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arXiv:1805.11139 · Computational Complexity

Quantum generalizations of the polynomial hierarchy with applications to QMA(2)

Sevag Gharibian, Miklós Sántha, Jamie Sikora, Aarthi Sundaram, Justin Yirka

Published Sep 20, 2022• 11 citations
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Abstract

The polynomial-time hierarchy (PH) has proven to be a powerful tool for providing separations in computational complexity theory (modulo standard conjectures such as PH do not collapse). Here, we study whether two quantum generalizations of PH can similarly prove separations in the quantum setting. The first generalization, $$\rm{QCPH}$$ , uses classical proofs, and the second, $$\rm{QPH}$$ , uses quantum proofs. For the former, we show quantum variants of the Karp-Lipton theorem and Toda's theorem. For the latter, we place its third level, $$\rm{Q\Sigma_3}$$ , into NEXP using the ellipsoid method for efficiently solving semidefinite programs. These results yield two implications for $$\rm{QMA(2)}$$ , the variant of Quantum Merlin-Arthur ( $$\rm{QMA}$$ ) with two unentangled proofs, a complexity class whose characterization has proven difficult. First, if $$\rm{QCPH = QPH}$$ (i.e., alternating quantifiers are sufficiently powerful so as to make classical and quantum proofs ``equivalent''), then QMA(2) is in the counting hierarchy (specifically, in $${\rm P}^{{\rm pp}^{{\rm pp}}}$$ ). Second, because $$\rm{QMA(2)}\subseteq \rm{Q\Sigma_3}$$ , $$\rm{QMA(2)}$$ is strictly contained in NEXP unless $$\rm{QMA(2)}=\rm{Q\Sigma_3}$$ (i.e., alternating quantifiers do not help in the presence of ``unentanglement'').

