Nullifiers of non-Gaussian cluster states through homodyne measurement

TL;DR

This work addresses certifying non-Gaussian resources in continuous-variable cluster-state quantum computing by introducing non-Gaussian nullifiers, which extend Gaussian nullifiers through an inverse symplectic transform $M^{-1}$ and Weyl-symmetric operators to access initial mode statistics from homodyne data. The authors derive an explicit non-Gaussian nullifier for photon-subtracted squeezed states, represent it as a finite polynomial in generalized quadratures with a Gaussian threshold $\min_G \mathrm{Tr}[\rho_G O_n] \approx 0.611$, and show how a value below this limit certifies non-Gaussianity. They experimentally demonstrate the approach with heralded photon-subtracted states (

Abstract

In continuous variable optical platforms, large-scale Gaussian cluster states have already been demonstrated, but non-Gaussian resources are essential to achieve universality and fault tolerance in measurement-based quantum computation. However, characterizing and certifying non-Gaussian cluster states remains an outstanding challenge. Here, we introduce a general framework for the characterization of non-Gaussian cluster states based on non-Gaussian nullifiers, extending the widely used Gaussian nullifier concept. We show that these nullifiers can be directly evaluated from homodyne measurement data, making them experimentally accessible. As an illustration, we derive and experimentally demonstrate non-Gaussian nullifiers for photon-subtracted squeezed states. Our results provide a practical and operational tool for certifying quantum non-Gaussianity in large-scale optical cluster states.

Figures (5)

• Figure 1: a Schematic illustration of a cluster state generation via $Cz$ gate and evaluation of the nullifiers $N_1$ and $N_2$\ref{['2null']} from homodyne measurement results $m_{x2},m_{p1}$ and $m_{x1},m_{p2}$, respectively. b Optical implementation using beam splitter and phase shifters. c Graph of the two-node cluster state.
• Figure 2: The value of the non-Gaussian nullifier \ref{['kitsq']} as a function of the quantum efficiency $\eta$ of the detector heralding the photon subtraction and as a function of antisqueezing $V_A$[dB] of the initial state, with squeezing fixed to -2 dB. The Gaussian threshold is indicated by the white stripe.
• Figure 3: Value of the non-Gaussian nullifier for photon-subtracted squeezed state cluster state, which was evaluated with imperfect knowledge of the cluster preparation. The state is prepared by a balanced beam splitter, whereas the nullifier is computed for a beam splitter with transitivity $t' = \frac{1}{\sqrt{2}} + \Delta$. The minimum can be seen when $\Delta=0$ and thus the linear operation in evaluation corresponds to an inverse of a balanced beam splitter. Three cases are shown, an ideal cluster state and then a cluster state created from photon-subtracted squeezed vacua that were subject to $10\%$ and $20\%$ losses.
• Figure 4: Mean value and standard deviation of the non-Gaussian nullifier estimated from a given number of measurements per homodyne phase simulated from statistics of an ideal photon-subtracted state. The red line at $0.611$ depicts the Gaussian minimum of the nullifier. Red points correspond to initial squeezing given by $r=0.1$, blue points correspond to $r=0.2$.
• Figure 5: Wigner functions of the experimentally prepared single photon state and squeezed single photon state.