Quantum Generalizations of the Polynomial Hierarchy with Applications to QMA(2)

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 does not collapse). Here, we study whether two quantum generalizations of PH can similarly prove separations in the quantum setting. The first generalization, QCPH, uses classical proofs, and the second, QPH, uses quantum proofs. For the former, we show quantum variants of the Karp-Lipton theorem and Toda\u27s theorem. For the latter, we place its third level, Q Sigma_3, into NEXP using the Ellipsoid Method for efficiently solving semidefinite programs. These results yield two implications for QMA(2), the variant of Quantum Merlin-Arthur (QMA) with two unentangled proofs, a complexity class whose characterization has proven difficult. First, if 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 P^{PP^{PP}}). Second, unless QMA(2)= Q Sigma_3 (i.e., alternating quantifiers do not help in the presence of "unentanglement"), QMA(2) is strictly contained in NEXP

Limitations of semidefinite programs for separable states and entangled games

Semidefinite programs (SDPs) are a framework for exact or approximate optimization that have widespread application in quantum information theory. We introduce a new method for using reductions to construct integrality gaps for SDPs. These are based on new limitations on the sum-of-squares (SoS) hierarchy in approximating two particularly important sets in quantum information theory, where previously no $\omega(1)$-round integrality gaps were known: the set of separable (i.e. unentangled) states, or equivalently, the $2 \rightarrow 4$ norm of a matrix, and the set of quantum correlations; i.e. conditional probability distributions achievable with local measurements on a shared entangled state. In both cases no-go theorems were previously known based on computational assumptions such as the Exponential Time Hypothesis (ETH) which asserts that 3-SAT requires exponential time to solve. Our unconditional results achieve the same parameters as all of these previous results (for separable states) or as some of the previous results (for quantum correlations). In some cases we can make use of the framework of Lee-Raghavendra-Steurer (LRS) to establish integrality gaps for any SDP, not only the SoS hierarchy. Our hardness result on separable states also yields a dimension lower bound of approximate disentanglers, answering a question of Watrous and Aaronson et al. These results can be viewed as limitations on the monogamy principle, the PPT test, the ability of Tsirelson-type bounds to restrict quantum correlations, as well as the SDP hierarchies of Doherty-Parrilo-Spedalieri, Navascues-Pironio-Acin and Berta-Fawzi-Scholz.Comment: 47 pages. v2. small changes, fixes and clarifications. published versio

Hardness of approximation for quantum problems

The polynomial hierarchy plays a central role in classical complexity theory. Here, we define a quantum generalization of the polynomial hierarchy, and initiate its study. We show that not only are there natural complete problems for the second level of this quantum hierarchy, but that these problems are in fact hard to approximate. Using these techniques, we also obtain hardness of approximation for the class QCMA. Our approach is based on the use of dispersers, and is inspired by the classical results of Umans regarding hardness of approximation for the second level of the classical polynomial hierarchy [Umans, FOCS 1999]. The problems for which we prove hardness of approximation for include, among others, a quantum version of the Succinct Set Cover problem, and a variant of the local Hamiltonian problem with hybrid classical-quantum ground states.Comment: 21 pages, 1 figure, extended abstract appeared in Proceedings of the 39th International Colloquium on Automata, Languages and Programming (ICALP), pages 387-398, Springer, 201

Uniform Diagonalization Theorem for Complexity Classes of Promise Problems including Randomized and Quantum Classes

Diagonalization in the spirit of Cantor's diagonal arguments is a widely used tool in theoretical computer sciences to obtain structural results about computational problems and complexity classes by indirect proofs. The Uniform Diagonalization Theorem allows the construction of problems outside complexity classes while still being reducible to a specific decision problem. This paper provides a generalization of the Uniform Diagonalization Theorem by extending it to promise problems and the complexity classes they form, e.g. randomized and quantum complexity classes. The theorem requires from the underlying computing model not only the decidability of its acceptance and rejection behaviour but also of its promise-contradicting indifferent behaviour - a property that we will introduce as "total decidability" of promise problems. Implications of the Uniform Diagonalization Theorem are mainly of two kinds: 1. Existence of intermediate problems (e.g. between BQP and QMA) - also known as Ladner's Theorem - and 2. Undecidability if a problem of a complexity class is contained in a subclass (e.g. membership of a QMA-problem in BQP). Like the original Uniform Diagonalization Theorem the extension applies besides BQP and QMA to a large variety of complexity class pairs, including combinations from deterministic, randomized and quantum classes.Comment: 15 page

