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arXiv:1907.12487 · Nature

Quantum error correction of a qubit encoded in grid states of an oscillator

Philippe Campagne-Ibarcq, Alec Eickbusch, Steven Touzard, E. Zalys-Geller, Nicholas Frattini, Volodymyr Sivak, Philip Reinhold, Shruti Puri, Shyam Shankar, Robert J. Schoelkopf, Luigi Frunzio, Mazyar Mirrahimi, Michel Devoret

Published Aug 19, 2020• 456 citations
arXivHTMLPDF
DOI
10.1038/s41586-020-2603-3

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Quantum error correction of a grid-state qubit with state preparation and measurement errors below 10−310^{-3}10−3
Grid state qubits offer a hardware-efficient approach to large-scale fault-tolerant quantum computing. They access the information redundancy required for quantum error correction by exploiting the large Hilbert space naturally available in harmonic oscillators. Superconducting architectures are particularly suitable to implement grid state qubits due to their fast and high-fidelity operations. Grid states in superconducting circuits enable quantum error correction (QEC) with performance beyond break-even. However, the state preparation and measurements (SPAM) errors of grid states has been a significant limitation to computational performances. In this work, we leverage high-performance QEC to enable repeat-until-success state preparation of both cardinal and magic states of the single-mode grid-state qubit. We combine this with an improved measurement protocol that corrects for both finite-energy envelope and auxiliary qubit readout errors, and increases robustness to photon loss. Our experiments, using both techniques, achieve a combined state-preparation and measurement error below 10−310^{-3}10−3. This represents two orders-of-magnitude improvement over the state of the art, bringing this platform on par with standard SPAM error levels measured in transmon qubits.
Bosonic quantum error correction enables hardware-efficient protection of quantum information by encoding logical qubits in harmonic oscillators. Bosonic grid states, such as Gottesman-Kitaev-Preskill (GKP) states, are particularly promising due to their potential to correct small displacements and boson loss. However, their generation remains challenging, typically relying on probabilistic protocols or auxiliary qubit systems. Here, we propose deterministic protocols for generating bosonic grid states using programmable nonlinear bosonic circuits composed solely of squeezing, displacement, and Kerr operations. We show that aiming to enforce GKP symmetries in the output of these circuits yields states with competitive performance with respect to current realizations, but whose quality saturates with increasing circuit depth due to imperfect symmetry restoration. Instead, we find that these bosonic circuits naturally give rise to a distinct class of states, that we label as phased-comb states, which are unitarily related to standard grid states but feature an intrinsic phase structure. We demonstrate that these states define a scalable bosonic quantum error-correcting code with near-optimal performance under boson loss comparable to that of approximate GKP states. We further analyze their logical operations and show how to implement a universal gate set for them. Our results establish programmable nonlinear bosonic circuits as a viable route towards the generation of scalable bosonic quantum error-correcting states beyond standard GKP encodings.
Encoding logical quantum information in harmonic oscillator modes is a promising and hardwareefficient approach to the realization of a quantum computer.In this work, we propose to encode logical qubits in grid states of an ensemble of harmonic oscillator modes.We first discuss general results about these multimode bosonic codes; how to design them, how to practically implement them in different experimental platforms, and how lattice symmetries can be leveraged to perform logical non-Clifford operations.We then introduce in detail two two-mode grid codes based on the hypercubic and D 4 lattices, respectively, showing how to perform a universal set of logical operations.We demonstrate numerically that multimode grid codes have, compared to their single-mode counterpart, increased robustness against propagation of errors from ancillas used for error correction.Finally, we highlight some interesting links between multidimensional lattices and single-mode grid codes concatenated with qubit codes.