Space-bounded quantum interactive proof systems

Abstract

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 pinching intermediate measurements per verifier action, 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 ${\sf P} = {\sf NP}$: - ${\sf QIPL}^{\rm HC}$, a subclass of ${\sf QIPL}$ defined by a high-concentration condition on yes instances, 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 (pinching) 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.