Product-state approximations to quantum ground states

The local Hamiltonian problem consists of estimating the ground-state energy (given by the minimum eigenvalue) of a local quantum Hamiltonian. It can be considered as a quantum generalization of constraint satisfaction problems (CSPs) and has a key role in quantum complexity theory, being the first and most natural QMA-complete problem known. An interesting regime for the local Hamiltonian problem is that of extensive error, where one is interested in estimating the mean ground-state energy to constant accuracy. The problem is NP-hard by the PCP theorem, but whether it is QMA-hard is an important open question in quantum complexity theory. A positive solution would represent a quantum analogue of the PCP theorem. A key feature that distinguishes quantum Hamiltonians from classical CSPs is that the solutions may involve complicated entangled states. In this paper, we demonstrate several large classes of Hamiltonians for which product (i.e. unentangled) states can approximate the ground state energy to within a small extensive error. First, we show the mere existence of a good product-state approximation for the ground-state energy of 2-local Hamiltonians with one of more of the following properties: (1) super-constant degree, (2) small expansion, or (3) a ground state with sublinear entanglement with respect to some partition into small pieces. The approximation based on degree is a new and surprising difference between quantum Hamiltonians and classical CSPs, since in the classical setting, higher degree is usually associated with harder CSPs. The approximation based on expansion is not new, but the approximation based on low entanglement was previously known only in the regime where the entanglement was close to zero. Since the existence of a low-energy product state can be checked in NP, this implies that any Hamiltonian used for a quantum PCP theorem should have: (1) constant degree, (2) constant expansion, (3) a ``volume law'' for entanglement with respect to any partition into small parts. Second, we show that in several cases, good product-state approximations not only exist, but can be found in deterministic polynomial time: (1) 2-local Hamiltonians on any planar graph, solving an open problem of Bansal, Bravyi, and Terhal, (2) dense k-local Hamiltonians for any constant k, solving an open problem of Gharibian and Kempe, and (3) 2-local Hamiltonians on graphs with low threshold rank, via a quantum generalization of a recent result of Barak, Raghavendra and Steurer. Our work involves two new tools which may be of independent interest. First, we prove a new quantum version of the de Finetti theorem which does not require the usual assumption of symmetry. Second, we describe a way to analyze the application of the Lasserre/Parrilo SDP hierarchy to local quantum Hamiltonians

The Importance of the Spectral Gap in Estimating Ground-State Energies

The field of quantum Hamiltonian complexity lies at the intersection of quantum many-body physics and computational complexity theory, with deep implications to both fields. The main object of study is the Local Hamiltonian problem, which is concerned with estimating the ground-state energy of a local Hamiltonian and is complete for the class QMA, a quantum generalization of the class NP. A major challenge in the field is to understand the complexity of the Local Hamiltonian problem in more physically natural parameter regimes. One crucial parameter in understanding the ground space of any Hamiltonian in many-body physics is the spectral gap, which is the difference between the smallest two eigenvalues. Despite its importance in quantum many-body physics, the role played by the spectral gap in the complexity of the Local Hamiltonian problem is less well-understood. In this work, we make progress on this question by considering the precise regime, in which one estimates the ground-state energy to within inverse exponential precision. Computing ground-state energies precisely is a task that is important for quantum chemistry and quantum many-body physics. In the setting of inverse-exponential precision (promise gap), there is a surprising result that the complexity of Local Hamiltonian is magnified from QMA to PSPACE, the class of problems solvable in polynomial space (but possibly exponential time). We clarify the reason behind this boost in complexity. Specifically, we show that the full complexity of the high precision case only comes about when the spectral gap is exponentially small. As a consequence of the proof techniques developed to show our results, we uncover important implications for the representability and circuit complexity of ground states of local Hamiltonians, the theory of uniqueness of quantum witnesses, and techniques for the amplification of quantum witnesses in the presence of postselection