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Fault-Tolerant Multi-Qubit Geometric Entangling Gates Using Photonic Cat Qubits

Ye‐Hong Chen, Roberto Stassi, Wei Qin, Adam Miranowicz, Franco Nori

Published Sep 10, 2021• 2 citations
OpenAlex

Abstract

We propose a protocol to implement multi-qubit geometric gates (i.e., the Molmer-Sorensen gate) using photonic cat qubits. These cat qubits stored in high-$Q$ resonators are promising for hardware-efficient universal quantum computing. Specifically, in the limit of strong two-photon drivings, phase-flip errors of the cat qubits are effectively suppressed, leaving only a bit-flip error to be corrected. A geometric evolution guarantees the robustness of the protocol against stochastic noise along the evolution path. Moreover, by changing detunings of the cavity-cavity couplings at a proper time, the protocol can be robust against control imperfections (e.g., the total evolution time) without introducing extra noises into the system. As a result, the gate can produce multi-mode entangled cat states in a short time with high fidelities.

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We propose a protocol to implement multi-qubit geometric gates (i.e., the Molmer-Sorensen gate) using photonic cat qubits. These cat qubits stored in high-QQQ resonators are promising for hardware-efficient universal quantum computing. Specifically, in the limit of strong two-photon drivings, phase-flip errors of the cat qubits are effectively suppressed, leaving only a bit-flip error to be corrected. A geometric evolution guarantees the robustness of the protocol against stochastic noise along the evolution path. Moreover, by changing detunings of the cavity-cavity couplings at a proper time, the protocol can be robust against control imperfections (e.g., the total evolution time) without introducing extra noises into the system. As a result, the gate can produce multi-mode entangled cat states in a short time with high fidelities.
We propose a theoretical protocol to implement multiqubit geometric gates (i.e., the M\o{}lmer-S\o{}rensen gate) using photonic cat-state qubits. These cat-state qubits stored in high-QQQ resonators are promising for hardware-efficient universal quantum computing. Specifically, in the limit of strong two-photon drivings, phase-flip errors of the cat-state qubits are effectively suppressed, leaving only a bit-flip error to be corrected. Because this dominant error commutes with the evolution operator, our protocol preserves the error bias, and, thus, can lower the code-capacity threshold for error correction. A geometric evolution guarantees the robustness of the protocol against stochastic noise along the evolution path. Moreover, by changing detunings of the cavity-cavity couplings at a proper time, the protocol can be robust against parameter imperfections (e.g., the total evolution time) without introducing extra noises into the system. As a result, the gate can produce multimode entangled cat states in a short time with high fidelities.
To realize fault-tolerant quantum computing, it is necessary to store quantum information in logical qubits with error correction functions, realized by distributing a logical state among multiple physical qubits or by encoding it in the Hilbert space of a high-dimensional system. Quantum gate operations between these error-correctable logical qubits, which are essential for implementation of any practical quantum computational task, have not been experimentally demonstrated yet. Here we demonstrate a geometric method for realizing controlled-phase gates between two logical qubits encoded in photonic fields stored in cavities. The gates are realized by dispersively coupling an ancillary superconducting qubit to these cavities and driving it to make a cyclic evolution depending on the joint photonic state of the cavities, which produces a conditional geometric phase. We first realize phase gates for photonic qubits with the logical basis states encoded in two quasiorthogonal coherent states, which have important implications for continuous-variable-based quantum computation. Then we use this geometric method to implement a controlled-phase gate between two binomially encoded logical qubits, which have an error-correctable function.
Stabilized cat qubits that possess biased noise channel with bit-flip errors exponentially smaller than phase-flip errors. Together with a set of bias-preserving (BP) gates, cat qubits are a promising candidate for realizing hardware efficient quantum error correction and fault-tolerant quantum computing. Compared to dissipatively stabilized cat qubits, the Kerr cat qubits can in principle support faster gate operations with higher gate fidelity, benefiting from the large energy gap that protects the code space. However, the leakage of the Kerr cats can increase the minor type of errors and compromise the noise bias. Both the fast implementation of gates and the interaction with environment can lead to such detrimental leakage if no sophisticated controls are applied. In this work, we introduce new fine-control techniques to overcome the above obstacles for Kerr cat qubits. To suppress the gate leakage, we use the derivative-based transition suppression technique to design derivative-based controls for the Kerr BP gates. We show that the fine-controlled gates can simultaneously have high gate fidelity and high noise bias and when applied to concatenated quantum error correction, can not only improve the logical error rate but also reduce resource overhead. To suppress the environment-induced leakage, we introduce colored single-photon dissipation, which can continuously cool the Kerr cats and suppress the minor errors while not enhancing the major errors.
We estimate and analyze the error rates and the resource overheads of the repetition cat qubit approach to universal and fault-tolerant quantum computation. The cat qubits stabilized by two-photon dissipation exhibit an extremely biased noise where the bit-flip error rate is exponentially suppressed with the mean number of photons. In a recent work we suggested that the remaining phase-flip error channel could be suppressed using a one-dimensional repetition code. Indeed, using only bias-preserving gates on the cat qubits, it is possible to build a universal set of fault-tolerant logical gates at the level of the repetition cat qubit. In this paper we perform Monte Carlo simulations of all the circuits implementing the protected logical gates, using a circuit-level error model. Furthermore, we analyze two different approaches to implement a fault-tolerant Toffoli gate on repetition cat qubits. These numerical simulations indicate that very low logical error rates could be achieved with a reasonable resource overhead, and with parameters that are within the reach of near-term circuit QED experiments.
Fault tolerant quantum computers greatly benefit from having both high fidelity entangling gates and a non-local qubit connectivity. However, most entangling gates for neutral atoms are local, requiring separated atoms to be moved using optical tweezers to yield entanglement between them. Here, we design a high fidelity Mølmer Sørensen-like entangling gate between neutral atoms coupled to a photonic cavity, enabling a programmable all-to-all connectivity through the shared cavity coupling. The gate is both robust to changes in cavity photon number, and enables selective entanglement between any subset of separated atoms through local laser addressing. This eliminates the need to physically move atoms to perform the gate, and therefore motional errors and erasure are suppressed. Using experimentally realistic parameters we demonstrate gate fidelities exceeding F>0.99F > 0.99F>0.99, with higher fidelities achievable through optimal control of the pump amplitudes.
We present a 1D repetition code based on the so-called cat qubits as a viable approach toward hardwareefficient universal and fault-tolerant quantum computation.The cat qubits that are stabilized by a twophoton driven-dissipative process exhibit a tunable noise bias where the effective bit-flip errors are exponentially suppressed with the average number of photons.We propose a realization of a set of gates on the cat qubits that preserve such a noise bias.Combining these base qubit operations, we build, at the level of the repetition cat qubit, a universal set of fully protected logical gates.This set includes single-qubit preparations and measurements, NOT, controlled-NOT, and controlled-controlled-NOT (Toffoli) gates.Remarkably, this construction avoids the costly magic state preparation, distillation, and injection.Finally, all required operations on the cat qubits could be performed with slight modifications of existing experimental setups.