Nonclassical nullifiers for quantum hypergraph states

Abstract

Quantum hypergraph states form a generalisation of the graph state formalism that goes beyond the pairwise (dyadic) interactions imposed by remaining inside the Gaussian approximation. Networks of such states are able to achieve universality for continuous variable measurement based quantum computation with only Gaussian measurements. For normalised states, the simplest hypergraph states are formed from $k$-adic interactions among a collection of $k$ harmonic oscillator ground states. However such powerful resources have not yet been observed in experiments and their robustness and scalability have not been tested. Here we develop and analyse necessary criteria for hypergraph nonclassicality based on simultaneous nonlinear squeezing in the nullifiers of hypergraph states. We put forward an essential analysis of their robustness to realistic scenarios involving thermalisation or loss and suggest several basic proof-of-principle options for experiments to observe nonclassicality in hypergraph states.

Figures (8)

• Figure 1: The nonclassical nullifiers of the dyadic (Gaussian cluster state), triadic and tetradic hypergraph states with no initial squeezing ($r=0$) (blue), with 3 dB momentum squeezing (yellow), with loss (dashed, $T_{\text{Dyad}}=0$, $T_{\text{Triad}}=0.46$, and $T_{\text{Tetrad}}=0.73$), or with thermalisation (dotted, $\bar{n}_{\text{Dyad}}=0.38$, $\bar{n}_{\text{Triad}}=0.16$ and $\bar{n}_{\text{Tetrad}}=0.09$). Each case has nonlinear strength $\gamma=1$, corresponding to the basic unweighted hypergraph states and highlighted by the vertical line. Hypergraph nonclassicality is evidenced by the squeezing of $\mathcal{N}_i$ below the level of the ground state (black). Furthermore the nullifier nonclassicality is not due to the initial Gaussian squeezing of the states, as the nonlinear squeezing goes below the variance achievable with local squeezing (dashed black). For pure states without loss or thermalisation (solid), initial squeezing in momentum always increases the nonlinear squeezing of the nullifier. Naturally, greater numbers of initially squeezed states are required for higher numbers of modes, $k$. However, note the different behaviour of the nonlinear squeezing for different numbers of modes $k$ as well as the type of noise process. In each case the amount of noise present is chosen to reflect the depth for the unweighted hypergraph state with initial 3 dB momentum squeezing. The dyad (Gaussian cluster state) remains nonclassical with arbitrary loss, in contrast to the increasing sensitivity of the non-Gaussian triad and tetrad. Furthermore for the hypergraph states loss and thermalisation counteract initial squeezing in momentum such that the initial squeezing must be optimised with respect to the hypergraph weight $\gamma$ and the type and strength of the loss or thermalisation. This can be seen in the insets where the fixed initial squeezing results in greater sensitivity to loss (yellow above blue).
• Figure 2: The minimal required interaction strength $\gamma$ to reach nonclassical nullifiers, aided by initial squeezing. Nullifier variances at $\lambda=\gamma$ for loss $T=0.85$ (dashed curves) and thermalisation $\bar{n}=0.05$ (dotted curves), for position squeezing (red), momentum squeezing (yellow), or no squeezing (blue). Nullifier variances below the ground state variance (black) and locally squeezed states threshold (dashed black) indicate nonclassicality. For the Gaussian dyad momentum squeezing is always beneficial, and position squeezing always harmful, regardless of loss or thermalisation. In distinct contrast, the response to loss and thermalisation for the non-Gaussian triad and tetrad depends on $\gamma$. As $\gamma$ increases the rate at which the nonlinear squeezing decreases depends on the kind of initial linear squeezing. For low $\gamma$, including the unweighted hypergraph state $\gamma=1$, momentum squeezing increases tolerance to loss and thermalisation. For larger $\gamma$ this behaviour transitions to a negative effect, where nonclassicality is lost faster than with no squeezing. Instead in this regime position squeezing increases the tolerance to loss and thermalisation, despite having lower nonlinear squeezing at the level of the pure states.
• Figure 3: Nonclassicality depths of hypergraph states. Greatest amount of loss $T_{Max}$ (full plot markers) and greatest thermal occupation number $\bar{n}_{Max}$ (empty plot markers) still showing hypergraph nonclassicality against hypergraph order (number of modes $k\ge3$) for momentum (yellow), position (red) squeezing or no (blue) squeezing, and optimal nonlinear strength $\gamma$. The extra robustness granted due to initial squeezing decreases as the number of modes increases.
• Figure 4: Maximum initial thermal noise $\Bar{n}$ before the hypergraph nonclassicality cannot be distinguished from Gaussian squeezing for the triad. Thermal noise can always be dealt with by increasing momentum squeezing or increasing hypergraph weight $\gamma$.
• Figure 5: Nullifier variances at $\lambda=\gamma$ for loss $T=0.85$ (dashed curves) and thermalisation $\bar{n}=0.05$ (dotted curves), for position squeezing (red), momentum squeezing (yellow), or no squeezing (blue). Nullifier variances below the ground state variance (black) and squeezed state threshold (dashed black) indicate nonclassicality. For the Gaussian sheared state (quadratic) momentum squeezing is always beneficial, and position squeezing always harmful, regardless of loss or thermalisation. The response to loss and thermalisation for the non-Gaussian cubic and quartic phase states depends on $\gamma$ in a similar fashion to that of the hypergraph states in Fig \ref{['LossATherm']}. Additionally, for the cubic and quartic phase states the squeezed state limit is in fact a quantum non-Gaussianity criterion moore_hierarchy_2022.