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A Computational Tsirelson's Theorem for the Value of Compiled XOR Games

David Cui, Giulio Malavolta, Arthur Mehta, Anand Natarajan, Connor Paddock, Simon Schmidt, Michael Walter, Tina Zhang

TL;DR

This work proves that the Kalai–KLVY compiler preserves the quantum value for any two-player XOR game up to a negligible, cryptography-based error, resolving an open question about general XOR games beyond CHSH. The authors fuse two main techniques—a nice degree-1 sum-of-squares certificate for XOR games and Tsirelson-type correspondences—to bound the compiled game's value by the original game's quantum value, underpinned by quantum homomorphic encryption security. Consequences include operator self-testing for compiled XOR games, a parallel repetition theorem with exponential decay, and a robust self-test for the compiled Magic Square game via anticommuting operator structure, all facilitated by a concrete SOS certificate that manifests rigidity. The results enable reliable delegated quantum computation with a single, cryptographically bounded quantum verifier and provide a framework for extending compiled-game analysis beyond XOR games in future work.

Abstract

Nonlocal games are a foundational tool for understanding entanglement and constructing quantum protocols in settings with multiple spatially separated quantum devices. In this work, we continue the study initiated by Kalai et al. (STOC '23) of compiled nonlocal games, played between a classical verifier and a single cryptographically limited quantum device. Our main result is that the compiler proposed by Kalai et al. is sound for any two-player XOR game. A celebrated theorem of Tsirelson shows that for XOR games, the quantum value is exactly given by a semidefinite program, and we obtain our result by showing that the SDP upper bound holds for the compiled game up to a negligible error arising from the compilation. This answers a question raised by Natarajan and Zhang (FOCS '23), who showed soundness for the specific case of the CHSH game. Using our techniques, we obtain several additional results, including (1) tight bounds on the compiled value of parallel-repeated XOR games, (2) operator self-testing statements for any compiled XOR game, and (3) a ``nice'' sum-of-squares certificate for any XOR game, from which operator rigidity is manifest.

A Computational Tsirelson's Theorem for the Value of Compiled XOR Games

TL;DR

This work proves that the Kalai–KLVY compiler preserves the quantum value for any two-player XOR game up to a negligible, cryptography-based error, resolving an open question about general XOR games beyond CHSH. The authors fuse two main techniques—a nice degree-1 sum-of-squares certificate for XOR games and Tsirelson-type correspondences—to bound the compiled game's value by the original game's quantum value, underpinned by quantum homomorphic encryption security. Consequences include operator self-testing for compiled XOR games, a parallel repetition theorem with exponential decay, and a robust self-test for the compiled Magic Square game via anticommuting operator structure, all facilitated by a concrete SOS certificate that manifests rigidity. The results enable reliable delegated quantum computation with a single, cryptographically bounded quantum verifier and provide a framework for extending compiled-game analysis beyond XOR games in future work.

Abstract

Nonlocal games are a foundational tool for understanding entanglement and constructing quantum protocols in settings with multiple spatially separated quantum devices. In this work, we continue the study initiated by Kalai et al. (STOC '23) of compiled nonlocal games, played between a classical verifier and a single cryptographically limited quantum device. Our main result is that the compiler proposed by Kalai et al. is sound for any two-player XOR game. A celebrated theorem of Tsirelson shows that for XOR games, the quantum value is exactly given by a semidefinite program, and we obtain our result by showing that the SDP upper bound holds for the compiled game up to a negligible error arising from the compilation. This answers a question raised by Natarajan and Zhang (FOCS '23), who showed soundness for the specific case of the CHSH game. Using our techniques, we obtain several additional results, including (1) tight bounds on the compiled value of parallel-repeated XOR games, (2) operator self-testing statements for any compiled XOR game, and (3) a ``nice'' sum-of-squares certificate for any XOR game, from which operator rigidity is manifest.
Paper Structure (20 sections, 28 theorems, 106 equations)

This paper contains 20 sections, 28 theorems, 106 equations.

Table of Contents

  1. Introduction
  2. Results
  3. Technical Outline
  4. Non-local games and the klvy compiler.
  5. Further directions
  6. Related work
  7. Preliminaries
  8. Cryptography
  9. Non-Local Games
  10. XOR Games
  11. Compiled Games
  12. Block encodings
  13. Bounding the compiled value of an XOR game
  14. SOS's and bounding the value of nonlocal games
  15. Bounding the compiled value using cryptographic pseudo-expectations
  16. ...and 5 more sections

Key Result

Theorem 1.1

Let $\mathcal{G}$ be any XOR game and $\mathcal{G}_\mathrm{comp}$ the corresponding compiled game. Then any prover running in time polynomial in the security parameter $\lambda$ wins $\mathcal{G}_{\mathrm{comp}}$ with probability at most $\omega_q^*(\mathcal{G}) + \mathrm{negl}(\lambda)$, where $\om

Theorems & Definitions (63)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Definition 2.9
  • ...and 53 more