We show that the NP-complete problems max cut and independent set can be formulated as the 2-local Hamiltonian problem as defined by Kitaev. The 5-local Hamiltonian problem was the first problem to be shown to be complete for the quantum complexity class QMA — the quantum analog of NP. Subsequently, it was shown that 3-locality is already sufficient for QMA-completeness. It is still not known whether the 2-local Hamiltonian problem is QMA-complete. Therefore it is interesting to determine what problems can be reduced to the 2-local Hamiltonian problem. Kitaev showed that 3-SAT can be formulated as a 3-local Hamiltonian problem. We extend his result by showing that 2-locality is sufficient in order to encompass NP.We show that the NP-complete problems max cut and independent set can be formulated as the 2-local Hamiltonian problem as defined by Kitaev. The 5-local Hamiltonian problem was the first problem to be shown to be complete for the quantum complexity class QMA — the quantum analog of NP. Subsequently, it was shown that 3-locality is already sufficient for QMA-completeness. It is still not known whether the 2-local Hamiltonian problem is QMA-complete. Therefore it is interesting to determine what problems can be reduced to the 2-local Hamiltonian problem. Kitaev showed that 3-SAT can be formulated as a 3-local Hamiltonian problem. We extend his result by showing that 2-locality is sufficient in order to encompass NP.

arXiv:quant-ph/0301087International Journal of Quantum InformationSep 1, 200316 citations

Localization goes long

Chris R. Laumann, Norman Y. Yao

Nature PhysicsAug 1, 20161 citation

The Commuting Local Hamiltonian Problem: Relativized Evidence Against BQP-Hardness

Itay Shalit, Mark Zhandry

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 $\mathsf{NP}$ for certain families of commuting local Hamiltonians. Nevertheless, there has been no formal evidence against the possibility that the general CLH problem is $\mathsf{QMA}$ -complete. CLH is complete for the complexity class $\mathsf{QIMA}$ , defined through quantum verifiers whose local gates commute. Therefore, CLH is $\mathsf{QMA}$ -complete if and only if $\mathsf{QIMA}=\mathsf{QMA}$ . In this work, we introduce a classical-oracle analogue $\mathsf{QIMA}^{\mathcal{O}}$ and construct a classical oracle $\mathcal{O}$ such that $\mathsf{BQP}^{\mathcal{O}}\not\subseteq\mathsf{QIMA}^{\mathcal{O}}$ . Since $\mathsf{BQP}^{\mathcal{O}}\subseteq\mathsf{QMA}^{\mathcal{O}}$ for any classical oracle $\mathcal{O}$ , this implies $\mathsf{QIMA}^{\mathcal{O}}\neq\mathsf{QMA}^{\mathcal{O}}$ for our constructed oracle. Thus, our result provides relativized evidence against the possibility that the general commuting local-Hamiltonian problem is $\mathsf{BQP}$ -hard, and hence also against the possibility that it is $\mathsf{QMA}$ -complete.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 $\mathsf{NP}$ for certain families of commuting local Hamiltonians. Nevertheless, there has been no formal evidence against the possibility that the general CLH problem is $\mathsf{QMA}$-complete. CLH is complete for the complexity class $\mathsf{QIMA}$, defined through quantum verifiers whose local gates commute. Therefore, CLH is $\mathsf{QMA}$-complete if and only if $\mathsf{QIMA}=\mathsf{QMA}$. In this work, we introduce a classical-oracle analogue $\mathsf{QIMA}^{\mathcal{O}}$ and construct a classical oracle $\mathcal{O}$ such that $\mathsf{BQP}^{\mathcal{O}}\not\subseteq\mathsf{QIMA}^{\mathcal{O}}$. Since $\mathsf{BQP}^{\mathcal{O}}\subseteq\mathsf{QMA}^{\mathcal{O}}$ for any classical oracle $\mathcal{O}$, this implies $\mathsf{QIMA}^{\mathcal{O}}\neq\mathsf{QMA}^{\mathcal{O}}$ for our constructed oracle. Thus, our result provides relativized evidence against the possibility that the general commuting local-Hamiltonian problem is $\mathsf{BQP}$-hard, and hence also against the possibility that it is $\mathsf{QMA}$-complete.

