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arXiv:2512.16654

Scalable tests of quantum contextuality from stabilizer-testing nonlocal games

Wanbing Zhao, H. W. Shawn Liew, Wen Wei Ho, Chunxiao Liu, Vir B. Bulchandani

quant-phcond-mat.stat-mechcond-mat.str-elSubmitted Dec 18, 2025• Announced Dec 19, 2025• 0 citations• Revised Oct 2, 2026 (v2)
v1 · Dec 18, 2025v2 · Oct 2, 2026
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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.

Comments
v2: 30+5 pages, 5 figures, minor revisions except for the addition of Theorem 2 proving an asymptotically perfect difference for a family of 2D CSS codes

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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 nnn-qubit stabilizer state defines a specific "stabilizer-testing" nnn-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 nnn-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 pcl∗<1p_{\mathrm{cl}}^* < 1pcl∗​<1, then pcl∗≤7/8p_{\mathrm{cl}}^* \leq 7/8pcl∗​≤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.
Contextuality is arguably the fundamental property that makes quantum mechanics different from classical physics. It is responsible for quantum computational speedups in both magic-state-injection-based and measurement-based models of computation, and can be directly probed in a many-body setting by multiplayer nonlocal quantum games. Here, we discuss a family of games that can be won with certainty when performing single-site Pauli measurements on a state that is a codeword of a Calderbank-Shor-Steane (CSS) error-correcting quantum code. We show that these games require deterministic computation of a code-dependent Boolean function, and that the classical probability of success is upper bounded by a generalized notion of nonlinearity/nonquadraticity. This success probability quantifies the state's contextuality, and is computed via the function's (generalized) Walsh-Hadamard spectrum. To calculate this, we introduce an efficient, many-body-physics-inspired method that involves identifying the symmetries of an auxiliary hypergraph state. We compute the classical probability of success for several paradigmatic CSS codes and relate it to both classical statistical mechanics models and to strange correlators of symmetry-protected topological states. We also consider CSS submeasurement games, which can only be won with certainty by sharing the appropriate codeword up to local isometries. These games therefore enable self-testing, which we illustrate explicitly for the 2D toric code. We also discuss how submeasurement games enable an extensive notion of contextuality in many-body states.
An ontological model of an operational theory is considered to be universally noncontextual if both preparation and measurement noncontextuality assumptions are satisfied in that model. In this report, we first generalize the logical proofs of quantum preparation and measurement contextuality for qubit system for any odd number of preparations and measurements. Based on the logical proof, we derive testable universally non-contextual inequalities violated by quantum theory. We then propose a class of two-party communication games and show that the average success probability of winning such games is solely linked to suitable Bell expression whose local bound is greater than universal non-contextual bound. Thus, for a given state, even if quantum theory does not exhibit non-locality, it may still reveal non-classicality by violating the universal non-contextual bound. Further, we consider a different communication game to demonstrate that for a given choices of observables in quantum theory, even if there is no logical proof of preparation and measurement contextuality exist, the universal quantum contextuality can be revealed through that communication game. Such a game thus test a weaker form of universal non-contextuality with minimal assumption.
Nonlocal games yield an unusual perspective on entangled quantum states. The defining property of such games is that a set of players in joint possession of an entangled state can win the game with higher probability than is allowed by classical physics. Here we construct a nonlocal game that can be won with certainty by 2N2N2N players if they have access to the ground state of the toric code on as many qubits. By contrast, the game cannot be won by classical players more than half the time in the large NNN limit. Our game differs from previous examples because it arranges the players on a lattice and allows them to carry out quantum operations in teams, whose composition is dynamically specified. This is natural when seeking to characterize the degree of quantumness of nontrivial many-body states, which potentially include states in much more varied phases of matter than the toric code. We present generalizations of the toric code game to states with ZM{\mathbb{Z}}_{M}ZM​ topological order.
The prevailing view is that quantum phenomena can be harnessed to tackle certain problems beyond the reach of classical approaches. Quantifying this capability as a quantum-classical separation and demonstrating it on current quantum processors has remained elusive. Using a superconducting qubit processor, we show that quantum contextuality enables certain tasks to be performed with success probabilities beyond classical limits. With a few qubits, we illustrate quantum contextuality with the magic square game, as well as quantify it through a Kochen--Specker--Bell inequality violation. To examine many-body contextuality, we implement the N-player GHZ game and separately solve a 2D hidden linear function problem, exceeding classical success rate in both. Our work proposes novel ways to benchmark quantum processors using contextuality-based algorithms.
Fault-tolerant quantum computation requires quantum error correction (QEC), which relies on entanglement to protect information from local noise. Achieving universality, however, demands overcoming the Eastin-Knill theorem. This is often accomplished through strategies like magic state distillation, a process that prepares computational resources—namely, magic states—whose power is now understood to be rooted in quantum contextuality, a fundamental nonclassical feature generalizing Bell nonlocality. Yet, the broader role of contextuality in enabling universality, including its significance as an inherent feature of QEC codes and protocols themselves, has remained largely unexplored. In this work, we develop a rigorous framework for contextuality in QEC and prove three main results. Fundamentally, we show that subsystem stabilizer codes with two or more gauge qubits are strongly contextual in their partial closure, while others are noncontextual, establishing a clear criterion for identifying contextual codes. Mathematically, we unify Abramsky-Brandenburger’s sheaf-theoretic and Kirby-Love’s tree-based definitions of contextuality, resolving a conjecture of Kim and Abramsky. Practically, we prove that many widely studied code-switching protocols which admit universal transversal gate sets, such as the doubled color codes introduced by Bravyi and Cross, are necessarily strongly contextual in their partial closure. Collectively, our results establish quantum contextuality as an intrinsic characteristic of fault-tolerant quantum codes and protocols, complementing entanglement and magic as resources for scalable quantum computation. For quantum coding theorists, this provides a new invariant: contextuality classifies which subsystem stabilizer codes can participate in universal fault-tolerant protocols. These findings position contextuality not only as a foundational concept but also as a practical guide for the design and analysis of future QEC architectures.
In the recent years self-testing has grown into a rich and active area of study with applications ranging from practical verification of quantum devices to deep complexity theoretic results. Self-testing allows a classical verifier to deduce which quantum measurements and on what state are used, for example, by provers Alice and Bob in a nonlocal game. Hence, self-testing as well as its noise-tolerant cousin – robust self-testing – are desirable features for a nonlocal game to have.Contrary to what one might expect, we have a rather incomplete understanding of if and how self-testing could fail to hold. In particular, could it be that every 2-party nonlocal game or Bell inequality with a quantum advantage certifies the presence of a specific quantum state? Also, is it the case that every self-testing result can be turned robust with enough ingeniuty and effort? We answer these questions in the negative by providing simple and fully explicit counterexamples. To this end, given two nonlocal games G1 and G2 , we introduce the (G1∨G2) -game, in which the players get pairs of questions and choose which game they want to play. The players win if they choose the same game and win it with the answers they have given. Our counterexamples are based on this game and we believe this class of games to be of independent interest.