We introduce two models of space-bounded quantum interactive proof systems, ${\sf QIPL}$ and ${\sf QIP_{\rm U}L}$. The ${\sf QIP_{\rm U}L}$ model, a space-bounded variant of quantum interactive proofs (${\sf QIP}$) introduced by Watrous (CC 2003) and Kitaev and Watrous (STOC 2000), restricts verifier actions to unitary circuits. In contrast, ${\sf QIPL}$ allows logarithmically many intermediate measurements per verifier action (with a high-concentration condition on yes instances), making it the weakest model that encompasses the classical model of Condon and Ladner (JCSS 1995). We characterize the computational power of ${\sf QIPL}$ and ${\sf QIP_{\rm U}L}$. When the message number $m$ is polynomially bounded, ${\sf QIP_{\rm U}L} \subsetneq {\sf QIPL}$ unless ${\sf P} = {\sf NP}$: - ${\sf QIPL}$ exactly characterizes ${\sf NP}$. - ${\sf QIP_{\rm U}L}$ is contained in ${\sf P}$ and contains ${\sf SAC}^1 \cup {\sf BQL}$, where ${\sf SAC}^1$ denotes problems solvable by classical logarithmic-depth, semi-unbounded fan-in circuits. However, this distinction vanishes when $m$ is constant. Our results further indicate that intermediate measurements uniquely impact space-bounded quantum interactive proofs, unlike in space-bounded quantum computation, where ${\sf BQL}={\sf BQ_{\rm U}L}$. We also introduce space-bounded unitary quantum statistical zero-knowledge (${\sf QSZK_{\rm U}L}$), a specific form of ${\sf QIP_{\rm U}L}$ proof systems with statistical zero-knowledge against any verifier. This class is a space-bounded variant of quantum statistical zero-knowledge (${\sf QSZK}$) defined by Watrous (SICOMP 2009). We prove that ${\sf QSZK_{\rm U}L} = {\sf BQL}$, implying that the statistical zero-knowledge property negates the computational advantage typically gained from the interaction.

翻译：我们引入了两种空间有界量子交互式证明系统模型：${\sf QIPL}$ 和 ${\sf QIP_{\rm U}L}$。${\sf QIP_{\rm U}L}$ 模型是 Watrous (CC 2003) 以及 Kitaev 和 Watrous (STOC 2000) 引入的量子交互式证明 (${\sf QIP}$) 的空间有界变体，它将验证者的操作限制为酉电路。相比之下，${\sf QIPL}$ 允许验证者每次操作进行对数次中间测量（并对是实例施加高浓度条件），这使其成为包含 Condon 和 Ladner (JCSS 1995) 经典模型的最弱模型。我们刻画了 ${\sf QIPL}$ 和 ${\sf QIP_{\rm U}L}$ 的计算能力。当消息数 $m$ 是多项式有界时，除非 ${\sf P} = {\sf NP}$，否则 ${\sf QIP_{\rm U}L} \subsetneq {\sf QIPL}$：- ${\sf QIPL}$ 精确刻画了 ${\sf NP}$。- ${\sf QIP_{\rm U}L}$ 包含于 ${\sf P}$ 中，并包含 ${\sf SAC}^1 \cup {\sf BQL}$，其中 ${\sf SAC}^1$ 表示可由经典对数深度、半无界扇入电路求解的问题。然而，当 $m$ 为常数时，这种区别消失。我们的结果进一步表明，中间测量对空间有界量子交互式证明的影响是独特的，这与空间有界量子计算不同，在后者中 ${\sf BQL}={\sf BQ_{\rm U}L}$。我们还引入了空间有界酉量子统计零知识 (${\sf QSZK_{\rm U}L}$)，这是 ${\sf QIP_{\rm U}L}$ 证明系统的一种特定形式，具有针对任何验证者的统计零知识性。此类是 Watrous (SICOMP 2009) 定义的量子统计零知识 (${\sf QSZK}$) 的空间有界变体。我们证明了 ${\sf QSZK_{\rm U}L} = {\sf BQL}$，这意味着统计零知识性抵消了通常从交互中获得的计算优势。