quant-pharXiv:2609.38957Oct 1, 20260 citations

The Complexity of the Separable Hamiltonian Problem

André Chailloux, Or Sattath

In this paper, we study variants of the canonical Local Hamiltonian problem where, in addition, the witness is promised to be separable. We define two variants of the Local Hamiltonian problem. The input for the Separable Local Hamiltonian problem is the same as the Local Hamiltonian problem, i.e. a local Hamiltonian and two energies a and b, but the question is somewhat different: the answer is YES if there is a separable quantum state with energy at most a, and the answer is NO if all separable quantum states have energy at least b. The Separable Sparse Hamiltonian problem is defined similarly, but the Hamiltonian is not necessarily local, but rather sparse. We show that the Separable Sparse Hamiltonian problem is QMA(2)-Complete, while Separable Local Hamiltonian is in QMA. This should be compared to the Local Hamiltonian problem, and the Sparse Hamiltonian problem which are both QMA-Complete. To the best of our knowledge, Separable Sparse Hamiltonian is the first non-trivial problem shown to be QMA(2)-Complete.In this paper, we study variants of the canonical Local Hamiltonian problem where, in addition, the witness is promised to be separable. We define two variants of the Local Hamiltonian problem. The input for the Separable Local Hamiltonian problem is the same as the Local Hamiltonian problem, i.e. a local Hamiltonian and two energies a and b, but the question is somewhat different: the answer is YES if there is a separable quantum state with energy at most a, and the answer is NO if all separable quantum states have energy at least b. The Separable Sparse Hamiltonian problem is defined similarly, but the Hamiltonian is not necessarily local, but rather sparse. We show that the Separable Sparse Hamiltonian problem is QMA(2)-Complete, while Separable Local Hamiltonian is in QMA. This should be compared to the Local Hamiltonian problem, and the Sparse Hamiltonian problem which are both QMA-Complete. To the best of our knowledge, Separable Sparse Hamiltonian is the first non-trivial problem shown to be QMA(2)-Complete.

arXiv:1111.5247IEEE Conference on Computational Complexity (CCC)Jun 1, 201210 citations

The complexity of the separable hamiltonian problem

André Chailloux, Or Sattath

In this paper, we study variants of the canonical Local-Hamiltonian problem where, in addition, the witness is promised to be separable. We define two variants of the Local-Hamiltonian problem. The input for the Separable-Local-Hamiltonian problem is the same as the Local-Hamiltonian problem, i.e. a local Hamiltonian and two energies a and b, but the question is somewhat different: the answer is YES if there is a separable quantum state with energy at most a, and the answer is NO if all separable quantum states have energy at least b. The Separable-Sparse-Hamiltonian problem is defined similarly, but the Hamiltonian is not necessarily local, but rather sparse. We show that the Separable-Sparse-Hamiltonian problem is QMA(2)-Complete, while Separable-Local-Hamiltonian is in QMA. This should be compared to the Local-Hamiltonian problem, and the Sparse-Hamiltonian problem which are both QMA-Complete. To the best of our knowledge, Separable-SPARSE-Hamiltonian is the first non-trivial problem shown to be QMA(2)-Complete.In this paper, we study variants of the canonical Local-Hamiltonian problem where, in addition, the witness is promised to be separable. We define two variants of the Local-Hamiltonian problem. The input for the Separable-Local-Hamiltonian problem is the same as the Local-Hamiltonian problem, i.e. a local Hamiltonian and two energies a and b, but the question is somewhat different: the answer is YES if there is a separable quantum state with energy at most a, and the answer is NO if all separable quantum states have energy at least b. The Separable-Sparse-Hamiltonian problem is defined similarly, but the Hamiltonian is not necessarily local, but rather sparse. We show that the Separable-Sparse-Hamiltonian problem is QMA(2)-Complete, while Separable-Local-Hamiltonian is in QMA. This should be compared to the Local-Hamiltonian problem, and the Sparse-Hamiltonian problem which are both QMA-Complete. To the best of our knowledge, Separable-SPARSE-Hamiltonian is the first non-trivial problem shown to be QMA(2)-Complete.

Jan 1, 201214 citations

\mathsf{NP}

for certain families of commuting local Hamiltonians. Nevertheless, there has been no formal evidence against the possibility that the general CLH problem is

\mathsf{QMA}

-complete. CLH is complete for the complexity class

\mathsf{QIMA}

, defined through quantum verifiers whose local gates commute. Therefore, CLH is

\mathsf{QMA}

-complete if and only if

\mathsf{QIMA}=\mathsf{QMA}

. In this work, we introduce a classical-oracle analogue

\mathsf{QIMA}^{\mathcal{O}}

and construct a classical oracle

\mathcal{O}

such that

\mathsf{BQP}^{\mathcal{O}}\not\subseteq\mathsf{QIMA}^{\mathcal{O}}

. Since

\mathsf{BQP}^{\mathcal{O}}\subseteq\mathsf{QMA}^{\mathcal{O}}

for any classical oracle

\mathcal{O}

, this implies

\mathsf{QIMA}^{\mathcal{O}}\neq\mathsf{QMA}^{\mathcal{O}}

for our constructed oracle. Thus, our result provides relativized evidence against the possibility that the general commuting local-Hamiltonian problem is

\mathsf{BQP}

-hard, and hence also against the possibility that it is

\mathsf{QMA}

-complete.