On the Pure Quantum Polynomial Hierarchy and Quantified Hamiltonian Complexity
Sabee Grewal, Dorian Rudolph
TL;DR
The paper argues that the natural quantum analogue of the polynomial hierarchy should be the pureQPH, showing $QMA(2) \subseteq pureQ\Sigma_{2} \subseteq Q\Sigma_{3} \subseteq NEXP$ and proving that $pureQPH = QPH$ via a dimension-independent disentangler that enables amplification with controlled completeness and soundness. It introduces a seven-round construction so that each round packs four unentangled proofs, along with a disentangler that reduces inputs to a small mixture of product states, thereby achieving a robust equality between pure and standard quantum hierarchies. The work also initiates quantified Hamiltonian complexity, defining natural problems (PSH variants) and proving $PSH-\Sigma_{i}$ is $pureQ\Sigma_{i}$-complete and $PSH-\Pi_{i}$ is $pureQ\Pi_{i}$-complete, with additional containment results for local/mixed and sparse variants. Together, these results establish natural complete problems for pureQPH and illuminate the power and structure of quantum quantifier alternations, providing new tools for understanding quantum verifications, amplification, and robust ground-state questions.
Abstract
We prove several new results concerning the pure quantum polynomial hierarchy (pureQPH). First, we show that QMA(2) is contained in pureQSigma2, that is, two unentangled existential provers can be simulated by competing existential and universal provers. We further prove that pureQSigma2 is contained in QSigma3, which in turn is contained in NEXP. Second, we give an error reduction result for pureQPH, and, as a consequence, prove that pureQPH = QPH. A key ingredient in this result is an improved dimension-independent disentangler. Finally, we initiate the study of quantified Hamiltonian complexity, the quantum analogue of quantified Boolean formulae. We prove that the quantified pure sparse Hamiltonian problem is pureQSigma-complete. By contrast, other natural variants (pure/local, mixed/local, and mixed/sparse) admit nontrivial containments but fail to be complete under known techniques. For example, we show that the exists-forall mixed local Hamiltonian problem lies in NP^QMA \cap coNP^QMA.