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arXiv:1311.7378

The commuting local Hamiltonian on locally-expanding graphs is in NP

Dorit Aharonov, Lior Eldar

Published Nov 28, 2013• 6 citations
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Abstract

The local Hamiltonian problem is famously complete for the class QMA, the quantum analogue of NP. The complexity of its semi-classical version, in which the terms of the Hamiltonian are required to commute (the CLH problem), has attracted considerable attention recently due to its intriguing nature, as well as in relation to growing interest in the qPCP conjecture. We show here that if the underlying bipartite interaction graph of the CLH instance is a good locally-expanding graph, namely, the expansion of any constant-size set is e-close to optimal, then approximating its ground energy to within additive factor O(e) lies in NP. The proof holds for k- local Hamiltonians for any constant k and any constant dimensionality of particles d. We also show that the approximation problem of CLH on such good local expanders is NP-hard. This implies that too good local expansion of the interaction graph constitutes an obstacle against quantum hardness of the approximation problem, though it retains its classical hardness. The result highlights new difficulties in trying to mimic classical proofs (in particular Dinur's PCP proof) in an attempt to prove the quantum PCP conjecture. A related result was discovered recently independently by Brandao and Harrow, for 2-local general Hamiltonians, bounding the quantum hardness of the approximation problem on good expanders, though no NP-hardness is known in that case.

