Quantum-Enhanced Deterministic Inference of $k$-Independent Set Instances on Neutral Atom Arrays

TL;DR

This work introduces deterministic error mitigation (DEM) to translate noisy quantum optimization outputs into comparable, cost-aware benchmarks. By modeling measurement noise as a Hamming-shell around the ideal configuration, the authors derive an entropy-controlled postprocessing cost $T(N,p) \sim 2^{N H_2(p)}$ and extend it to asymmetric SPAM via an effective rate $p_{\mathrm{eff}}$. Experimental validation on neutral-atom MIS experiments shows that DEM costs scale with $H_2(p_{\mathrm{eff}})$ and can be lower than classical baselines, enabling hardware-vs-classical comparisons with hundreds of atoms at current error rates. Tarjan-DEM further corroborates the entropy-driven scaling under pruning, emphasizing that the leading cost is set by the noise-induced search space rather than graph geometry. Overall, DEM provides a rigorous, hardware-relevant framework for benchmarking quantum optimization against classical algorithms by normalizing solution quality and classical inference cost.

Abstract

Noisy quantum annealing experiments on Rydberg atom arrays produce measurement outcomes that deviate from ideal distributions, complicating performance evaluation. To enable a data-driven benchmarking methodology for quantum devices that accounts for both solution quality and the classical computational cost of inference from noisy measurements, we introduce deterministic error mitigation (DEM), a shot-level inference procedure informed by experimentally characterized noise. We demonstrate this approach using the decision version of the $k$-independent set problem. Within a Hamming-shell framework, the DEM candidate volume is governed by the binary entropy of the bit-flip error rate, yielding an entropy-controlled classical postprocessing cost. Using experimental measurement data, DEM reduces postprocessing overhead relative to classical inference baselines. Numerical simulations and experimental results from neutral atom devices validate the predicted scaling with system size and error rate. These scalings indicate that one hour of classical computation on an Intel i9 processor corresponds to neutral atom experiments with up to $N=250-450$ atoms at effective error rates, enabling a direct, cost-based comparison between noisy quantum experiments and classical algorithms.

Figures (4)

• Figure 1: (a) Projective measurements of a Rydberg-atom MIS experiment produce a defective bitstring $z$ that may violate the independence constraint. Deterministic error mitigation (DEM) searches candidate bitstrings within increasing Hamming-distance shells around $z$. At distance $i$, candidate $w$ are generated by flipping $i$ bits of $z$ and checked for independence and target size $k$. For a $3\times3$ ($N=9$ and $k=5$), the measured bitstring $z=101\,001\,101$, yields a valid solution $w = 101\,010\,101$, returning a YES instance of the $k$-independent set decision problem. (b) Geometrically, DEM expands Hamming shells around $z$, enlarging the candidate volume by $\binom{9}{1}=9$ at $i=1$ and $\binom{9}{2}=36$ at $i=2$, until a valid $k$-independent set is found.
• Figure 2: (a) Geometry of experimentally realized $L\times L$ square Rydberg atom arrays with spacing $5.4(1)$$\mu$m and blockade radius $R_b=7.2(1)$$\mu$m. The MIS experiment uses a constant Rabi frequency $\Omega=1.0$ MHz and a linear detuning sweep from $\Delta_i=-2.0$ MHz to $\Delta_f=+1.0$ MHz over $T=3.0~\mu$s. (b) Measured DEM postprocessing cost versus system size $N=L^2$. Colored markers show costs from experimental data (Pasqal Fresnel, QuEra Aquila), grey markers the classical baseline, and solid lines the theoretical cost prediction from calibrated bit-flip error rates. (c) Emulated DEM postprocessing cost versus asymmetric bit-flip error probabilities $p_{01}$ and $p_{10}$ for $4\times 4$ and $4 \times 6$ lattices. Box plots show numerical emulation results, with solid lines indicating the analytic cost model $T(N,p_{01},p_{10})$.
• Figure 3: (a) Operation counts for BF-DEM and Tarjan-DEM versus graph size $N$, measured on Pasqal's Fresnel device. Tarjan-DEM (light gray markers and line) shows substantially improved scaling compared to BF-DEM. (b) The empirical exponent base $c$, defined by $\mathrm{Nodes} \propto c^N$, is plotted as a function of the effective error rate $p$ for 2D square lattices ($L\times L$ with $L=4,\dots,8$). The navy dashed curve indicates the BF-DEM reference, $c_{\mathrm{BF}}(p)=2^{H_2(p)}$; the black dashed curve shows the pruning-aware interpolation $c_{\mathrm{th}}(p)=c_0^{H_2(p)}$ with $c_0 \equiv 2^{1/3}$. The gray solid line is the fitted Tarjan-DEM base $c_{\mathrm{emp}}(p)$ extracted from the branch-and-reduce search tree.
• Figure 4: (a) DEM operation cost versus symmetric bit-flip error rate $p$ for MIS instances on square lattices at different filling fractions (FF), realized by embedding a fixed $16$-vertex subgraph into larger lattices. Box charts show emulated operation costs, and solid lines show theoretical predictions. (b) DEM cost for King's-lattice graphs at different filling fractions. The same dependence on $p$ persists, indicating that DEM cost is governed primarily by the effective error rate rather than detailed graph geometry.