The complexity class Quantum Statistical Zero-Knowledge ($\mathsf{QSZK}$) captures computational difficulties of quantum state testing with respect to the trace distance for efficiently preparable mixed states (Quantum State Distinguishability Problem, QSDP), as introduced by Watrous (FOCS 2002). However, this class faces the same parameter issue as its classical counterpart, because of error reduction for the QSDP (the polarization lemma), as demonstrated by Sahai and Vadhan (JACM, 2003). In this paper, we introduce quantum analogues of triangular discrimination, which is a symmetric version of the $\chi^2$ divergence, and investigate the quantum state testing problems for quantum triangular discrimination and quantum Jensen-Shannon divergence (a symmetric version of the quantum relative entropy). These new $\mathsf{QSZK}$-complete problems allow us to improve the parameter regime for testing quantum states in trace distance. Additionally, we prove that the quantum state testing for trace distance with negligible errors is in $\mathsf{PP}$ while the same problem without error is in $\mathsf{BQP}_1$. This indicates that the length-preserving polarization for the QSDP implies that $\mathsf{QSZK}$ is in $\mathsf{PP}$.
翻译:复杂的量子统计零知识等级 (mathsf<unk> SZK}$) 记录了Watroth(FOCS 2002) 引入的高效预设混合国家(QSDP,QSDP) 的追踪距离(量子国区别问题,QSDP) 的量子状态测试的计算困难(量子国区别问题,QSDP ) 的计算困难。 然而,由于QSDP(两极分化利玛) 的误差减少,这与古典对应的参数问题相同,正如Sahai 和 Vadhan (JACM, 2003) 所显示的。在本论文中,我们引入了三角歧视的量子状态测试(量子国差异的对称版),这是$chnical-Jensen-Shannon差异(量子相对酶的对应版本) 的量子测试问题。 这些新的 $mathf<unk> SZ=PP} 以美元计值差差的量子国检验是Q\mafs=QSmaxxxxxxx 问题, 问题意味着“xrumaluplationalupilance”</s>