[Paper Review] Unique Games with Entangled Provers are Easy

Julia Kempe, Oded Regev+1 more|ArXiv.org|Oct 2, 2007

Complexity and Algorithms in Graphs35 references4 citations

TL;DR

This paper demonstrates that unique games with entangled provers can be efficiently approximated using semidefinite programming, contradicting the quantum version of the Unique Games Conjecture. The authors introduce a novel quantum rounding technique to transform SDP solutions into quantum strategies, proving that entanglement significantly simplifies the problem classically believed to be hard.

ABSTRACT

We consider one-round games between a classical verifier and two provers who share entanglement. We show that when the constraints enforced by the verifier are `unique' constraints (i.e., permutations), the value of the game can be well approximated by a semidefinite program. Essentially the only algorithm known previously was for the special case of binary answers, as follows from the work of Tsirelson in 1980. Among other things, our result implies that the variant of the unique games conjecture where we allow the provers to share entanglement is false. Our proof is based on a novel `quantum rounding technique', showing how to take a solution to an SDP and transform it to a strategy for entangled provers. Using our approximation by a semidefinite program we also show a parallel repetition theorem for unique entangled games.

Motivation & Objective

• To investigate the computational complexity of unique games when provers share entanglement, challenging the quantum variant of the Unique Games Conjecture.
• To develop a method for approximating the quantum value of unique games using semidefinite programming.
• To establish that entanglement reduces the hardness of unique games, contrary to classical complexity expectations.
• To prove a parallel repetition theorem for unique games with entangled provers using the new SDP approximation.

Proposed method

• The authors introduce a new semidefinite program (SDP 3) that relaxes the quantum value of unique games, incorporating constraints on state norms and inner products.
• They design a quantum rounding technique that maps a solution to the SDP into a valid quantum strategy for entangled provers.
• The method leverages the structure of unique constraints (permutation-based acceptance) to bound the success probability in terms of SDP feasibility and inner product fidelity.
• The proof uses bounds on the difference between success probabilities in different strategy constructions, relying on triangle inequality and norm inequalities.
• A key technical step involves showing that the SDP for the product game is a relaxation of the tensor product SDP, enabling parallel repetition analysis.
• The authors apply Theorem 5.5 to show that the value of the tensor product SDP equals the product of individual SDP values, enabling exponential decay bounds.

Experimental results

Research questions

• RQ1 Can the quantum value of unique games be efficiently approximated when provers share entanglement?
• RQ2 Does the existence of entanglement invalidate the quantum version of the Unique Games Conjecture?
• RQ3 Can a semidefinite program be constructed to tightly bound the quantum value of unique games?
• RQ4 Is there a quantum analog of classical parallel repetition for unique games with entangled provers?
• RQ5 How does quantum rounding differ from classical rounding in the context of entangled games?

Key findings

• The quantum value of unique games with entangled provers can be approximated within an additive error of O(√ε) using a semidefinite program, where ε is the deviation from perfect satisfaction.
• The quantum unique games conjecture is false, as entangled provers can achieve high success probabilities even when the classical value is bounded away from 1.
• A parallel repetition theorem is established: the value of the m-fold repeated game decays exponentially as (1 - ε²/16)^m, where ε is the inverse gap from the single-game value.
• The SDP relaxation (SDP 3) is strictly feasible and its value matches the quantum value up to a small error, enabling efficient approximation.
• The quantum rounding technique successfully converts an SDP solution into a quantum strategy, with the success probability bounded by the fidelity of the underlying state vectors.
• The tensor product of SDP solutions for two games is a feasible solution for the SDP of their product game, ensuring consistency in repeated games.