a cross-resonance gate or an iswap gate, can be realized without a detrimental effect from the static $ZZ$ interaction. To illustrate this, we demonstrate that an iswap gate with a fast gate speed and dramatically lower conditional phase error can be achieved. By scaling up to a large-scale transmon quantum processor, especially for cases with fixed coupling, addressing errors, idling errors, and crosstalk that arise from the static $ZZ$ interaction can also be strongly suppressed.

Figure 1

(a) Circuit layout and level diagram of a coupled two-qubit system. Truncation to qubit space gives rise an effective two-qubit Hamiltonian with interqubit $XY$ coupling $J$ and static $ZZ$ coupling $\zeta$. (b) Example circuit diagram of two transmons coupled via a coupler circuit combining a capacitor and an ancilla transmon.

Figure 2

(a),(b) Level diagram of the coupled transmon system shown in Fig. 1. (c) Landscapes of $XY$ coupling $J$ and (d) $ZZ$ coupling $\zeta$ (perturbation theory, making RWA) as a function of coupler frequency ${\omega }_c$ and direct coupling strength $g_{12}$ with qubit frequency ${\tilde{\omega }}_{1(2)}/2\pi =5.114(4.914)\mathrm{GHz}$, qubit or coupler anharmonicity ${\alpha }_{1(2)}/2\pi =-330\mathrm{MHz}$, ${\alpha }_c/2\pi =0\mathrm{MHz}$ (linear coupler), and qubit-coupler interaction strength $g_{1c(2c)}/2\pi =98(83)\mathrm{MHz}$ [7, 35]. Dashed curves in (c),(d) correspond to contours of $J=0$ and $\zeta =0$, respectively. Open circle (square) in (c),(d) marks the set of parameters chosen to suppress $XY$ ($ZZ$) coupling.

Figure 3

Landscapes of $XY$ coupling $J$ (perturbation theory, without making RWA) and $ZZ$ coupling $\zeta$ (numerical diagonalization) as a function of coupler frequency ${\omega }_c$ and direct coupling strength $g_{12}$ with coupler anharmonicity (a) ${\alpha }_c/2\pi =0$ (linear coupler), (b) ${\alpha }_c/2\pi =-200\mathrm{MHz}$, (c) ${\alpha }_c/2\pi =-400\mathrm{MHz}$, and (d) ${\alpha }_c/2\pi =-600\mathrm{MHz}$. Other system parameters are similar to those used in Fig. 2. For each coupler anharmonicity, two landscapes are superposed in a single panel, where the region corresponding to $ZZ$ coupling with $\zeta$ below 20 kHz is omitted and replaced with the corresponding values of $XY$ coupling $J$ (region outlined with black bold lines). Insets of (a)–(d) show the maintained $XY$ coupling strength obtained with the corresponding parameter set marked in (a)–(d). Solid (dashed) line represents the results derived from perturbation theory (period $T$ of the simulated cross-resonance oscillation).

Figure 4

(a) Landscapes of numerically calculated $XY$ coupling strength $J$ and $ZZ$ coupling strength $\zeta$ (data points with $ZZ$ coupling strength below $20\mathrm{kHz}$ are omitted and replaced with the corresponding values of the $XY$ coupling, $J$, i.e., the region outlined with black bold lines) as a function of coupler frequency ${\omega }_c$ and direct coupling strength $g_{12}$ in the two-transmon system [Fig. 1] with qubit frequency ${\omega }_{1(2)}/2\pi =6.5\mathrm{GHz}$, qubit anharmonicity ${\alpha }_{1(2)}/2\pi =-250\mathrm{MHz}$, coupler anharmonicity ${\alpha }_c/2\pi =-400\mathrm{MHz}$, and frequency-dependent transmon-coupler coupling strength $g_{1c(2c)}/2\pi =125\sqrt{{\omega }_c/{\omega }_1}(130\sqrt{{\omega }_c/{\omega }_2})\mathrm{MHz}$ [39]. Parameter set marked by an open circle denotes the idling frequency point for the coupler, and the dashed line indicates the direct coupling strength adopted to mitigate leakage during the iswap gate operation. (b) Typical control pulse for implementing an iswap gate, where the full width at half maximum is defined as the hold time. Dashed (solid) line shows the control pulse for a system operated in the dispersive (quasi-dispersive) regime, as marked by the open square (triangle) in (a). (c),(d) Swap error, ${\epsilon }_{\mathrm{swap}}$, and leakage, $L_1$, as a function of hold time with systems operated in the dispersive regime and the quasi-dispersive regime, respectively.

Figure 5

Level diagram of the coupled transmon system. (a) One-excitation subspace. (b) Two-excitation subspace. Green solid lines denote qubit level, while pink dashed lines represent nonqubit levels.

Figure 6

Cross-resonance oscillation of target transmon $Q_2$ with controlled transmon $Q_1$ in its ground state. Left (Right) panel shows population oscillation for target transmon $Q_2$ as a function of strength ${\mathrm{\Omega }}_d$ of the driving force applied on transmon $Q_1$ and time with parameter set $({\omega }_c,g_{12})$ marked with open circle (square) in Figs. 2. Other system parameters are similar to those used in Fig. 2. Horizontal dashed lines indicate the driving strength, ${\mathrm{\Omega }}_d/2\pi =10\mathrm{MHz}$, which is used to infer the effective $XY$ coupling strength $J$ [36].

