Attributed-graphs kernel implementation using local detuning of neutral-atoms Rydberg Hamiltonian

Abstract

We extend the quantum-feature kernel framework, which relies on measurements of graph-dependent observables, along three directions. First, leveraging neutral-atom quantum processing units (QPUs), we introduce a scheme that incorporates attributed graphs by embedding edge features into atomic positions and node features into local detuning fields of a Rydberg Hamiltonian. We demonstrate both theoretically and empirically that local detuning enhances kernel expressiveness. Second, in addition to the existing quantum evolution kernel (QEK), which uses global observables, we propose the generalized-distance quantum-correlation (GDQC) kernel, based on local observables. While the two kernels show comparable performance, we show that GDQC can achieve higher expressiveness. Third, instead of restricting to observables at single time steps, we combine information from multiple stages of the quantum evolution via pooling operations. Using extensive simulations on two molecular benchmark datasets, MUTAG and PTC\_FM, we find: (a) QEK and GDQC perform competitively with leading classical algorithms; and (b) pooling further improves performance, enabling quantum-feature kernels to surpass classical baselines. These results show that node-feature embedding and kernel designs based on local observables advance quantum-enhanced graph machine learning on neutral-atom devices.

Figures (4)

• Figure 1: Emulation results for the molecule in panel (a). Panels (b)-(d) display the time evolution of the three observables defined in Eq. (\ref{['eq:observables']}): the excitation probability $P_k$, the site occupations $n_i = \langle \hat{n}_i \rangle$, and the pairwise correlations $C_{i,j} = \langle \hat{n}_i \hat{n}_j \rangle$. Solid (dashed) curves correspond to evolution under the global-detuning (local-detuning) Hamiltonian, with detuning scale set to $\delta_{0}=2\pi\,\mathrm{rad}\,/\mathrm{\mu s}$; see Eqs. (\ref{['eq:H_global']}) and (\ref{['eq:H_local']}). For comparison, the inset in (d) shows results for a smaller detuning scale, $\delta_{0}=\pi\,\mathrm{rad}\,/\mathrm{\mu s}$. For the molecule and parameters shown, differences between global and local detuning become apparent by $t \gtrsim 250 \, \mathrm{ns}$. The molecule in (a) is taken from the PTC_FM$^*$ dataset and has SMILES string "CC1(C)COC1=O". The gray dotted vertical line marks $t=0.4\,\mathrm{\mu s}$, which is the time step used for the visualization in Fig. \ref{['fig:GDQC_illustration']}.
• Figure 2: Illustration for the construction of the generalized-distance quantum-correlation feature vector $\boldsymbol{\chi}\left[C^{\mathcal{G}}\left(t\right),D^{\mathcal{G}}\right]$ according to the algorithm \ref{['algo:GDQC']} for the indicated graph of size $N=7$. The figure shows the measured correlation $C^{\mathcal{G}}_{i,j}$ (upper bar) together with the respective distance measure $d_{\mathcal{G}}(i,j)$ (lower bar) for each of the $N^2=49$ node pairs $\left(i,j \right)$. Note the symmetries $C^{\mathcal{G}}_{j,i}=C^{\mathcal{G}}_{i,j}$ and $d_{\mathcal{G}}(j,i)=d_{\mathcal{G}}(i,j)$. Also, notice how the pairs that are at distance 1 showcase very low values due to the blockade phenomenon. Having chosen $N_{\mathrm{bins}}^C = 5$ correlation bins, the node pairs $\left(i,j \right)$ are binned according to their correlation-matrix values (indicated by