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.
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.
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]
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 k-local CLHs, for any constant k; 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 4-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.
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 for certain families of commuting local Hamiltonians. Nevertheless, there has been no formal evidence against the possibility that the general CLH problem is QMA-complete.
CLH is complete for the complexity class QIMA, defined through quantum verifiers whose local gates commute. Therefore, CLH is QMA-complete if and only if QIMA=QMA. In this work, we introduce a classical-oracle analogue QIMAO and construct a classical oracle O such that BQPO⊆QIMAO.
Since BQPO⊆QMAO for any classical oracle O, this implies QIMAO=QMAO for our constructed oracle. Thus, our result provides relativized evidence against the possibility that the general commuting local-Hamiltonian problem is BQP-hard, and hence also against the possibility that it is QMA-complete.
We define a general formulation of quantum PCPs, which captures adaptivity and multiple unentangled provers, and give a detailed construction of the quantum reduction to a local Hamiltonian with a constant promise gap. The reduction turns out to be a versatile subroutine to prove properties of quantum PCPs, allowing us to show: (i) Non-adaptive quantum PCPs can simulate adaptive quantum PCPs when the number of proof queries is constant. In fact, this can even be shown to hold when the non-adaptive quantum PCP picks the proof indices simply uniformly at random from a subset of all possible index combinations, answering an open question by Aharonov, Arad, Landau and Vazirani (STOC '09). (ii) If the q -local Hamiltonian problem with constant promise gap can be solved in QCMA , then QPCP[q]⊆QCMA for any q∈O(1) . (iii) If QMA(k) has a quantum PCP for any k≤poly(n) , then QMA(2)=QMA , connecting two of the longest-standing open problems in quantum complexity theory. Moreover, we also show that there exist (quantum) oracles relative to which certain quantum PCP statements are false. Hence, any attempt to prove the quantum PCP conjecture requires, just as was the case for the classical PCP theorem, (quantumly) non-relativizing techniques.
Commuting Hamiltonians lie at the boundary between classical constraint satisfaction and quantum many-body physics, exhibiting rich quantum structure while remaining more tractable than general noncommuting models. In contrast, physical Hamiltonians are rarely exactly commuting, which naturally motivates the study of almost commuting Hamiltonians. Despite their relevance, the implications of approximate commutation are only poorly understood.
In this work, we show how to efficiently approximate any almost commuting 2-local qubit Hamiltonian by a commuting one: we give a new locality-preserving algorithmic rounding technique that maps any 2-local Hamiltonian H=∑i=1mhi with ∥[hi,hj]∥≤ε to a nearby Hamiltonian H^ whose terms pair-wise commute, and which is within overall distance ∥H−H^∥=O(mε1/3). As a consequence, we show that δ-approximations to the ground energy for ε-almost commuting 2-local qubit Hamiltonians lie in NP when δ≫mε1/3, extending the classical containment well beyond the commuting setting. Finally, we present two applications of our rounding framework: Gibbs sampling and fast Hamiltonian simulation for almost commuting systems.
A major goal of quantum complexity theory is to understand which computational problems can be solved with access to certain quantum resources. The subfield of Hamiltonian complexity specifically considers computational problems that ask about properties of local Hamiltonians, which are of critical importance in quantum complexity because they can be viewed as quantum generalizations of classical constraint satisfaction problems. In this work, we study the complexity of certain restricted variants of the Quantum-k-Sat problem, a quantum analog of the NP-complete k-Sat problem. We introduce new variants of Quantum-k-Sat which place a basis restriction on the input Hamiltonian H = Σᵢ hᵢ . Each variant is defined by a fixed collection of bases B₁, . . . , Bᵣ of n-qubit space. We require that each Hamiltonian term hi must be diagonal in one of these bases. Our results resolve the complexity of certaim basis-restricted variants of Quantum-k-Sat. First we show the Quantum-6-Sat problem with Hamiltonian terms restricted to be diagonal in an X/Z mixed basis is QMA₁-complete. Second, we combine basis restriction with the restriction of commutativity, and show the following easiness result, which applies generally to higher-level quantum systems (qudits) and bases Q and R (which are real-valued and satisfy an overlap condition): The commmuting Quantum-Sat problem on qudits, where Hamiltonian terms are either diagonal in the Q basis, the R basis, or a single mixed Q/R basis, is in NP.