Efficient and accurate two-qubit-gate operation in a high-connectivity transmon lattice utilizing a tunable coupling to a shared mode

Abstract

Increasing connectivity and decreasing qubit-state delocalization without compromising the speed and accuracy of elementary gate operations are topical challenges in the development of large-scale superconducting quantum computers. In this theoretical work, we study a special honeycomb qubit lattice where each qubit inside a unit cell is coupled to every other one via two dedicated tunable couplers and a common central element. This results in an effective multi-mode interaction enabling tunable, on-demand, all-to-all connectivity between each qubit pair within the unit cell. We provide a thorough analysis of the unit cell, including a proposal for a novel and efficient conditional-Z gate scheme which takes advantage of the effective multi-mode coupling. We develop an experimentally viable pulse protocol for a single-step gate implementation which considerably improves the gate speed compared to the previous two-qubit-gate realizations suggested for architectures utilizing a center mode. We also show numerical results on how the presence of spectator qubits affects the average two-qubit-gate fidelity, and analyse how the multi-mode coupling structure mitigates the delocalization-induced crosstalk during simultaneous single-qubit gates within the unit cell. We also provide analytical estimates for the errors caused by relaxation and dephasing during a two-qubit-gate operation, including noise terms for the multi-mode coupling structure. Our multi-mode coupling architecture results in a good balance between increased connectivity and available parallelism, especially when several interacting unit cells form a quantum processing unit. We anticipate that the obtained results pave the way towards high-connectivity quantum processors with efficient and low-overhead quantum algorithms.

Figures (10)

• Figure 1: Schematic of a quantum-processing unit with the honeycomb topology utilizing multi-mode couplers. (a) Schematic of the honeycomb lattice with multi-mode couplers. Ideally, there are no direct qubit-qubit couplings along the unit cell boundaries (dashed lines), and each qubit is capacitively coupled only to the three nearest multi-mode couplers. We have highlighted the nearest-neighbour qubits (blue) and multi-mode couplers (green) for qubit $\text{q}_1$ (orange). (b) Schematic of the unit cell, where $\mathcal{C}$ denotes the multi-mode coupler. (c) Schematic of the multi-mode coupler, with $\text{c}_i$ representing the $i$th tunable coupler.
• Figure 2: ZZ couplings $\zeta_{\ell\ell'}$ in the minimal system as a function of the frequencies $\omega_{\text{c}_1}$ and $\omega_{\text{c}_2}$ of the tunable couplers $\text{c}_1$ and $\text{c}_2$. ZZ coupling between (a) the qubit $\text{q}_1$ and the center mode $\text{c}$, (b) the qubit $\text{q}_2$ and the center mode $\text{c}$. The ZZ coupling between the qubit $\text{q}_j$ and the center mode is independent on the frequencies of the tunable couplers of the other qubits. (c) The ZZ coupling between the qubit $\text{q}_1$ and the center mode $\text{c}$ as a function of the frequencies $\omega_{\text{c}_1}$ and $\omega_{\text{q}_1}$. Black dots correspond to the chosen idling frequencies, see Tab. \ref{['tab:parameters']}. The direct ZZ coupling between the qubits $\text{q}_1$ and $\text{q}_2$ is less than 10kHz for the chosen parameters. These results generalize for additional qubits.
• Figure 3: Population dynamics for the states performing the gate obtained with the simple model in Eq. \ref{['eq:hamiltonians']}, and with the complete Hamiltonian in Eq. \ref{['eq:minimal_hamiltonian']} including pulse shapes and tunable couplers. We use $\mathinner{|{\psi(0)}\rangle} = (\mathinner{|{000}\rangle} + \mathinner{|{100}\rangle} + \mathinner{|{010}\rangle} + \mathinner{|{110}\rangle})/2$ as the initial state, and the labeling convention is $\mathinner{|{n_{\text{q}_1}n_{\text{q}_2}n_{\text{c}}}\rangle}$. (a) Single-excitation states of the analytical model. (b) Two-excitation states of the analytical model. (c) Single-excitation states of the complete system. (d) Two excitation states of the complete system. The CZ-gate infidelity in the full model is $1-\mathcal{F} = 6.2\cdot10^{-7}$. In the complete system also the coupler states momentarily gain population (data not shown). (e) Accumulation of the conditional phase $\phi_{\rm CP}$ during the gate using the analytical result in Eq. \ref{['eq:analyt_cp']} (solid dark blue) and the full numerical solution (dashed pink), which gives for the conditional phase $\phi_{\rm CP}(\tau)=3.141$ at the end of the gate.
• Figure 4: Schematic of the pulse schedule for a $\tau=60.0ns$ pulse with $\sigma_{\text{q}}=1.0ns$ and $\sigma_{\text{c}}=3.4ns$. First the qubits $\text{q}_1$ and $\text{q}_2$ are pulsed to their respective operation points, $\text{q}_2$ close to resonance with the center mode $\text{c}$ and $\text{q}_1$ anharmonicity $|\alpha_{\text{q}_1}|$ above. Then the tunable couplers $\text{c}_1$ and $\text{c}_2$ are pulsed closer to the center mode $\text{c}$. As predicted by Eq. \ref{['eq:coupling_restriction']}, the coupler $\text{c}_1$ goes slightly higher than $\text{c}_2$, generating a stronger effective coupling.
• Figure 5: Effect of the number of qubits $N$ in the system on the CZ gate fidelity as a function of coupler-pulse filter width $\sigma_{\text{c}}$. We have used $\sigma_{\text{q}}=1.0ns$ and the gate duration $\tau=60.0ns$. (a) Gate infidelity for the initial state $\mathinner{|{\psi(0)}\rangle} = (\hat{I} + \hat{a}_{\text{q}_1}^\dag)(\hat{I} + \hat{a}_{\text{q}_2}^\dag)\mathinner{|{\text{vac}}\rangle}/2$. (b) Average gate infidelity. Dashed lines are extrapolated from the existing data for each value of $\sigma_\text{c}$, since the numerical optimization of the average gate fidelity becomes heavy for larger systems. (c) Qubit occupation difference at the beginning and at the end of the gate as a function of the number $N$ of qubits. For each $N$, we have used $\sigma_{\text{c}}$ giving the smallest infidelity. (d) Coupler occupation difference at the beginning and the end of the gate as a function of the qubit number. (e) Optimized amplitudes for $\text{q}_1$ (solid) and $\text{q} _2$ (dashed) for different system sizes, colours defined in the labels of (a). (f) Optimized amplitudes for $\text{c}_1$ (solid) and $\text{c}_2$ (dashed) for different system sizes.