Many-body contextuality and self-testing quantum matter via nonlocal games

TL;DR

The paper develops a comprehensive framework to quantify contextuality in many-body quantum matter via CSS XOR games, linking optimal classical success to the nonlinearity of a code-determined Boolean function and to the Walsh-Hadamard spectrum. It introduces a hypergraph-state formalism to efficiently compute these spectra and demonstrates how to bound classical strategies for paradigmatic phases (GHZ, 1D cluster, 2D toric code), highlighting deep connections to graph/hypergraph states, partition functions, and strange correlators. The work then extends to submeasurement games that enable self-testing of CSS codewords, particularly the toric code, and provides an extensive, translationally invariant approach to quantify contextuality across many-body systems, with clear avenues for generalization to subsystem codes and noisy devices. Overall, the methodology offers a practical, extensible route to diagnose and quantify contextuality as a resource for quantum advantage in complex quantum matter.

Abstract

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.

Figures (6)

• Figure 1: Overview of the structure of the technical results of this paper. Broadly, the paper first covers XOR games. Section \ref{['sec:XORgames']} introduces XOR games, and connects the optimal classical success fraction to properties of Boolean functions (via theorems \ref{['thm:nonlinearity']}, \ref{['thm:CSS-success-fraction']}, and \ref{['thm:clifford-equivalence']}) and to contextuality of Pauli measurements. Section \ref{['sec:walsh-from-graphs']} introduces many-body-physics-inspired techniques for computing the Walsh spectrum via theorems \ref{['thm:walsh-from-symmetries']} and \ref{['thm:bell-pairs']}, while Sec. \ref{['sec:examples']} puts these tools to use by computing the success fraction of various CSS XOR games. Finally, Sec. \ref{['sec:submeasurement']} introduces submeasurement games and connects them to self-testing.
• Figure 2: Upper bound \ref{['eqn:NCF-bound']} on the performance of strategies that make use of the empirical model that derives from measurement of a quantum state compared with the performance of the Pauli strategy (Prop. \ref{['prop:perfect-quantum-strat']}). States are deformed by the nonunitary operator $A(\theta) = e^{\theta Z} e^{2i\theta Y}$ and the noncontextual fraction ($\mathsf{NCF}$) is computed using the linear-programming method described in Ref. Abramsky2017. Left: The deformed GHZ state $A(\theta) \ket{\text{GHZ}}$ with three qubits, for which $\omega = 3/4$. Right: The deformed 1D cluster state $A(\theta)\ket{C}$ with four qubits, for which $\omega = 7/8$.
• Figure 3: Illustration of graph equivalence under linear transformations. Top row: The left graph, the complete graph $K_{5}$, represents the Boolean function \ref{['eqn:GHZ-all-to-all']} computed in the GHZ game for a specific choice of stabilizer generators with six qubits. There exists a linear transformation that transforms $K_5$ to a linear graph [corresponding to Eq. \ref{['eqn:ghz-function']}]. Finally, there exists a sequence of linear transformations that relates the linear graph to a ladder rung graph with two rungs. Bottom row: Explicit elementary operations that transform the linear graph (left) to the ladder rung graph (right). The source and target qubits that are acted upon in accordance with Eq. \ref{['eqn:CX-transform']} are illustrated by the $\mathsf{CX}_{ij}$ gates. The neighborhood $N(j)$ of the target $j$ that enters Eq. \ref{['eqn:graph-transform']} is illustrated in orange. In each step, the connections between the control, $i$, and the neighbors ($\neq i$) of the target, $j$, are toggled.
• Figure 4: An illustration of Algorithm \ref{['alg:standard-form']} for a generic, randomly generated adjacency matrix, constructed by taking a matrix with entries $g_{ij}$ that are iid using the uniform distribution over $\{ 0, 1 \}$, then writing $B_{ij} = g_{ij} \oplus g_{ji}$. Left: The initial graph, in which each vertex is connected to many other vertices. Middle: An intermediate reduced graph in which every vertex is connected to at most one other vertex. Right: The graph in standard form (Def. \ref{['def:standard-form']}).
• Figure 5: Graphical illustration of the function computed by the toric-code game on a lattice corresponding to a generic tesselation. Left: Generic tesselation $\mathcal{L}$ constructed from a Voronoi diagram (gray) and the associated dual medial graph $M^*(\mathcal{L})$ (black). Right: Locations of $\mathsf{CZ}$ and $\mathsf{CCZ}$ gates given the dual medial graph. For every edge of $M^*(\mathcal{L})$, there exists a $\mathsf{CZ}$ gate. Given a face of $M^*(\mathcal{L})$, there exists a $\mathsf{CCZ}$ gate for every triple of sites.

Theorems & Definitions (60)

• Definition 2.1: CSS XOR game
• Remark 2.1
• Definition 2.2: Hamming weight
• Lemma 2.1: Ref. Hoban2011, Eq. (4)
• Example 2.1: GHZ game ghz1989GHSZ1990MerminPolynomialsBrassard2003multipartybrassard2005recastingbrassard2005pseudotelepathy
• Lemma 2.2
• proof
• Proposition 1: Pauli strategy
• proof
• Proposition 2