The early Gottesman, Kitaev, and Preskill (GKP) proposal for encoding a qubit in an oscillator has recently been followed by cat- and binomial-code proposals. Numerically optimized codes have also been proposed, and we introduce codes of this type here. These codes have yet to be compared using the same error model; we provide such a comparison by determining the entanglement fidelity of all codes with respect to the bosonic pure-loss channel (i.e., photon loss) after the optimal recovery operation. We then compare achievable communication rates of the combined encoding-error-recovery channel by calculating the channel's hashing bound for each code. Cat and binomial codes perform similarly, with binomial codes outperforming cat codes at small loss rates. Despite not being designed to protect against the pure-loss channel, GKP codes significantly outperform all other codes for most values of the loss rate. We show that the performance of GKP and some binomial codes increases monotonically with increasing average photon number of the codes. In order to corroborate our numerical evidence of the cat-binomial-GKP order of performance occurring at small loss rates, we analytically evaluate the quantum error-correction conditions of those codes. For GKP codes, we find an essential singularity in the entanglement fidelity in the limit of vanishing loss rate. In addition to comparing the codes, we draw parallels between binomial codes and discrete-variable systems. First, we characterize one- and two-mode binomial as well as multiqubit permutation-invariant codes in terms of spin-coherent states. Such a characterization allows us to introduce check operators and error-correction procedures for binomial codes. Second, we introduce a generalization of spin-coherent states, extending our characterization to qudit binomial codes and yielding a multiqudit code.@en

We analyze the performance of decoders for the 2D and 4D toric code which are local by construction. The 2D decoder is a cellular automaton decoder formulated by Harrington [1] which explicitly has a finite speed of communication and computation. For a model of independent X and Z errors and faulty syndrome measurements with identical probability, we report a threshold of 0:133% for this Harrington decoder. We implement a decoder for the 4D toric code which is based on a decoder by Hastings [2]. Incorporating a method for handling faulty syndromes we estimate a threshold of 1:59% for the same noise model as in the 2D case. We compare the performance of this decoder with a decoder based on a 4D version of Toom’s cellular automaton rule as well as the decoding method suggested by Dennis et al. [3].@en

We show that one can determine both parameters of a displacement acting on an oscillator with an accuracy which scales inversely with the square root of the number of photons in the oscillator. Our results are obtained by using a grid state as a sensor state for detecting small translations in phase space (displacements). Grid states were first proposed [D. Gottesman, Phys. Rev. A 64, 012310 (2001)PLRAAN1050-294710.1103/PhysRevA.64.012310] for encoding a qubit into an oscillator: an efficient preparation protocol of such states, using a coupling to a qubit, was later developed [B. M. Terhal and D. Weigand, Phys. Rev. A 93, 012315 (2016)1050-294710.1103/PhysRevA.93.012315]. We compare the performance of the grid state with the quantum compass or cat code state and place our results in the context of the two-parameter quantum Cramér-Rao lower bound on the variances of the displacement parameters. We show that the accessible information about the displacement for a grid state increases with the number of photons in the state when we measure and prepare the state using a phase estimation protocol. This is in contrast with the accessible information in the quantum compass state which we show is always upper bounded by a constant, independent of the number of photons. We present numerical simulations of a phase estimation based preparation protocol of a grid state in the presence of photon loss, nonlinearities, and qubit measurement, using no post-selection, showing how the two effective squeezing parameters which characterize the grid state change during the preparation. The idea behind the phase estimation protocol is a simple maximal-information gain strategy.@en