Complexity Theory for Quantum Promise Problems
Nai-Hui Chia, Kai-Min Chung, Tzu-Hsiang Huang, Jhih-Wei Shih
TL;DR
The paper develops a comprehensive framework for quantum promise problems, introducing pure-state and mixed-state promise classes (e.g., pBQP, mBQP, pQMA, mQMA, pQCMA, mQCMA, pQSZKhv, pQIP, pPSPACE, and their mixed-state counterparts) and studying their structural properties, completeness, and separations. It proves unconditional separations such as p/mQIP[2] ⊄ p/mUNBOUND and p/mBQP/qpoly ⊄ p/mBQP/poly, and identifies complete problems via Quantum OR and related constructions that extend classical complete problems to the quantum-input setting. The work also connects quantum promise complexity to quantum cryptography and property testing, delivering results such as an unconditional secure commitment scheme in the quantum auxiliary-input model, and demonstrating exponential savings in interactive quantum property testing for certain pure- and mixed-state properties. Technical innovations include novel mQMA-complete problem constructions using distributional Local Hamiltonians, Hadamard-test energy estimation with
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.