Complexity Theory for Quantum Promise Problems

Abstract: We begin by establishing structural results for several fundamental quantum complexity classes: p/mBQP, p/mQ(C)MA, $\text{p/mQSZK}{\text{hv}}$, p/mQIP, p/mBQP/qpoly, p/mBQP/poly, and p/mPSPACE. This includes identifying complete problems, as well as proving containment and separation results among these classes. Here, p/mC denotes the corresponding quantum promise complexity class with pure (p) or mixed (m) quantum input states for any classical complexity class C. Surprisingly, our findings uncover relationships that diverge from their classical analogues -- specifically, we show unconditionally that p/mQIP$\neq$p/mPSPACE and p/mBQP/qpoly$\neq$p/mBQP/poly. This starkly contrasts the classical setting, where QIP$=$PSPACE and separations such as BQP/qpoly$\neq$BQP/poly are only known relative to oracles. For applications, we address interesting questions in quantum cryptography, quantum property testing, and unitary synthesis using this new framework. In particular, we show the first unconditional secure auxiliary-input quantum commitment with statistical hiding, solving an open question in [Qia24,MNY24], and demonstrate the first pure quantum state property testing problem that only needs exponentially fewer samples and runtime in the interactive model than the single-party model, which is analogous to Chiesa and Gur [CG18] studying interactive mode for distribution testing. Also, our works offer new insights into Impagliazzo's five worlds view. Roughly, by substituting classical complexity classes in Pessiland, Heuristica, and Algorithmica with mBQP and mQCMA or $\text{mQSZK}\text{hv}$, we establish a natural connection between quantum cryptography and quantum promise complexity theory.