Local Hamiltonians are a quantum analogue to SAT in classical complexity. As a model of “inter-
mediate” complexity, 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, while possibly being more amenable to analysis. Despite its simpler
nature, the exact complexity of the commuting local Hamiltonian problem (CLH) remains elusive. A
number of works have shown that increasingly expressive families of commuting local Hamiltonians ad-
mit completely 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 and the Structure
Lemma. 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. Specifically, 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
Lov´asz Local Lemma.

Updated proofs and diagrams.

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 existence of area laws. Although they are a simplified model of quantum computation, the status of the commuting local Hamiltonian problem remains largely unknown. A number of works have shown that increasingly expressive families of commuting local Hamiltonians admit completely classical verifiers. Despite intense work, the largest class of commuting local Hamiltonians we can place in NP are those on a square lattice, where each lattice site is a qutrit. Even worse, many of the techniques used to analyze these problems rely heavily on the geometry of the square lattice and the properties of the numbers 2 and 3 as local dimensions. 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 and the Structure Lemma. Our rounding technique is much more flexible than previous work, and allows us to show that a larger family of commuting local Hamiltonians is in NP, albiet with the restriction that all terms are rank-1. Specifically, we prove the following two results:
1. Commuting local Hamiltonians in 2D that are rank-1 are contained in NP, independent of the qudit dimension. Note that this family of commuting local Hamiltonians has no restriction on the local dimension or the locality.
2. We prove that rank-1, 3D commuting Hamiltonians with qudits on edges are in NP. To our knowledge this is the first time a family of 3D commuting local Hamiltonians has been contained in NP.