David F. Locher

Bosonic quantum error correction codes encode logical qubits in the Hilbert space of one or multiple harmonic oscillators. A prominent class of bosonic codes is that of Gottesman-Kitaev-Preskill (GKP) codes of which implementations have been demonstrated with trapped ions and microwave cavities. In this paper, we investigate theoretically the preparation and error correction of a GKP qubit in a vibrational mode of a neutral atom stored in an optical dipole trap. This platform has recently shown remarkable progress in simultaneously controlling the motional and electronic degrees of freedom of trapped atoms. The protocols we develop make use of motional states and, additionally, internal electronic states of the trapped atom to serve as an ancilla qubit. We compare optical tweezer arrays and optical lattices and find that the latter provide more flexible control over the confinement in the out-of-plane direction, which can be utilized to optimize the conditions for the implementation of GKP codes. Concretely, the different frequency scales that the harmonic oscillators in the axial and radial lattice directions exhibit and a small oscillator anharmonicity prove to be beneficial for robust encodings of GKP states. Finally, we underpin the experimental feasibility of the proposed protocols by numerically simulating the preparation of GKP qubits in an optical lattice with realistic parameters.

Logical qubits can be protected from decoherence by performing QEC cycles repeatedly. Algorithms for fault-tolerant QEC must be compiled to the specific hardware platform under consideration in order to practically realize a quantum memory that operates for in principle arbitrary long times. All circuit components must be assumed as noisy unless specific assumptions about the form of the noise are made. Modern QEC schemes are challenging to implement experimentally in physical architectures where in-sequence measurements and feed-forward of classical information cannot be reliably executed fast enough or even at all. Here we provide a novel scheme to perform QEC cycles without the need of measuring qubits that is fully fault-tolerant with respect to all components used in the circuit. Our scheme can be used for any low-distance CSS code since its only requirement towards the underlying code is a transversal CNOT gate. Similarly to Steane-type EC, we coherently copy errors to a logical auxiliary qubit but then apply a coherent feedback operation from the auxiliary system to the logical data qubit. The logical auxiliary qubit is prepared fault-tolerantly without measurements, too. We benchmark logical failure rates of the scheme in comparison to a flag-qubit based EC cycle. We map out a parameter region where our scheme is feasible and estimate physical error rates necessary to achieve the break-even point of beneficial QEC with our scheme. We outline how our scheme could be implemented in ion traps and with neutral atoms in a tweezer array. For recently demonstrated capabilities of atom shuttling and native multi-atom Rydberg gates, we achieve moderate circuit depths and beneficial performance of our scheme while not breaking fault tolerance. These results thereby enable practical fault-tolerant QEC in hardware architectures that do not support mid-circuit measurements.