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SIAM Journal on Computing

Entangled Games Are Hard to Approximate

Julia Kempe, Hirotada Kobayashi, Keiji Matsumoto, Ben Toner, Thomas Vidick

Published Jan 1, 2011• 47 citations
Publisher

Abstract

We establish the first hardness results for the problem of computing the value of one-round games played by a verifier and a team of provers who can share quantum entanglement. In particular, we show that it is NP-hard to approximate within an inverse polynomial the value of a one-round game with (i) a quantum verifier and two entangled provers or (ii) a classical verifier and three entangled provers. Previously it was not even known if computing the value exactly is NP-hard. We also describe a mathematical conjecture, which, if true, would imply hardness of approximation of entangled-prover games to within a constant. Using our techniques we also show that every language in PSPACE has a two-prover one-round interactive proof system with perfect completeness and soundness $1-1/\,\mathrm{poly}$ even against entangled provers. We start our proof by describing two ways to modify classical multiprover games to make them resistant to entangled provers. We then show that a strategy for the modified game that uses entanglement can be “rounded” to one that does not. The results then follow from classical inapproximability bounds. Our work implies that, unless $\mathrm{P}=\mathrm{NP}$, the values of entangled-prover games cannot be computed by semidefinite programs that are polynomial in the size of the verifier's system, a method that has been successful for more restricted quantum games.

DOI
10.1137/090751293

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SIAM Journal on Computing•Jan 1, 2010•73 citations
We establish the first hardness results for the problem of computing the value of one-round games played by a verifier and a team of provers who can share quantum entanglement. In particular, we show that it is NP-hard to approximate within an inverse polynomial the value of a one-round game with (i) a quantum verifier and two entangled provers or (ii) a classical verifier and three entangled provers. Previously it was not even known if computing the value exactly is NP-hard. We also describe a mathematical conjecture, which, if true, would imply hardness of approximation of entangled-prover games to within a constant. Using our techniques we also show that every language in PSPACE has a two-prover one-round interactive proof system with perfect completeness and soundness 1−1/ poly1-1/\,\mathrm{poly}1−1/poly even against entangled provers. We start our proof by describing two ways to modify classical multiprover games to make them resistant to entangled provers. We then show that a strategy for the modified game that uses entanglement can be “rounded” to one that does not. The results then follow from classical inapproximability bounds. Our work implies that, unless P=NP\mathrm{P}=\mathrm{NP}P=NP, the values of entangled-prover games cannot be computed by semidefinite programs that are polynomial in the size of the verifier's system, a method that has been successful for more restricted quantum games.
We establish the first hardness results for the problem of computing the value of one-round games played by a verifier and a team of provers who can share quantum entanglement. In particular, we show that it is NP-hard to approximate within an inverse polynomial the value of a one-round game with (i) quantum verifier and two entangled provers or (ii) classical verifier and three entangled provers. Previously it was not even known if computing the value exactly is NP-hard. We also describe a mathematical conjecture, which, if true, would imply hardness of approximation to within a constant. We start our proof by describing two ways to modify classical multi-prover games to make them resistant to entangled provers. We then show that a strategy for the modified game that uses entanglement can be ``rounded'' to one that does not. The results then follow from classical inapproximability bounds. Our work implies that, unless P=NP, the values of entangled-prover games cannot be computed by semidefinite programs that are polynomial in the size of the verifier's system, a method that has been successful for more restricted quantum games.
We establish the first hardness results for the problem of computing the value of one-round games played by a referee and a team of players who can share quantum entanglement. In particular, we show that it is NP-hard to approximate within an inverse polynomial the value of a one-round game with (i) quantum referee and two entangled players or (ii) classical referee and three entangled players. Previously it was not even known if computing the value exactly is NP-hard. We also describe a mathematical conjecture, which, if true, would imply hardness of approximation to within a constant.We start our proof by describing two ways to modify classical multi-player games to make them resistant to entangled players. We then show that a strategy for the modified game that uses entanglement can be "rounded'' to one that does not. The results then follow from classical inapproximability bounds. Our work implies that, unless P = NP, the values of entangled-player games cannot be computed by semidefinite programs that are polynomial in the size of the referee's system, a method that has been successful for more restricted quantum games.
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.
We show that the value of a general two-prover quantum game cannot be computed by a semi-definite program ofvpolynomial size (unless P=NP), a method that has been successful in more restricted quantum games. More precisely, we show that proof of membership in the NP-complete problem GAP-3D-Matching can be obtained by a 2-prover, 1-round quantum interactive proof system where the provers share entanglement, with perfect completeness and soundness s=1-2^(-O(n)), and such that the space of the verifier and the size of the messages are O(log n). This implies that QMIP^*_{log n,1,1-2^(-O(n))} \nsubseteq P unless P = NP and provides the first non-trivial lower bound on the power of entangled quantum provers, albeit with an exponentially small gap. The gap achievable by our proof system might in fact be larger, provided a certain conjecture on almost commuting versus nearly commuting projector matrices is true.
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.
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 (SDP). 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 into a strategy for entangled provers. Using our approximation by an SDP, we also show a parallel repetition theorem for unique entangled games.