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arXiv:1905.11635

Complexity lower bounds for computing the approximately-commuting operator value of non-local games to high precision

Matthew Coudron, William Slofstra

Published May 28, 2019• 4 citations
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Abstract

We study the problem of approximating the commuting-operator value of a two-player non-local game. It is well-known that it is $\mathrm{NP}$-complete to decide whether the classical value of a non-local game is 1 or $1- ε$. Furthermore, as long as $ε$ is small enough, this result does not depend on the gap $ε$. In contrast, a recent result of Fitzsimons, Ji, Vidick, and Yuen shows that the complexity of computing the quantum value grows without bound as the gap $ε$ decreases. In this paper, we show that this also holds for the commuting-operator value of a game. Specifically, in the language of multi-prover interactive proofs, we show that the power of $\mathrm{MIP}^{co}(2,1,1,s)$ (proofs with two provers, one round, completeness probability $1$, soundness probability $s$, and commuting-operator strategies) can increase without bound as the gap $1-s$ gets arbitrarily small. Our results also extend naturally in two ways, to perfect zero-knowledge protocols, and to lower bounds on the complexity of computing the approximately-commuting value of a game. Thus we get lower bounds on the complexity class $\mathrm{PZK}$-$\mathrm{MIP}^{co}_δ(2,1,1,s)$ of perfect zero-knowledge multi-prover proofs with approximately-commuting operator strategies, as the gap $1-s$ gets arbitrarily small. While we do not know any computable time upper bound on the class $\mathrm{MIP}^{co}$, a result of the first author and Vidick shows that for $s = 1-1/\text{poly}(f(n))$ and $δ= 1/\text{poly}(f(n))$, the class $\mathrm{MIP}^{co}_δ(2,1,1,s)$, with constant communication from the provers, is contained in $\mathrm{TIME}(\exp(\text{poly}(f(n))))$. We give a lower bound of $\mathrm{coNTIME}(f(n))$ (ignoring constants inside the function) for this class, which is tight up to polynomial factors assuming the exponential time hypothesis.

