Scalable tests of quantum contextuality from stabilizer-testing nonlocal games

TL;DR

The paper establishes a unified, coding-theory–based framework for stabilizer-testing games derived from stabilizer codewords to certify quantum contextuality. It proves a universal bound p_{ ext{cl}}^{*} ≤ 7/8 for full stabilizer-query games and derives state-specific asymptotic bounds: p_{ ext{cl}}^{*} → 3/4 for GHZ and toric-code states and p_{ ext{cl}}^{*} → 1/2 for cyclic cluster states, with precise asymptotics obtained via nonlinearity analysis and transfer-matrix methods. The cyclic cluster state's classical value decays exponentially toward 1/2, enabling contextuality witnesses with fidelity-per-qubit as low as ~0.89, which is compatible with current experimental capabilities, and the results are linked to MBQC contextuality and resource-state fidelity. These contributions provide robust, scalable criteria for quantum advantage in many-body stabilizer codes and open avenues for extending the approach to higher-dimensional or continuous-variable systems and MBQC frameworks.

Abstract

Soon after the dawn of quantum error correction, DiVincenzo and Peres observed that stabilizer codewords could give rise to simple proofs of quantumness via contextuality. This discovery can be recast in the language of nonlocal games: every $n$-qubit stabilizer state defines a specific "stabilizer-testing" $n$-player nonlocal game, which quantum players can win with probability one. If quantum players can moreover outperform all possible classical players, then the state is contextual. However, the classical values of stabilizer-testing games are largely unknown for scalable examples beyond the $n$-qubit GHZ state. We introduce several new methods for upper-bounding the classical values of these games. We first prove a general coding-theory bound for all stabilizer-testing games: if the classical value $p_{\mathrm{cl}}^* < 1$, then $p_{\mathrm{cl}}^* \leq 7/8$, i.e., there is no classical strategy that can perform as well as the optimal quantum strategy even in an asymptotic sense. We then show how to tighten this bound for the most common scalable examples, namely GHZ, toric-code and cyclic cluster states. In particular, we establish an asymptotically tight upper bound for cyclic cluster states using transfer-matrix methods. This leads to the striking conclusion that measuring an exponentially small fidelity to the cyclic cluster state will suffice to witness its contextuality.

Figures (4)

• Figure 1: (i) A star operator $A_v$ of the toric code that acts on four adjacent qubits with Pauli $X$ operators (blue). (ii) The intersection of a star and plaquette operator that multiplies the Pauli representation Eq. \ref{['eq:Paulirep']} by a global phase of $-1$ and generates a pair of Pauli $Y$ edges (red). (iii) A plaquette operator $B_p$ of the toric code that acts on four adjacent qubits with Pauli $Z$ operators (orange).
• Figure 2: Visualizing the adjacency graph of the quadratic form obtained from the Boolean derivative of the toric-code parity function. (a) Schematic illustration of the three types of terms contributing to the derivatives in Eqs. \ref{['eq:derivative_cycle_0']} and \ref{['eq:derivative_cycle']}, following the visual conventions of Fig. \ref{['fig:toric_code_stabilizers']}. Overlaps between triplets of distinct stabilizers give rise to quadratic terms while overlaps between pairs of distinct stabilizers give rise to linear terms in these expressions. (b) Choice of a Boolean derivative direction $\mathbf{a}$ that couples to every other star. The stars in the support of $\mathbf{a}$ form a set of $\lfloor L/2\rfloor^2$ equally spaced, pairwise non-adjacent vertices of the direct lattice, leading to plaquette--plaquette interactions around each star, illustrated in the figure by pink bonds on the dual lattice. The plaquette--plaquette block of the resulting quadratic form has rank at least $2\lfloor L/2\rfloor^2$, leading to the bound in Eq. (57).
• Figure 3: Invariant polytope $P$ defined by Eq. \ref{['p1p2p3']}.
• Figure 4: Comparison between our rigorous bound, Eq. \ref{['eq:clusterstatefidelity']}, on the fidelity $\mathcal{F} > \mathcal{F}_c = 2p_{\mathrm{cl}}^*-1$ to the cyclic cluster state required to witness contextuality (red line) and a plausible Iqbal_2024kumar2025quantumclassicalseparationboundedresourcetasks estimate of the global fidelity $\mathcal{F} = \epsilon^n,\, \epsilon = 0.98$ to the $n$-qubit cluster state attainable on state-of-the-art quantum devices (black dotted line). The broad region above the red curve and below the black dotted line suggests that witnessing contextuality of the cluster state in many-body regimes of $n=50-100$ qubits should be feasible on current quantum devices. For completeness, we include the rigorous lower bound Eq. \ref{['eq:pcllowerbound']} (blue line) and exact numerical results for $n \leq 16$ (black dots and Table \ref{['tab:cycliccluster']}) for $\mathcal{F}_c = 2p_{\mathrm{cl}}^*-1$ that suggest tightness of this lower bound.

Theorems & Definitions (43)

• Definition 1
• Definition 2
• Definition 3
• Definition 4: Asymptotic difference
• Definition 5: $(\mathcal{C}_n,\mathcal{M})$ stabilizer testing game
• Theorem 1
• proof
• Lemma 1
• proof
• Proposition 1