Estimation of the minimum eigenvalue of a quantum Hamiltonian can be formalised as the Local Hamiltonian problem. We study the natural special case of the Local Hamiltonian problem where the same 2-local interaction, with differing weights, is applied across each pair of qubits. First we consider antiferromagnetic/ferromagnetic interactions, where the weights of the terms in the Hamiltonian are restricted to all be of the same sign. We show that for symmetric 2-local interactions with no 1-local part, the problem is either QMA-complete or in StoqMA. In particular the antiferromagnetic Heisenberg and antiferromagnetic XY interactions are shown to be QMA-complete. We also prove StoqMA-completeness of the antiferromagnetic transverse field Ising model. Second, we study the Local Hamiltonian problem under the restriction that the interaction terms can only be chosen to lie on a particular graph. We prove that nearly all of the QMA-complete 2-local interactions remain QMA-complete when restricted to a 2D square lattice. Finally we consider both restrictions at the same time and discover that, with the exception of the antiferromagnetic Heisenberg interaction, all of the interactions which are QMA-complete with positive coefficients remain QMA-complete when restricted to a 2D triangular lattice.
The calculation of ground-state energies of physical systems can be formalised as the k-local Hamiltonian problem, which is the natural quantum analogue of classical constraint satisfaction problems. One way of making the problem more physically meaningful is to restrict the Hamiltonian in question by picking its terms from a fixed set S. Examples of such special cases are the Heisenberg and Ising models from condensed-matter physics. In this work we characterise the complexity of this problem for all 2-local qubit Hamiltonians. Depending on the subset S, the problem falls into one of the following categories: in P, NP-complete, polynomial-time equivalent to the Ising model with transverse magnetic fields, or QMA-complete. The third of these classes contains NP and is contained within StoqMA. The characterisation holds even if S does not contain any 1-local terms, for example, we prove for the first time QMA-completeness of the Heisenberg and XY interactions in this setting. If S is assumed to contain all 1-local terms, which is the setting considered by previous work, we have a characterisation that goes beyond 2-local interactions: for any constant k, all k-local qubit Hamiltonians whose terms are picked from a fixed set S correspond to problems either in P, polynomial-time equivalent to the Ising model with transverse magnetic fields, or QMA-complete. These results are a quantum analogue of Schaefer's dichotomy theorem for boolean constraint satisfaction problems.
The calculation of ground-state energies of physical systems can be formalised as the k-local Hamiltonian problem, which is the natural quantum analogue of classical constraint satisfaction problems. One way of making the problem more physically meaningful is to restrict the Hamiltonian in question by picking its terms from a fixed set S. Examples of such special cases are the Heisenberg and Ising models from condensed-matter physics. In this work we characterise the complexity of this problem for all 2-local qubit Hamiltonians. Depending on the subset S, the problem falls into one of the following categories: in P; NP-complete; polynomial-time equivalent to the Ising model with transverse magnetic fields; or QMA-complete. The third of these classes has been shown to be StoqMA-complete by Bravyi and Hastings. The characterisation holds even if S does not contain any 1-local terms; for example, we prove for the first time QMA-completeness of the Heisenberg and XY interactions in this setting. If S is assumed to contain all 1-local terms, which is the setting considered by previous work, we have a characterisation that goes beyond 2-local interactions: for any constant k, all k-local qubit Hamiltonians whose terms are picked from a fixed set S correspond to problems either in P; polynomial-time equivalent to the Ising model with transverse magnetic fields; or QMA-complete. These results are a quantum analogue of Schaefer's dichotomy theorem for boolean constraint satisfaction problems.
The calculation of ground-state energies of physical systems can be formalized as the
k-local Hamiltonian problem, which is a natural quantum analogue of classical constraint satisfaction problems. One way of making the problem more physically meaningful is to restrict the Hamiltonian in question by picking its terms from a fixed set
S and scaling them by arbitrary weights. Examples of such special cases are the Heisenberg and Ising models from condensed-matter physics. In this work we characterize the complexity of this problem for all 2-local qubit Hamiltonians. Depending on the subset
S, the problem falls into one of the following categories: in
P;
NP-complete; polynomial-time equivalent to the Ising model with transverse magnetic fields; or
QMA-complete. The third of these classes has been shown to be
StoqMA-complete by Bravyi and Hastings. The characterization holds even if
S does not contain any 1-local terms; for example, we prove for the first time
QMA-completeness of the Heisenberg and XY interactions in this setting. If
S is assumed to contain all 1-local terms, which is the setting considered by previous work, we have a characterization that goes beyond 2-local interactions: for any constant
k, all
k-local qubit Hamiltonians whose terms are picked from a fixed set
S correspond to problems either in
P; polynomial-time equivalent to the Ising model with transverse magnetic fields; or
QMA-complete. These results are a quantum analogue of the maximization variant of Schaefer's dichotomy theorem for Boolean constraint satisfaction problems.
The problem 2-LOCAL HAMILTONIAN has been shown to be complete for the quantum computational class QMA, see quant-ph/0406180. In this paper we show that this important problem remains QMA-complete when the interactions of the 2-local Hamiltonian are restricted to a two-dimensional (2-D) square lattice. Our results are partially derived with novel perturbation gadgets that employ mediator qubits which allow us to manipulate 2-local interactions. As a side result, we obtain that quantum adiabatic computation using 2-local interactions restricted to a 2-D square lattice is equivalent to the circuit model of quantum computation.
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
k-Local Hamiltonian to SIH
3k+3 up to a known global energy scale and inverse-polynomial approximation.
Sharper constructions give QMA-completeness of SIH
3 and of geometrically local SIH
8 with bounded degree and interaction range. In the stoquastic setting, we obtain StoqMA-completeness of Stoq-SIH
3 and QMA-completeness of 1-Pinned Stoq-SIH
4. 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 problem 2-LOCAL HAMILTONIAN has been shown to be complete for the quantum computational class QMA. In this paper we show that this important problem remains QMA-complete when the interactions of the 2-local Hamiltonian are between qubits on a two-dimensional (2-D) square lattice. Our results are partially derived with novel perturbation gadgets that employ mediator qubits which allow us to manipulate k-local interactions. As a side result, we obtain that quantum adiabatic computation using 2-local interactions restricted to a 2-D square lattice is equivalent to the circuit model of quantum computation. Our perturbation method also shows how any stabilizer space associated with a k-local stabilizer (for constant k) can be generated as an approximate ground-space of a 2-local Hamiltonian.