Figure 7

Landscapes of $XY$ coupling, $J$, and $ZZ$ coupling, $\zeta$, as a function of coupler frequency, ${\omega }_c$, and direct coupling strength, $g_{12}$, with qubit frequency ${\omega }_{1(2)}/2\pi =5.114(5.014)\mathrm{GHz}$, and coupler anharmonicity (a) ${\alpha }_c/2\pi =0$ MHz, (b) ${\alpha }_c/2\pi =-200$ MHz, (c) ${\alpha }_c/2\pi =-400$ MHz, (d) ${\alpha }_c/2\pi =-600$ MHz. Other system parameters are similar to those used in Fig. 3, where the qubit detuning is ${\mathrm{\Delta }}_{12}/2\pi =200\mathrm{MHz}$. Here, the two-qubit detuning is ${\mathrm{\Delta }}_{12}/2\pi =100\mathrm{MHz}$.

Figure 8

Landscapes of $XY$ coupling, $J$, and $ZZ$ coupling, $\zeta$, as a function of coupler frequency, ${\omega }_c$, and direct coupling strength, $g_{12}$, with qubit frequency ${\omega }_{1(2)}/2\pi =5.114(5.014)\mathrm{GHz}$, and coupler anharmonicity (a) ${\alpha }_c/2\pi =0$ MHz, (b) ${\alpha }_c/2\pi =-200$ MHz, (c) ${\alpha }_c/2\pi =-400$ MHz, (d) ${\alpha }_c/2\pi =-600$ MHz. Other system parameters are similar to those used in Fig. 3, where the qubit detuning is ${\mathrm{\Delta }}_{12}/2\pi =200\mathrm{MHz}$. Here, the two-qubit detuning is ${\mathrm{\Delta }}_{12}/2\pi =50\mathrm{MHz}$.

Figure 9

Landscapes of $XY$ coupling, $J$, and $ZZ$ coupling, $\zeta$, as a function of coupler frequency, ${\omega }_c$, and direct coupling strength, $g_{12}$, for on-resonantly-coupled qubits. (a) ${\alpha }_c/2\pi =-100\mathrm{MHz}$, (b) ${\alpha }_c/2\pi =-200\mathrm{MHz}$, (c) ${\alpha }_c/2\pi =-400\mathrm{MHz}$ [see also Fig. 4], and (d) ${\alpha }_c/2\pi =-600\mathrm{MHz}$. Other system parameters are similar to those used in Fig. 4.

Figure 10

Off-resonance case. Landscapes of $XY$ coupling, $J$ (perturbation theory), and $ZZ$ coupling, $\zeta$ (numerical diagonalization), as a function of coupler frequency, ${\omega }_{\pm }$, with a qubit frequency of ${\omega }_{1(2)}/2\pi =6.143(6.421)\mathrm{GHz}$, a qubit anharmonicity of ${\alpha }_{1(2)}/2\pi =-330\mathrm{MHz}$, a coupler anharmonicity of ${\alpha }_{+}/2\pi =0\mathrm{MHz}$, a transmon-coupler coupling strength of $g_{1-(2-)}/2\pi =85\mathrm{MHz}$, $g_{1+(2+)}/2\pi =102\mathrm{MHz}$ [5], and coupler anharmonicities of (a) ${\alpha }_{-}/2\pi =-100\mathrm{MHz}$, (b) ${\alpha }_c/2\pi =-300\mathrm{MHz}$, (c) ${\alpha }_c/2\pi =-500\mathrm{MHz}$, and (d) ${\alpha }_c/2\pi =-700\mathrm{MHz}$. For each coupler anharmonicity, the two landscapes are superposed in a single panel, where the region corresponding to $ZZ$ coupling with $\zeta$ below 20 kHz is omitted and replaced with the corresponding values of $XY$ coupling $J$ (region outlined with black dotted lines).

Figure 11

Off-resonance case. Landscapes of $XY$ coupling, $J$ (perturbation theory), and $ZZ$ coupling, $\zeta$ (numerical diagonalization), as a function of coupler frequency, ${\omega }_{\pm }$, with a coupler anharmonicity of ${\alpha }_{-}/2\pi =-100\mathrm{MHz}$. (a) ${\alpha }_{+}/2\pi =0$, (b) ${\alpha }_{+}/2\pi =-200\mathrm{MHz}$, (c) ${\alpha }_{+}/2\pi =-400\mathrm{MHz}$, and (d) ${\alpha }_{+}/2\pi =-600\mathrm{MHz}$. Other system parameters are similar to those used in Fig. 10.

Figure 12

On-resonance case. Landscapes of $XY$ coupling, $J$ (numerical diagonalization), and $ZZ$ coupling, $\zeta$ (numerical diagonalization), as a function of coupler frequency, ${\omega }_{\pm }$, with a qubit frequency of ${\omega }_{1(2)}/2\pi =6.421\mathrm{GHz}$. Other system parameters are similar to those used in Fig. 11.