A convergent sum-of-squares hierarchy for compiled nonlocal games

• URL: http://arxiv.org/abs/2507.17581v1
• Date: Wed, 23 Jul 2025 15:16:38 GMT
• Title: A convergent sum-of-squares hierarchy for compiled nonlocal games
• Authors: David Cui, Chirag Falor, Anand Natarajan, Tina Zhang,
• Abstract summary: We study "compiled" nonlocal games played between a classical verifier and a single quantum prover.<n>We show that the success probability of a quantum prover in the compiled game is bounded by the game's quantum commuting-operator value.<n>We extend the niceness framework and construct a hierarchy of semidefinite programs that searches exclusively over nice certificates.
• Score: 1.5029560229270191
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We continue the line of work initiated by Kalai et al. (STOC '23), studying "compiled" nonlocal games played between a classical verifier and a single quantum prover, with cryptography simulating the spatial separation between the players. The central open question in this area is to understand the soundness of this compiler against quantum strategies, and apart from results for specific games, all that is known is the recent "qualitative" result of Kulpe et al. (STOC '25) showing that the success probability of a quantum prover in the compiled game is bounded by the game's quantum commuting-operator value in the limit as the cryptographic security parameter goes to infinity. In this work, we make progress towards a quantitative understanding of quantum soundness for general games, by giving a concrete framework to bound the quantum value of compiled nonlocal games. Building on the result of Kulpe et al. together with the notion of "nice" sum-of-squares certificates, introduced by Natarajan and Zhang (FOCS '23) to bound the value of the compiled CHSH game, we extend the niceness framework and construct a hierarchy of semidefinite programs that searches exclusively over nice certificates. We show that this hierarchy converges to the optimal quantum value of the game. Additionally, we present a transformation to make any degree-1 sum-of-squares certificate nice. This approach provides a systematic method to reproduce all known bounds for special classes of games together with Kulpe et al.'s bound for general games from the same framework.

Related papers

• Quantitative Quantum Soundness for Bipartite Compiled Bell Games via the Sequential NPA Hierarchy [3.34301287453961]
  We show the first quantitative quantum soundness bounds for every bipartite compiled Bell game.<n>More generally, for all bipartite games we show that the compiled score cannot significantly exceed the bounds given by a newly formalized sequential Navascu'es-Pironio-Ac'in hierarchy.
  arXiv  Detail & Related papers  (2025-07-22T20:31:41Z)
• A Game-Theoretic Quantum Algorithm for Solving Magic Squares [2.09260520196733]
  We present a variational framework for the Magic Square Game (MSG), a two-player non-local game with perfect quantum advantage.<n>We construct a value Hamiltonian that encodes the game's parity and consistency constraints, then optimize parameterized quantum circuits to minimize this cost.
  arXiv  Detail & Related papers  (2025-05-19T17:12:53Z)
• A bound on the quantum value of all compiled nonlocal games [49.32403970784162]
  A cryptographic compiler converts any nonlocal game into an interactive protocol with a single computationally bounded prover.<n>We establish a quantum soundness result for all compiled two-player nonlocal games.
  arXiv  Detail & Related papers  (2024-08-13T08:11:56Z)
• Quantum bounds for compiled XOR games and $d$-outcome CHSH games [1.099532646524593]
  We show that the compilation procedure of Kalai et al. preserves the quantum bound for two classes of games. For any pair of qubit measurements, there exists an XOR game such that its optimal winning probability serves as a self-test for that particular pair of measurements.
  arXiv  Detail & Related papers  (2024-03-08T18:20:21Z)
• A Computational Tsirelson's Theorem for the Value of Compiled XOR Games [9.818381970014284]
  We show that the compiler proposed by Kalai et al. is sound for any two-player XOR game. Using our techniques, we obtain several additional results, including tight bounds on the compiled value of parallel-repeated XOR games.
  arXiv  Detail & Related papers  (2024-02-27T08:24:21Z)
• A Quadratic Speedup in Finding Nash Equilibria of Quantum Zero-Sum Games [95.50895904060309]
  We introduce the Optimistic Matrix Multiplicative Weights Update (OMMWU) algorithm and establish its average-iterate convergence complexity as $mathcalO(d/epsilon)$ to $epsilon$-Nash equilibria.<n>This quadratic speed-up sets a new benchmark for computing $epsilon$-Nash equilibria in quantum zero-sum games.
  arXiv  Detail & Related papers  (2023-11-17T20:38:38Z)
• Photonic implementation of the quantum Morra game [69.65384453064829]
  We study a faithful translation of a two-player quantum Morra game, which builds on previous work by including the classical game as a special case. We propose a natural deformation of the game in the quantum regime in which Alice has a winning advantage, breaking the balance of the classical game. We discuss potential applications of the quantum Morra game to the study of quantum information and communication.
  arXiv  Detail & Related papers  (2023-11-14T19:41:50Z)
• Simple Tests of Quantumness Also Certify Qubits [69.96668065491183]
  A test of quantumness is a protocol that allows a classical verifier to certify (only) that a prover is not classical. We show that tests of quantumness that follow a certain template, which captures recent proposals such as (Kalai et al., 2022) can in fact do much more. Namely, the same protocols can be used for certifying a qubit, a building-block that stands at the heart of applications such as certifiable randomness and classical delegation of quantum computation.
  arXiv  Detail & Related papers  (2023-03-02T14:18:17Z)
• On the relation between completely bounded and $(1,cb)$-summing maps with applications to quantum XOR games [65.51757376525798]
  We show that given a linear map from a general operator space into the dual of a C$*$-algebra, its completely bounded norm is upper bounded by a universal constant times its $(''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''
  arXiv  Detail & Related papers  (2021-12-09T21:06:52Z)
• Quantum guessing games with posterior information [68.8204255655161]
  A quantum guessing game with posterior information uses quantum systems to encode messages and classical communication to give partial information after a quantum measurement has been performed. We formalize symmetry of guessing games and characterize the optimal measurements in cases where the symmetry is related to an irreducible representation.
  arXiv  Detail & Related papers  (2021-07-25T19:10:26Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.