DOI
10.48550/arxiv.1905.11635

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We study the problem of approximating the commuting-operator value of a two-player non-local game. It is well-known that it is NP\mathrm{NP}NP-complete to decide whether the classical value of a non-local game is 1 or 1−ε1- ε1−ε. Furthermore, as long as εεε is small enough, this result does not depend on the gap εεε. In contrast, a recent result of Fitzsimons, Ji, Vidick, and Yuen shows that the complexity of computing the quantum value grows without bound as the gap εεε decreases. In this paper, we show that this also holds for the commuting-operator value of a game. Specifically, in the language of multi-prover interactive proofs, we show that the power of MIPco(2,1,1,s)\mathrm{MIP}^{co}(2,1,1,s)MIPco(2,1,1,s) (proofs with two provers, one round, completeness probability 111, soundness probability sss, and commuting-operator strategies) can increase without bound as the gap 1−s1-s1−s gets arbitrarily small. Our results also extend naturally in two ways, to perfect zero-knowledge protocols, and to lower bounds on the complexity of computing the approximately-commuting value of a game. Thus we get lower bounds on the complexity class PZK\mathrm{PZK}PZK-MIPδco(2,1,1,s)\mathrm{MIP}^{co}_δ(2,1,1,s)MIPδco​(2,1,1,s) of perfect zero-knowledge multi-prover proofs with approximately-commuting operator strategies, as the gap 1−s1-s1−s gets arbitrarily small. While we do not know any computable time upper bound on the class MIPco\mathrm{MIP}^{co}MIPco, a result of the first author and Vidick shows that for s=1−1/poly(f(n))s = 1-1/\text{poly}(f(n))s=1−1/poly(f(n)) and δ=1/poly(f(n))δ= 1/\text{poly}(f(n))δ=1/poly(f(n)), the class MIPδco(2,1,1,s)\mathrm{MIP}^{co}_δ(2,1,1,s)MIPδco​(2,1,1,s), with constant communication from the provers, is contained in TIME(exp⁡(poly(f(n))))\mathrm{TIME}(\exp(\text{poly}(f(n))))TIME(exp(poly(f(n)))). We give a lower bound of coNTIME(f(n))\mathrm{coNTIME}(f(n))coNTIME(f(n)) (ignoring constants inside the function) for this class, which is tight up to polynomial factors assuming the exponential time hypothesis.
We study the problem of approximating the commuting-operator value of a two-player non-local game. It is well-known that it is NP-complete to decide whether the classical value of a non-local game is 1 or 1- epsilon, promised that one of the two is the case. Furthermore, as long as epsilon is small enough, this result does not depend on the gap epsilon. In contrast, a recent result of Fitzsimons, Ji, Vidick, and Yuen shows that the complexity of computing the quantum value grows without bound as the gap epsilon decreases. In this paper, we show that this also holds for the commuting-operator value of a game. Specifically, in the language of multi-prover interactive proofs, we show that the power of MIP^{co}(2,1,1,s) (proofs with two provers, one round, completeness probability 1, soundness probability s, and commuting-operator strategies) can increase without bound as the gap 1-s gets arbitrarily small. Our results also extend naturally in two ways, to perfect zero-knowledge protocols, and to lower bounds on the complexity of computing the approximately-commuting value of a game. Thus we get lower bounds on the complexity class PZK-MIP^{co}_{delta}(2,1,1,s) of perfect zero-knowledge multi-prover proofs with approximately-commuting operator strategies, as the gap 1-s gets arbitrarily small. While we do not know any computable time upper bound on the class MIP^{co}, a result of the first author and Vidick shows that for s = 1-1/poly(f(n)) and delta = 1/poly(f(n)), the class MIP^{co}_delta(2,1,1,s), with constant communication from the provers, is contained in TIME(exp(poly(f(n)))). We give a lower bound of coNTIME(f(n)) (ignoring constants inside the function) for this class, which is tight up to polynomial factors assuming the exponential time hypothesis.
The class MIP∗\mathsf{MIP}^*MIP∗ is the set of languages decidable by multiprover interactive proofs with quantum entangled provers. It was recently shown by Ji, Natarajan, Vidick, Wright and Yuen that MIP∗\mathsf{MIP}^*MIP∗ is equal to RE\mathsf{RE}RE, the set of recursively enumerable languages. In particular this shows that the complexity of approximating the quantum value of a non-local game GGG is equivalent to the complexity of the Halting problem. In this paper we investigate the complexity of deciding whether the quantum value of a non-local game GGG is exactly 111. This problem corresponds to a complexity class that we call zero gap MIP∗\mathsf{MIP}^*MIP∗, denoted by MIP0∗\mathsf{MIP}^*_0MIP0∗​, where there is no promise gap between the verifier's acceptance probabilities in the YES and NO cases. We prove that MIP0∗\mathsf{MIP}^*_0MIP0∗​ extends beyond the first level of the arithmetical hierarchy (which includes RE\mathsf{RE}RE and its complement coRE\mathsf{coRE}coRE), and in fact is equal to Π20Π_2^0Π20​, the class of languages that can be decided by quantified formulas of the form ∀y ∃z R(x,y,z)\forall y \, \exists z \, R(x,y,z)∀y∃zR(x,y,z). Combined with the previously known result that MIP0co\mathsf{MIP}^{co}_0MIP0co​ (the commuting operator variant of MIP0∗\mathsf{MIP}^*_0MIP0∗​) is equal to coRE\mathsf{coRE}coRE, our result further highlights the fascinating connection between various models of quantum multiprover interactive proofs and different classes in computability theory.
The class MIP^* is the set of languages decidable by multiprover interactive proofs with quantum entangled provers. It was recently shown by Ji, Natarajan, Vidick, Wright and Yuen that MIP^* is equal to RE, the set of recursively enumerable languages. In particular this shows that the complexity of approximating the quantum value of a non-local game G is equivalent to the complexity of the Halting problem. In this paper we investigate the complexity of deciding whether the quantum value of a non-local game G is exactly 1. This problem corresponds to a complexity class that we call zero gap MIP^*, denoted by MIP₀^*, where there is no promise gap between the verifier’s acceptance probabilities in the YES and NO cases. We prove that MIP₀^* extends beyond the first level of the arithmetical hierarchy (which includes RE and its complement coRE), and in fact is equal to Π₂⁰, the class of languages that can be decided by quantified formulas of the form ∀ y ∃ z R(x,y,z). Combined with the previously known result that MIP₀^{co} (the commuting operator variant of MIP₀^*) is equal to coRE, our result further highlights the fascinating connection between various models of quantum multiprover interactive proofs and different classes in computability theory.
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We study the complexity of computing the commuting-operator value ω∗ω^*ω∗ of entangled XOR games with any number of players. We introduce necessary and sufficient criteria for an XOR game to have ω∗=1ω^* = 1ω∗=1, and use these criteria to derive the following results: 1. An algorithm for symmetric games that decides in polynomial time whether ω∗=1ω^* = 1ω∗=1 or ω∗<1ω^* < 1ω∗<1, a task that was not previously known to be decidable, together with a simple tensor-product strategy that achieves value 1 in the former case. The only previous candidate algorithm for this problem was the Navascués-Pironio-Acín (also known as noncommutative Sum of Squares or ncSoS) hierarchy, but no convergence bounds were known. 2. A family of games with three players and with ω∗<1ω^* < 1ω∗<1, where it takes doubly exponential time for the ncSoS algorithm to witness this (in contrast with our algorithm which runs in polynomial time). 3. A family of games achieving a bias difference 2(ω∗−ω)2(ω^* - ω)2(ω∗−ω) arbitrarily close to the maximum possible value of 111 (and as a consequence, achieving an unbounded bias ratio), answering an open question of Briët and Vidick. 4. Existence of an unsatisfiable phase for random (non-symmetric) XOR games: that is, we show that there exists a constant CkunsatC_k^{\text{unsat}}Ckunsat​ depending only on the number kkk of players, such that a random kkk-XOR game over an alphabet of size nnn has ω∗<1ω^* < 1ω∗<1 with high probability when the number of clauses is above CkunsatnC_k^{\text{unsat}} nCkunsat​n. 5. A lower bound of Ω(nlog⁡(n)/log⁡log⁡(n))Ω(n \log(n)/\log\log(n))Ω(nlog(n)/loglog(n)) on the number of levels in the ncSoS hierarchy required to detect unsatisfiability for most random 3-XOR games. This is in contrast with the classical case where the nnn-th level of the sum-of-squares hierarchy is equivalent to brute-force enumeration of all possible solutions.
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