A New Hybrid Cat Qubit Combines Speed and Resilience for Improved Quantum Computers
Ronan Gautier, QUANTIC team Inria & Jérémie Guillaud, Alice&Bob
Recently, self-correcting qubits have been actively investigated for their promise of much simpler hardware for fault-tolerant universal quantum computation. In particular, cat qubits are expected to drastically reduce the overheads required for error correction thanks to their promise of an exponential qubit lifetime.
In the recent years, two main approaches have been developed to realize cat qubits. The first approach, introduced by Alice&Bob scientific advisor Mazyar Mirrahimi and co-workers in , is based on engineered dissipation and robustly suppresses bit errors in the presence of a broad range of perturbations. It also features a universal set of quantum gates that preserve the typical error bias of cat qubits (see the previous work by our team ), but their physical implementations require processes with limited speed which induce significant phase errors. While these errors can be suppressed through concatenation with a repetition code, their important rates lead to challenging requirements for the ratio between the engineered dissipation and the noise rates.
The second approach is built on the Kerr nonlinearity induced by Josephson junctions  and features fast gate implementations thanks to its conservative cat qubit confinement [5,6]. A recent paper however showed how Kerr-based cat qubits are thoroughly limited by thermal noise leading to leakage of the cat qubit, and proposed a solution to overcome this difficulty based on a colored cavity dissipation . Although promising, this solution requires careful experimental engineering of the bath beyond the routine Purcell filters, and does not appear to attain the same exponential error bias as dissipative cat qubits.
“In our latest work, we show how to combine dissipative and Hamiltonian cat qubit confinement to make the most out of each approach” says Alain Sarlette.
In their latest paper, the QUANTIC team continue to explore this principle of a combined dissipative and Hamiltonian confinement of cat qubits, but using the engineered dissipation approach. The first motivation of the paper is to show how the combination of Kerr confinement with dissipative schemes systematically features limited performance, either in the fidelity of quantum gates or in the qubit error bias. This limitation is due to the structure of the Kerr Hamiltonian energy spectrum that features high energy differences between subsequent states of even and odd photon number parities. Such states dephase with respect to each other inducing bit errors, unless additional dissipative schemes at least as strong as the Kerr nonlinearity can counteract this dephasing.
The second motivation of the paper is to introduce a new cat qubit confinement scheme coined the Two-Photon Exchange (TPE). It is based on a coupling between the cat qubit resonator and a two-level buffer mode, and, similarly as for Kerr-based cat qubits, it features a very stable steady state given by the cat qubit code-space. Contrary to their Kerr counterparts, TPE-based cat qubits feature a bounded distinction between states of opposite parities, and can thus combine extremely well with engineered dissipation. Such a combined scheme is also implementable experimentally using only minor modification of the cat qubit experiment of 2019 (from the co-founder and CTO of Alice&Bob)  thanks to the similarities between TPE and engineered dissipation.
With this combined Hamiltonian and dissipative scheme, the authors show improved performances of certain single-qubit gates and of two-qubit CNOT gates. While the CNOT gate speed and phase error probability can be ideally improved by up to two orders of magnitude, an improvement by a factor of 5 to 10 appears within the reach of the near-term experiments. Importantly, the combined scheme also offers a robust bit suppression against various perturbations with a similar performance as purely dissipative cat qubits.
As Mazyar Mirrahimi explains: “Overall, combining the Hamiltonian and dissipative confinements feature numerous advantages, and results in a cat qubit protected against a wide range of noise sources, yet compatible with fast gates”.
Because of its increased robustness to noise and fast gate operation, this new hybrid cat qubit is a leading candidate to build a logical qubit by concatenation with a repetition code. Thus, this new scheme could play an important role in the race towards fully functional quantum error correction and large scale quantum computation.
 Gautier, R., Sarlette, A., Mirrahimi, M. “Combined Dissipative and Hamiltonian Confinement of Cat Qubits.” arXiv preprint arXiv:2112.05545 (2021).
 Mirrahimi, M., et al. “Dynamically protected cat-qubits: a new paradigm for universal quantum computation.” New Journal of Physics 16 (2014): 045014.
 Guillaud, J., et al. “Repetition cat qubits for fault-tolerant quantum computation.” Physical Review X 9 (2019): 041053.
 Puri, S. et al., “Engineering the quantum states of light in a Kerr-nonlinear resonator by two-photon driving”, npj Quantum Information 3, 18 (2017).
 Puri, S., et al. “Bias-preserving gates with stabilized cat qubits.” Science advances 6.34 (2020): eaay5901.
 Xu, Q., et al. “Engineering fast bias-preserving gates on stabilized cat qubits.” arXiv preprint arXiv:2105.13908 (2021).
 Putterman, H., et al. “Colored Kerr cat qubits.” arXiv preprint arXiv:2107.09198 (2021).
 Lescanne, R., et al. “Exponential suppression of bit-flips in a qubit encoded in an oscillator.” Nature Physics 16.5 (2020): 509-513.