DOI
10.1007/s00037-022-00231-8

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The polynomial-time hierarchy (PH) has proven to be a powerful tool for providing separations in computational complexity theory (modulo standard conjectures such as PH do not collapse). Here, we study whether two quantum generalizations of PH can similarly prove separations in the quantum setting. The first generalization, QCPH\rm{QCPH}QCPH , uses classical proofs, and the second, QPH\rm{QPH}QPH , uses quantum proofs. For the former, we show quantum variants of the Karp-Lipton theorem and Toda's theorem. For the latter, we place its third level, QΣ3\rm{Q\Sigma_3}QΣ3​ , into NEXP using the ellipsoid method for efficiently solving semidefinite programs. These results yield two implications for QMA(2)\rm{QMA(2)}QMA(2) , the variant of Quantum Merlin-Arthur ( QMA\rm{QMA}QMA ) with two unentangled proofs, a complexity class whose characterization has proven difficult. First, if QCPH=QPH\rm{QCPH = QPH}QCPH=QPH (i.e., alternating quantifiers are sufficiently powerful so as to make classical and quantum proofs equivalent''), then QMA(2) is in the counting hierarchy (specifically, in $${\rm P}^{{\rm pp}^{{\rm pp}}}$$ ). Second, because $$\rm{QMA(2)}\subseteq \rm{Q\Sigma_3}$$ , $$\rm{QMA(2)}$$ is strictly contained in NEXP unless $$\rm{QMA(2)}=\rm{Q\Sigma_3}$$ (i.e., alternating quantifiers do not help in the presence of unentanglement'').
We prove several new results concerning the pure quantum polynomial hierarchy pureQPH. First, we show that QMA(2) ⊆ pureQΣ_2, i.e., two unentangled existential provers can be simulated by competing existential and universal provers. We further prove that pureQΣ_2 ⊆ QΣ_3 ⊆ 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 pureQΣ_i-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 ∃∀-mixed local Hamiltonian problem lies in NP^QMA ∩ coNP^QMA.
We investigate structural properties of quantum proof systems by establishing collapse results that uncover simplifications in their complexity landscape. We extend classical results such as the Karp-Lipton theorem to quantum polynomial hierarchy with quantum proofs and establish uniqueness preservation for quantum-classical probabilistically checkable proof systems. Our main contributions are threefold. First, we prove that restricting quantum-classical PCP systems to uniqueness does not reduce computational power: UniqueQCPCP=QCPCP\mathsf{UniqueQCPCP} = \mathsf{QCPCP}UniqueQCPCP=QCPCP under BQ\mathsf{BQ}BQ-operator and randomized reductions, demonstrating robustness similar to the UniqueQCMA=QCMA\mathsf{UniqueQCMA} = \mathsf{QCMA}UniqueQCMA=QCMA result. Second, we establish a non-uniform quantum analogue of the Karp-Lipton theorem, showing that if QMA⊆BQP/qpoly\mathsf{QMA} \subseteq \mathsf{BQP}/\mathsf{qpoly}QMA⊆BQP/qpoly, then QPH⊆QΣ2/qpoly\mathsf{QPH} \subseteq \mathsf{QΣ}_2/\mathsf{qpoly}QPH⊆QΣ2​/qpoly, extending the classical collapse theorem to quantum complexity with quantum advice. Third, we introduce a consistent variant of the quantum polynomial hierarchy (CQPH\mathsf{CQPH}CQPH) with consistency constraints across interaction rounds while maintaining product-state proofs, proving its unconditional collapse CQPH=CQΣ2\mathsf{CQPH} = \mathsf{CQΣ}_2CQPH=CQΣ2​. This contrasts with prior work on quantum-entangled polynomial hierarchy, showing that consistency rather than entanglement drives the collapse. These results contribute to understanding structural boundaries in quantum complexity theory and the interplay between constraint types in quantum proof systems.
The Polynomial-Time Hierarchy (PH\mathsf{PH}PH) is a staple of classical complexity theory, with applications spanning randomized computation to circuit lower bounds to ''quantum advantage'' analyses for near-term quantum computers. Quantumly, however, despite the fact that at least \emph{four} definitions of quantum PH\mathsf{PH}PH exist, it has been challenging to prove analogues for these of even basic facts from PH\mathsf{PH}PH. This work studies three quantum-verifier based generalizations of PH\mathsf{PH}PH, two of which are from [Gharibian, Santha, Sikora, Sundaram, Yirka, 2022] and use classical strings (QCPH\mathsf{QCPH}QCPH) and quantum mixed states (QPH\mathsf{QPH}QPH) as proofs, and one of which is new to this work, utilizing quantum pure states (pureQPH\mathsf{pureQPH}pureQPH) as proofs. We first resolve several open problems from [GSSSY22], including a collapse theorem and a Karp-Lipton theorem for QCPH\mathsf{QCPH}QCPH. Then, for our new class pureQPH\mathsf{pureQPH}pureQPH, we show one-sided error reduction for pureQPH\mathsf{pureQPH}pureQPH, as well as the first bounds relating these quantum variants of PH\mathsf{PH}PH, namely QCPH⊆pureQPH⊆EXPPP\mathsf{QCPH}\subseteq \mathsf{pureQPH} \subseteq \mathsf{EXP}^{\mathsf{PP}}QCPH⊆pureQPH⊆EXPPP.
The complexity class QMA(2), featuring two unentangled quantum provers, occupies a central yet poorly understood position in quantum complexity theory. While superficially a modest extension of QMA, unentanglement fundamentally changes both the algorithmic and complexitytheoretic landscape, connecting QMA(2) to deep questions in separability testing, polynomial optimization, and hardness of approximation. Over the past two decades, QMA(2) has inspired a rich body of work spanning restricted verification models, semidefinite and Sum-of-Squares hierarchies, and surprising links to classical problems such as the Unique Games Conjecture. This survey provides an overview of these developments, emphasizing structural insights, algorithmic techniques, and known barriers, while highlighting how progress on QMA(2) continues to illuminate the interplay between quantum information, convex optimization, and classical complexity theory.
Despite the importance of the non-uniform Polynomial-Time Hierarchy (PH/poly) in classical complexity understanding the collapse conditions of the Polynomial-Time Hierarchy (PH), a quantum equivalent of PH/poly has yet to be studied in the literature. We introduce the non-uniform computational Quantum Polynomial-Time Hierarchy (QCPH/mpoly), the quantum equivalent of PH/poly, and show that it collapses if and only if QCPH, the quantum equivalent of PH introduced by Gharibian et al. (comput. complex. 2022), also collapses. We also show that QCPH collapses if coQCMA is contained in QCMA/mpoly. These results are analogous to those of Yap (TCS 1983) commonly used to invoke the collapse of PH in classical complexity. QCPH/mpoly is analogous to QCPH with non-uniform quantum verifier circuits.
This paper furthers existing evidence that quantum computers are capable of computations beyond classical computers. Specifically, we strengthen the collapse of the polynomial hierarchy to the second level if: (i) Quantum computers with postselection are as powerful as classical computers with postselection (PostBQP=PostBPP\mathsf{PostBQP=PostBPP}PostBQP=PostBPP), (ii) any one of several quantum sampling experiments (BosonSampling\mathsf{BosonSampling}BosonSampling, IQP\mathsf{IQP}IQP, DQC1\mathsf{DQC1}DQC1) can be approximately performed by a classical computer (contingent on existing assumptions). This last result implies that if any of these experiment's hardness conjectures hold, then quantum computers can implement functions classical computers cannot (FBQP≠FBPP\mathsf{FBQP\neq FBPP}FBQP=FBPP) unless the polynomial hierarchy collapses to its 2nd level. These results are an improvement over previous work which either achieved a collapse to the third level or were concerned with exact sampling, a physically impractical case. The workhorse of these results is a new technical complexity-theoretic result which we believe could have value beyond quantum computation. In particular, we prove that if there exists an equivalence between problems solvable with an exact counting oracle and problems solvable with an approximate counting oracle, then the polynomial hierarchy collapses to its second level, indeed to ZPPNP\mathsf{ZPP^{NP}}ZPPNP.