the coloring), leading to an intermediate correlation-matrix binning vector $\boldsymbol{\chi}^C_{\mathrm{bin}} =\left( 36, 10, 2, 1, 0\right)^\intercal$. Next, the node pairs are further binned according to the graph distance $d_{\mathcal{G}}(i,j)$. For example, the 36 elements from the interval $[0, 0.2]$ (faint red values) are redistributed according to the four involved graph distance values, leading to a sequence $\{2, 14, 14, 6\}$. All bins combined, this procedure generates the $(N_{\mathrm{bins}}^C N_{\mathrm{bins}}^D)$-dimensional $\mathrm{GDQC}$ feature vector $\boldsymbol{\chi}$ which, in the present example becomes $\boldsymbol{\chi} = (2, 2, 2, 1, 0, 14, 0, 0, 0, 0, 14, 4, 0, 0, 0, 6, 4, 0, 0, 0)^\intercal /\sqrt{473}$. The presented correlation matrix $C^{\mathcal{G}}_{i,j}=C^{\mathcal{G}}_{i,j}\left(t = 0.4\, \mathrm{\mu s}\right)$ corresponds to the time step indicated in Fig. \ref{['fig:dynamics']}.
• Figure 3: Weighted F1 scores associated with SVMs trained on the quantum-feature kernels, $\mathrm{QEK}$ (\ref{['eq:QEK_kernel']}) and $\mathrm{GDQC}$ (\ref{['eq:GDQC_t']}), for the two datasets, (a) MUTAG and (b) PTC_FM$^*$. The superscripts $\mathrm{(loc)}$ and $\mathrm{(glob)}$ refer to whether the governing Hamiltonian has node features embedded [$\hat{\mathcal{H}}_{\mathrm{loc}}^{\mathcal{G}}$ (\ref{['eq:H_local']})] or not [$\hat{\mathcal{H}}_{\mathrm{glob}}^{\mathcal{G}}$ (\ref{['eq:H_global']})]. The training and evaluation procedure is separately conducted at each time steps across the time evolution. For comparison, the corresponding randomly stratified baseline, where each class is predicted proportionally to its frequency in the dataset are 0.56 for MUTAG and 0.53 for PTC_FM$^*$, significantly below the presented F1 scores. The $\mathrm{GDQC}$ results are derived for a number of bins $N_{\mathrm{bins}}^C = 10$. The light blue and red horizontal dashed lines correspond to the classical early-time baselines for $\mathrm{QEK}$ [see Eq. (\ref{['eq:P_vector_early_time_approx']})] and $\mathrm{GDQC}$ [see Eq. (\ref{['eq:chi_bin_classical']})], respectively.
• Figure 4: Relative rank of the Gram matrix $\mathrm{rank}(K)/N_\mathrm{data}$ (\ref{['eq:Gram_matrix']}) associated with the $\mathrm{GDQC}$ kernel $\kappa_{\mathrm{GDQC}}$ (\ref{['eq:GDQC_t']}) versus the F1 score of the corresponding kernel-based SVM for varying binning number $N_{\mathrm{bins}}^{C}$. The quantities are plotted as a function of the time steps across the entire time evolution with $t_\mathrm{max}=1\, \mathrm{\mu s}$. (a)-(d) corresponds to the dataset MUTAG; (e)-(h) corresponds to PTC_FM$^*$. The identifier $\mathrm{(loc)}$ [$\mathrm{(glob)}$] imply that the involved time evolution is governed by the Hamiltonian $\hat{\mathcal{H}}_{\mathrm{loc}}^{\mathcal{G}}$ (\ref{['eq:H_local']}) [$\hat{\mathcal{H}}_{\mathrm{glob}}^{\mathcal{G}}$ (\ref{['eq:H_global']})], whereby $\hat{\mathcal{H}}_{\mathrm{loc}}^{\mathcal{G}}$ has node features embedded. For comparison, the black dashed curve with x-markers represents the data associated with the $\mathrm{QEK}$ kernel $\kappa_{\mathrm{QEK}}$ (\ref{['eq:QEK_kernel']}), which does not depend on binning. The horizontal gray dotted line corresponds to the theoretical relative rank value computed from the number of distinct graph-isomorphism classes.

Theorems & Definitions (9)

• Proposition 1
• Proposition 2
• Proposition 3
• Proposition 4
• proof
• proof
• Lemma 5
• proof
• proof