Scalable tests of quantum contextuality from stabilizer-testing nonlocal games

Every n-qubit stabilizer state defines a specific “stabilizertesting” n-player nonlocal game, which quantum players can win with probability one, and if they outperform all possible classical players, then the state is contextual. This work introduces systematic methods for upper-bounding the classical values of these games to provide rigorous proofs of quantum contextuality for stabilizer states.

The Gist

The paper establishes a general coding-theory bound on the classical values of all stabilizer-testing games, proving that if a game admits non-zero quantum advantage, its classical value is at most 7/8, and further refines this to asymptotic upper bounds of 3/4 for GHZ and toric-code states.

The Stabilizer-Testing Game Framework

The core object of study is the stabilizer-testing game G(S), defined by an arbitrary stabilizer codeword ψ0⟩ with stabilizer group S. The game involves a referee choosing a random stabilizer M ∈ S and handing players the corresponding Pauli operator question αj(M) for each qubit j. Players must then return bits bj ∈ 0 or 1 to collectively win if Xn j=1 for all j, where c(M) labels the parity bit of M. The paper distinguishes between quantum strategies (sharing an entangled resource state ψ⟩ and a protocol P) and classical strategies (deterministic functions mapping questions to output bits).

Quantum Advantage and Contextuality

The game admits non-zero quantum advantage if there is a non-zero Hamming distance between V and c(x), where V is the vector subspace of quadratic Boolean functions corresponding to deterministic classical strategies, and c(x) is the n-input Boolean “parity function” defined by the Pauli measurement outcomes. This condition implies that a non-zero quantum advantage yields a proof of quantum contextuality for ψ0⟩ because the set of classical strategies constrained to return bj = 0 corresponds to local hidden variables, leading to p∗ l.h.v.(G) ≤ p∗ cl(G).

Upper Bounds via Coding Theory and Nonlinearity

The paper develops several complementary approaches for upper bounding the classical values, exploiting Boolean functions and Reed-Muller codes. A general coding-theory bound shows that if the game queries a full stabilizer group, then the classical value p∗ cl ≤ 7/8 whenever the game queries a full stabilizer group. This is refined to an asymptotic upper bound of 3/4 as n → ∞ for GHZ and toric-code states. Furthermore, using nonlinearity profiles, the paper derives bounds such as p∗ cl ≤ 1 − 2(-r) nl2(c) for the full-query regime and p∗ cl ≤ 1 − 2(-r+t) nl1(c') for the partial-query regime.

Asymptotic Behavior and Experimental Relevance

The paper analyzes the asymptotic difference ∆ = 2h1 - limn→∞ p∗ cl(Gn)i, showing that for the cyclic cluster state, this difference diverges as n → ∞ (∆cluster = 1), implying asymptotically perfect difference. This leads to a striking conclusion: imperfect fidelities-per-qubit ϵ ≳ 0.89 to the cyclic cluster state is sufficient to demonstrate its quantum contextuality for sufficiently large n, suggesting feasibility on state-of-the-art devices.

Transfer Matrix Methods for Cyclic Cluster States

For cyclic cluster states, an efficiently computable transfer-matrix expression is introduced using the one-dimensional and translationally invariant nature of the state. This allows for deriving asymptotically tight bounds on p∗ cl(G) for all n ≥ 3. The results show that p∗ cl(G) − 1/2 = O(0.885n) as n → ∞, yielding behavior qualitatively similar to but quantitatively distinct from the GHZ state parity game. This analysis also provides an upper bound of p∗ cl(toric) ≤ 3/4 + 2−⌊L/2⌋ 2−2 for the toric code.

Connection to Measurement-Based Quantum Computing (MBQC)

The results connect to MBQC by interpreting the default quantum strategy as an MBQC where input bit strings are mapped to measurement settings. The analysis shows that computing the cyclic, cubic Boolean function R3(x) in l2-MBQC yields a success probability psucc ∼ 1/2 + 1/2(0.884…)n, which is sufficient to witness contextuality of the quantum computation for large n.

Conclusion and Future Directions

The work provides a systematic theory of stabilizer-testing games that unifies related constructions and resolves the large n behavior of cyclic cluster states.

Based on the provided scientific paper, here are specific improvements that could be made to AI systems, categorized by the domain they would impact:

AI System Improvements Derived from This Paper

The core contribution of this research is establishing rigorous theoretical bounds (upper and lower) on the classical winning probabilities of stabilizer-testing nonlocal games, which serves as a fundamental probe for quantum contextuality. These results can be leveraged to improve AI systems in several specialized areas:

1. Theorems/Bounds for Quantum State Verification and Benchmarking

2. Improved Robustness Analysis for Many-Body Quantum Systems

3. Contextual Learning in Measurement-Based Quantum Computation (MBQC)

Specific Improvements and Capabilities:

1. Theorems/Bounds for Quantum State Verification and Benchmarking:

2. An AI system could be developed to automatically analyze the structure of a given stabilizer state (e.g., a toric code or cyclic cluster state) and immediately determine its theoretical maximum classical advantage, providing an upper bound on the classical value of any associated stabilizer-testing game.

3. This system would allow researchers to rapidly assess whether a specific quantum state is contextual by checking if the required experimental fidelity (derived from the paper's bounds, e.g., for cyclic cluster states, requiring an exponential small fidelity per qubit) is achievable on current hardware.

4. The AI can specifically utilize the derived bound of 7/8 for full-query games to rapidly screen stabilizer codes and identify those that are guaranteed to exhibit quantum advantage in a nonlocal game setting.

5. Improved Robustness Analysis for Many-Body Quantum Systems:

6. An AI system could be trained on the results concerning the asymptotic difference, denoted as the asymptotic difference (Definition 4), between quantum and classical winning probabilities in scalable stabilizer-testing games.

7. This system could predict whether a family of many-body quantum states will exhibit robust nonclassicality (i.e., if the asymptotic difference is non-zero) as the number of qubits increases, helping researchers prioritize which physical models are most promising for robust quantum phenomena versus those that degrade into classical behavior in the thermodynamic limit.

8. The system could use the derived lower bounds (e.g., Eq. 75 for cyclic cluster states) to establish a robustness threshold for experimental verification: it can predict the minimum fidelity required to prove contextuality, accounting for both quantum error correction and measurement noise simultaneously, significantly reducing experimental overhead planning.

9. Contextual Learning in Measurement-Based Quantum Computation (MBQC):

10. An AI system could be designed to analyze the Boolean functions generated by stabilizer states (like the cyclic cubic function R3(x) for cluster states) and predict their contextuality thresholds in an MBQC setting, based on their nonquadraticity or cubic polynomial nature.

11. This system could automatically determine if a specific measurement-based quantum computation (MBQC) protocol, defined by its classical side-processing structure, is likely to yield a contextual result (i.e., whether the success probability exceeds the threshold derived from the nonlinearity profile of the function).

12. The AI can predict how non-bent functions (like R3(x)) will behave in MBQC compared to bent functions (like those related to GHZ states), allowing for optimized choice of resource states or measurement bases that maximize the quantum advantage in a given computation task.