Fast and Robust Remote Two-Qubit Gates on Distributed Qubits

TL;DR

A remote quantum geometric gate scheme via parametric modulation that inherits the intrinsic robustness of geometric phases is proposed, offering a promising path toward the realization of fault-tolerant quantum computation.

Abstract

Distributed quantum computing offers a potential solution to the complexity of superconducting chip hardware layouts and error correction algorithms. High-quality gates between distributed chips enable the simplification of existing error correction algorithms. This article proposes and demonstrates a remote quantum geometric gate scheme via parametric modulation. Our scheme inherits the intrinsic robustness of geometric phases. Meanwhile, by employing gradient-based optimization algorithms(Adaptive Moment Estimation) from deep learning, we design control waveforms that significantly suppress population leakage. We experimentally realize the rapid remote SWAP and $\sqrt{\text{SWAP}}$ gates with high fidelity, completing operation in about 30 ns. The gate error of SWAP ($\sqrt{\text{SWAP}}$) is 1.16\% (0.91\%) after excluding the effect of energy relaxation. The simulation demonstrate that this scheme can be implemented in the distributed chips connected by cables extending several meters. Our results highlight the effectiveness of the proposed protocol in enabling modular quantum processors, offering a promising path toward the realization of fault-tolerant quantum computation.

Figures (4)

• Figure 1: Demonstration of Remote Connectivity Scheme. (a) Sample structure. Qubits were capacitively coupled to the 15cm Al cable. In the actual device, the qubits are connected to cables via couplers. In this experiment, the coupler is treated as a capacitor. Therefore, for clarity, it is omitted from the schematic diagram. We applied parametric modulation on two qubits and constructed coupling between qubits and cable modes. (b) Holonomic Evolution. Cyclic evolution in the base space induces non-Abelian geometric phases in the U(2) bundle. (c) Timing sequence of our experiment. The initial state was built by two arbitrary driving pulses on $Q_1$ and $Q_2$. Two parametric modulation pulses were simultaneously applied on $Q_1$ and $Q_2$ while realizing holonomic evolution. (d) Demonstration of Adam Optimizer. Given a guessed waveform, the Adam optimizer leverages its adaptive learning rate to avoid local optima, enabling the loss function to converge to the global optimum and effectively suppress cable mode leakage.
• Figure 2: Remote Holonomic SWAP and $\sqrt{\text{SWAP}}$ Gate. (a)(c) The population dynamics of the remote holonomic SWAP and $\sqrt{\text{SWAP}}$ gates. Orange, green and blue solid lines refer to the numerical results of $Q_1$, $Q_2$ and cable mode $M_2$ population, and dots refer to the experimental results. (b) Qubit tomography result of SWAP gate, the final state fidelity achieved $99.91\pm0.06\%$ considering the decoherence. (d) Subspace tomography result of $\sqrt{\text{SWAP}}$ gate. For single qubit tomography, $Q_2$ finally received 47.48% population. For subspace tomography, we consider the subspace $\{|10\rangle,|01\rangle\}$. The final state fidelity achieved $97.54\pm0.42\%$, considering the decoherence.
• Figure 3: Error and Robustness feature of holonomic gates. (a) Time sequence for the error estimation. We applied a series of SWAP ($\sqrt{\text{SWAP}}$) gates on the system and then measured the population of the qubit. (b)-(c) Result of the SWAP and $\sqrt{\text{SWAP}}$ gate. The blue solid line represents the fitted result of the numerical simulation that takes energy dissipation into account, while the purple solid line corresponds to the fitted result of the experimental data. After deducing the energy dissipation effect, the SWAP ($\sqrt{\text{SWAP}}$) average gate error can reach 1.16% (0.91%). (d) Robustness simulation results of the detuning-population relation considering decoherence. $\delta_1$ and $\delta_2$ refer to the flux pulse detuning of the two qubits. The left (right) panel represented to the holonomic (dynamic) SWAP gate. (e) The solid lines are the population corresponding to the red dashed lines in (d). At a detuning of 3 MHz, the dynamic SWAP gate exhibits 6.13% population loss, while our scheme results in merely 3.46% loss. The blue and orange dots (solid lines) are the experiment results (numerical simulations).
• Figure 4: Waveform Optimization and Leakage Suppression. (a) Illustration of the coupling between qubits and cable modes. We considered the nearest five cable modes. The coupling strengths $g_{1n}, (-1)^ng_{2n}$ and detunings $\Delta_1, \Delta_2$ between the qubits and each mode are labeled. (b) The cable mode leakage of square, Gaussian, cosine and Adam optimized waveforms at different cable lengths, represented by the red, yellow, green and blue solid lines, respectively. The cable length ranges from 12cm to 172cm, corresponding to the FSR from 35MHz to 500MHz. Left (Right) panel represents the in-refrigerator (refrigerator-to-refrigerator) connection condition. The cable length (FSR) used in our experiments is approximately 15cm (403MHz), which is indicated by the gray dashed box. Adaptive Moment Estimation (Adam) was applied to the chosen guessed waveform to suppress cable mode leakage. The application of Adam led to a suppression of leakage by one order of magnitude, demonstrating its effectiveness in improving waveform performance. 1m (FSR = 60MHz) and 1.5m (FSR = 40MHz) were represented by blue and purple dashed box, respectively. (c)-(d) Leakage Distribution Analysis. The leakage distribution was analyzed for cable length of 1m and 1.5m. At both lengths, the leakage in the target mode ($M_2$) was effectively suppressed. A slight increase in leakage was observed in the neighboring modes; however, this effect was minor and did not noticeably degrade the overall system performance.