Nonlocal Games as Cross-Platform Quantum Benchmarks: Exceeding unconditional classical bounds on trapped-ion processors

Abstract

Nonlocal games provide application-level benchmarks for quantum hardware whose classical performance bounds are information-theoretic, holding against all classical strategies regardless of computational resources. We implement a 14-vertex graph coloring game, the smallest graph exhibiting a quantum-classical separation for this game type, on four trapped-ion quantum processors across three institutions. One system achieved a win rate that surpasses the classical bound with statistical significance, marking the first violation of a classical bound in a graph coloring nonlocal game on quantum hardware. The remaining systems achieved win rates comparable to the best superconducting processors evaluated on the same game, further illustrating the potential of nonlocal games as cross-architecture quantum benchmarks.

Figures (3)

• Figure 1: (a) Schematic of a nonlocal game. Players A and B each control two qubits. A shared entangled state is prepared (purple, "State Prep."), after which the players are separated and each receives a question (a vertex or edge of $G_{14}$) from a referee. Each player applies a question-dependent measurement unitary ($\hat{U}_A$, $\hat{U}_B$) and measures; the two-bit outcomes are interpreted as color assignments, and the referee evaluates whether the coloring rule is satisfied. (b) Transpiled circuit for the $G_{14}$ game on four qubits. The purple-shaded region prepares two Bell pairs using two Mølmer-Sørensen (MS) gates (green). The orange-shaded region implements the question-dependent measurement unitaries, with one MS gate per player (4 MS gates total per circuit). Gray boxes are single-qubit rotations. One such circuit exists for each of the 51 game questions (14 vertices + 37 edges). (c) State-dependent fluorescence detection. Resonant laser light (cyan) drives a cycling transition on ions in $\ket{1}$, producing scattered photons collected by a multi-channel photomultiplier tube (PMT). Bright (dark) ions correspond to $\ket{1}$ ($\ket{0}$). (d) Coherent gate operations via stimulated Raman transitions. A global beam (pink, from above) illuminates all ions, while individual addressing beams pass through a multi-channel acousto-optical modulator (AOM) and are focused onto single ions, enabling independent phase, frequency, and amplitude control for single- and two-qubit gates.
• Figure 2: Win rates for the $G_{14}$ game on four trapped-ion systems. Win rates are computed using the weighted average in Eq. \ref{['eq:weighted_winrate']}. Error bars denote 95% confidence intervals computed using concentration inequalities (see Supplementary Note 3). The Blue system achieved $\omega = 0.982(3)$, exceeding the classical limit $\omega_c = 86/88 \approx 0.977$. Solid bars show raw win rates without error mitigation. The dashed bar shows the Silver win rate after SPAM correction via the method in Ref. shen2012correcting, improving from $0.952(3)$ to $0.969$ (see Section \ref{['subsec:error_sources']}).
• Figure 3: Comparison of $G_{14}$ benchmarking results across quantum hardware platforms. Prior experiments on superconducting devices (circles) are from Ref. furches2025application, with colored points indicating comparable circuit depth to this work. Trapped-ion results from this study (triangles) include systems at Duke University (Blue, Gold), the University of Maryland (Silver), and IonQ (Aria). The gray diagonal line marks $\omega_e = \omega_v$; all devices fall above this line, reflecting the asymmetric game rules that make edge questions more noise-robust than vertex questions. The purple curve shows the locus of $(\omega_v, \omega_e)$ pairs satisfying $\omega = \omega_c = 86/88$ under the weighted average in Eq. \ref{['eq:weighted_winrate']}; the shaded region above it is the quantum advantage region. Only the Blue system falls within it. The approximately linear trend across devices reflects an asymmetric sensitivity to noise inherent in the game rules: vertex questions are violated by any single-bit error, whereas edge questions tolerate certain bit flips furches2025application, so both win rates are governed by a common noise parameter and vary in a nearly fixed ratio across platforms.