DOI
10.48550/arxiv.1311.7378

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The local Hamiltonian problem is famously complete for the class QMA, the quantum analogue of NP. The complexity of its semi-classical version, in which the terms of the Hamiltonian are required to commute (the CLH problem), has attracted considerable attention recently due to its intriguing nature, as well as in relation to growing interest in the qPCP conjecture. We show here that if the underlying bipartite interaction graph of the CLH instance is a good locally-expanding graph, namely, the expansion of any constant-size set is e-close to optimal, then approximating its ground energy to within additive factor O(e) lies in NP. The proof holds for k- local Hamiltonians for any constant k and any constant dimensionality of particles d. We also show that the approximation problem of CLH on such good local expanders is NP-hard. This implies that too good local expansion of the interaction graph constitutes an obstacle against quantum hardness of the approximation problem, though it retains its classical hardness. The result highlights new difficulties in trying to mimic classical proofs (in particular Dinur's PCP proof) in an attempt to prove the quantum PCP conjecture. A related result was discovered recently independently by Brandao and Harrow, for 2-local general Hamiltonians, bounding the quantum hardness of the approximation problem on good expanders, though no NP-hardness is known in that case.
The commuting local-Hamiltonian (CLH) problem is a restriction of the local-Hamiltonian problem, in which the terms of the Hamiltonian are required to pairwise commute. A long line of work has shown that the problem lies in NP\mathsf{NP}NP for certain families of commuting local Hamiltonians. Nevertheless, there has been no formal evidence against the possibility that the general CLH problem is QMA\mathsf{QMA}QMA-complete. CLH is complete for the complexity class QIMA\mathsf{QIMA}QIMA, defined through quantum verifiers whose local gates commute. Therefore, CLH is QMA\mathsf{QMA}QMA-complete if and only if QIMA=QMA\mathsf{QIMA}=\mathsf{QMA}QIMA=QMA. In this work, we introduce a classical-oracle analogue QIMAO\mathsf{QIMA}^{\mathcal{O}}QIMAO and construct a classical oracle O\mathcal{O}O such that BQPO⊈QIMAO\mathsf{BQP}^{\mathcal{O}}\not\subseteq\mathsf{QIMA}^{\mathcal{O}}BQPO⊆QIMAO. Since BQPO⊆QMAO\mathsf{BQP}^{\mathcal{O}}\subseteq\mathsf{QMA}^{\mathcal{O}}BQPO⊆QMAO for any classical oracle O\mathcal{O}O, this implies QIMAO≠QMAO\mathsf{QIMA}^{\mathcal{O}}\neq\mathsf{QMA}^{\mathcal{O}}QIMAO=QMAO for our constructed oracle. Thus, our result provides relativized evidence against the possibility that the general commuting local-Hamiltonian problem is BQP\mathsf{BQP}BQP-hard, and hence also against the possibility that it is QMA\mathsf{QMA}QMA-complete.
The complexity of the commuting local Hamiltonians (CLH) problem still remains a mystery after two decades of research of quantum Hamiltonian complexity; it is only known to be contained in NP for few low parameters. Of particular interest is the tightly related question of understanding whether groundstates of CLHs can be generated by efficient quantum circuits. The two problems touch upon conceptual, physical and computational questions, including the centrality of non-commutation in quantum mechanics, quantum PCP and the area law. It is natural to try to address first the more physical case of CLHs embedded on a 2D lattice but this problem too remained open, apart from some very specific cases. Here we consider a wide class of two dimensional CLH instances; these are kkk-local CLHs, for any constant kkk; they are defined on qubits set on the edges of any surface complex, where we require that this surface complex is not too far from being "Euclidean". Each vertex and each face can be associated with an arbitrary term (as long as the terms commute). We show that this class is in NP, and moreover that the groundstates have an efficient quantum circuit that prepares them. This result subsumes that of Schuch [2011] which regarded the special case of 444-local Hamiltonians on a grid with qubits, and by that it removes the mysterious feature of Schuch's proof which showed containment in NP without providing a quantum circuit for the groundstate and considerably generalizes it. We believe this work and the tools we develop make a significant step towards showing that 2D CLHs are in NP.
Commuting local Hamiltonians provide a testing ground for studying many of the most interesting open questions in quantum information theory, including the quantum PCP conjecture and the nature of entanglement. However, unlike the general local Hamiltonian problem, the exact complexity of the commuting local Hamiltonian problem (CLH) remains unknown. A number of works have shown that increasingly expressive families of commuting local Hamiltonians admit classical verifiers. Despite intense work, proofs placing CLH in NP rely heavily on an underlying 2D lattice structure, or a very constrained local dimension and locality. In this work, we present a new technique to analyze the complexity of various families of commuting local Hamiltonians: guided reductions. Intuitively, these are a generalization of typical reduction where the prover provides a guide so that the verifier can construct a simpler Hamiltonian. The core of our reduction is a new rounding technique based on a combination of Jordan’s Lemma for pairs of projectors and the Structure Lemma for C^* algebras. Our rounding technique is much more flexible than previous work and allows us to remove constraints on local dimension in exchange for a rank-1 assumption. Using our rounding technique, we prove the following two results: 1) 2D-CLH for rank-1 instances are contained in NP, independent of the qudit dimension. It is notable that this family of commuting local Hamiltonians has no restriction on the local dimension or the locality of the Hamiltonian terms. 2) 3D-CLH for rank-1 instances are in NP. To our knowledge this is the first time a family of {3D} commuting local Hamiltonians has been contained in NP. Our results apply to Hamiltonians with large qudit degree and remain non-trivial despite the quantum Lovász Local Lemma. [Andris Ambainis et al., 2012]
We study the complexity of Local Hamiltonian problems under the restriction that every local term is an unweighted occurrence of the same fixed interaction. In the resulting Single-Interaction Hamiltonian (SIH) model, once the interaction is fixed, the Hamiltonian is specified entirely by its ordered interaction hypergraph. Our main technical tool is an exact compiler that converts any fixed finite alphabet of local interactions into a single interaction. On a designated auxiliary subspace, the compiled Hamiltonian reproduces the source Hamiltonian exactly, while the complete spectrum below a chosen separation scale is preserved with multiplicity. Combined with a finite-alphabet normalization, this gives a universal polynomial-time reduction from kkk-Local Hamiltonian to SIH3k+3_{3k+3}3k+3​ up to a known global energy scale and inverse-polynomial approximation. Sharper constructions give QMA-completeness of SIH3_33​ and of geometrically local SIH8_88​ with bounded degree and interaction range. In the stoquastic setting, we obtain StoqMA-completeness of Stoq-SIH3_33​ and QMA-completeness of 1-Pinned Stoq-SIH4_44​. We also establish hardness results for uniformly system-size-dependent interactions and derive single-interaction formulations of the Hamiltonian quantum PCP conjecture, frustration-free, exponentially precise, and guided Local Hamiltonian problems. These results show that independently chosen local matrices and independently tunable coupling strengths are not necessary for a broad range of Hamiltonian-complexity phenomena.
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.