单回合量子裁判博弈的复杂性限制
Complexity limitations on one-turn quantum refereed games
摘要：
本文研究了量子裁判博弈的复杂性理论方面，量子裁判博弈是两个竞争的玩家之间的抽象博弈，将量子态发送给裁判，裁判对两个状态进行有效的联合测量，以确定哪个玩家获胜。复杂性类$\mathrm{QRG}(1)$包含这样的决策问题，对于这些决策问题，其中一个参与者总是可以在yes实例中以高概率获胜，而另一个参与者总是可以在no实例中以高概率获胜，而不管对手的策略如何。该类通常包含$\mathrm{QMA} \cup \text{co-}\mathrm{QMA}$，并且已知包含在$\mathrm{PSPACE}$中。我们证明了更强的包容性，这类的两个限制性的变种。具体来说，如果一个参与者被限制发送一个经典的（概率）状态而不是量子状态，因此产生的复杂性类$\mathrm{CQRG}(1)$包含在$\exists\cdot\mathrm{PP}$中（应用于$\mathrm{PP}$的非确定性多项式时间算子）;而如果两个参与者都发送量子态但裁判被迫首先测量其中一个状态，并将该测量的经典结果合并到第二状态的测量中，所得到的类$\mathrm{MQRG}(1)$包含在$\mathrm{P}\cdot\mathrm{PP}$（应用于$\mathrm{PP}$的无界误差概率多项式时间算子）中。
英文摘要：
This paper studies complexity theoretic aspects of quantum refereed games, which are abstract games between two competing players that send quantum states to a referee, who performs an efficiently implementable joint measurement on the two states to determine which of the player wins. The complexity class $\mathrm{QRG}(1)$ contains those decision problems for which one of the players can always win with high probability on yes-instances and the other player can always win with high probability on no-instances, regardless of the opposing player's strategy. This class trivially contains $\mathrm{QMA} \cup \text{co-}\mathrm{QMA}$ and is known to be contained in $\mathrm{PSPACE}$. We prove stronger containments on two restricted variants of this class. Specifically, if one of the players is limited to sending a classical (probabilistic) state rather than a quantum state, the resulting complexity class $\mathrm{CQRG}(1)$ is contained in $\exists\cdot\mathrm{PP}$ (the nondeterministic polynomial-time operator applied to $\mathrm{PP}$); while if both players send quantum states but the referee is forced to measure one of the states first, and incorporates the classical outcome of this measurement into a measurement of the second state, the resulting class $\mathrm{MQRG}(1)$ is contained in $\mathrm{P}\cdot\mathrm{PP}$ (the unbounded-error probabilistic polynomial-time operator applied to $\mathrm{PP}$).
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