Exact simulation of realistic Gottesman-Kitaev-Preskill cluster states

Abstract

We describe a method for simulating and characterizing realistic Gottesman-Kitaev-Preskill (GKP) cluster states, rooted in the representation of resource states in terms of sums of Gaussian distributions in phase space. We apply our method to study the generation of single-mode GKP states via cat state breeding, and the formation of multimode GKP cluster states via linear optical circuits and homodyne measurements. We characterize resource states by referring to expectation values of their stabilizers, and witness operators constructed from them. Our method reproduces the results of standard Fock-basis simulations, while being more efficient, and being applicable in a broader parameter space. We also comment on the validity of the heuristic Gaussian random noise (GRN) model, through comparisons with our exact simulations: We find discrepancies in the stabilizer expectation values when homodyne measurement is involved in cluster state preparation, yet we find a close agreement between the two approaches on average.

Figures (13)

• Figure 1: Sketch of $\mathcal{M}=3$ rounds of a cat breeding protocol. The scheme involves interfering squeezed cat states at a balanced beamsplitter, with one of the outputs subjected to homodyne detection (here we have taken the outcome $p=0$). The unmeasured mode becomes a closer approximation to a GKP state with each round of breeding. Details can be found in Ref. Vasconcelos:10catbreed_PhysRevA.97.022341.
• Figure 2: Implementation of a GKP CZ operation through static linear components. $\hat{R}(\theta)$ denotes a $\theta$ phase shift, and the arrow represents a beamsplitter, following the notation used in staticLO_PRXQuantum.2.040353.
• Figure 3: Sketch of the cluster state generation circuits we consider in this manuscript: $M+N$ single-mode GKP states are sent through a linear unitary circuit (box labelled $U$). $M$ of the output modes are then subjected to homodyne measurement to yield a $N$-mode cluster state.
• Figure 4: Single-qubit stabilizer EVs for a single-mode GKP state, as a function of the parameters of the cat breeding protocol used to generate it. The stabilizer EVs increase as the squeezing of the input cat states and/or the number of iterations of cat state breeding are increased, reflecting the increasing quality of the GKP state.
• Figure 5: Wigner functions of approximate GKP states generated by the breeding protocol described in Section \ref{['section:sumGauss']}, for various $\mathcal{M}$ and $\xi$ (number of breeding iterations and cat squeezing, respectively).

Theorems & Definitions (2)

• proof
• proof