Despite immense progress in quantum Hamiltonian complexity in the past decade, little is known about the computational complexity of quantum physics at the thermodynamic limit. In fact, even defining the problem properly is not straight forward. We study the complexity of estimating the ground energy of a fixed, translationally-invariant (TI) Hamiltonian in the thermodynamic limit, to within a given precision; this precision (given by n the number of bits of the approximation) is the sole input to the problem. Understanding the complexity of this problem captures how difficult it is for a physicist to measure or compute another digit in the approximation of a physical quantity in the thermodynamic limit. We show that this problem is contained in FEXP QMA-EXP and is hard for FEXP NEXP . This means that the problem is doubly exponentially hard in the size of the input. As an ingredient in our construction, we study the problem of computing the ground energy of translationally invariant finite 1D chains. A single Hamiltonian term, which is a fixed parameter of the problem, is applied to every pair of particles in a finite chain. In the finite case, the length of the chain is the sole input to the problem and the task is to compute an approximation of the ground energy. No thresholds are provided as in the standard formulation of the local Hamiltonian problem. We show that this problem is contained in FP QMA-EXP and is hard for FP NEXP . Our techniques employ a circular clock structure in which the ground energy is calibrated by the length of the cycle. This requires more precise expressions for the ground energies of the resulting matrices than were required for previous QMA-completeness constructions and even exact analytical bounds for the infinite case which we derive using techniques from spectral graph theory. To our knowledge, this is the first use of the circuit-to-Hamiltonian construction that shows hardness for a function class.
Despite immense progress in quantum Hamiltonian complexity in the past decade, little is known about the computational complexity of quantum physics at the thermodynamic limit. In fact, even defining the problem properly is not straight forward. We study the complexity of estimating the ground energy of a fixed, translationally-invariant (TI) Hamiltonian in the thermodynamic limit, to within a given precision; this precision (given by n the number of bits of the approximation) is the sole input to the problem. Understanding the complexity of this problem captures how difficult it is for a physicist to measure or compute another digit in the approximation of a physical quantity in the thermodynamic limit. We show that this problem is contained in FEXP QMA-EXP and is hard for FEXP NEXP . This means that the problem is doubly exponentially hard in the size of the input. As an ingredient in our construction, we study the problem of computing the ground energy of translationally invariant finite 1D chains. A single Hamiltonian term, which is a fixed parameter of the problem, is applied to every pair of particles in a finite chain. In the finite case, the length of the chain is the sole input to the problem and the task is to compute an approximation of the ground energy. No thresholds are provided as in the standard formulation of the local Hamiltonian problem. We show that this problem is contained in FP QMA-EXP and is hard for FP NEXP . Our techniques employ a circular clock structure in which the ground energy is calibrated by the length of the cycle. This requires more precise expressions for the ground energies of the resulting matrices than were required for previous QMA-completeness constructions and even exact analytical bounds for the infinite case which we derive using techniques from spectral graph theory. To our knowledge, this is the first use of the circuit-to-Hamiltonian construction that shows hardness for a function class.
Despite immense progress in quantum Hamiltonian complexity in the past decade, little is known about the computational complexity of quantum physics at the thermodynamic limit. In fact, even defining the problem properly is not straight forward. We study the complexity of estimating the ground energy of a fixed, translationally-invariant (TI) Hamiltonian in the thermodynamic limit, to within a given precision; this precision (given by n the number of bits of the approximation) is the sole input to the problem. Understanding the complexity of this problem captures how difficult it is for a physicist to measure or compute another digit in the approximation of a physical quantity in the thermodynamic limit. We show that this problem is contained in FEXPQMA-EXP and is hard for FEXPNEXP. This means that the problem is doubly exponentially hard in the size of the input.
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 SIH3k+3 up to a known global energy scale and inverse-polynomial approximation.
Sharper constructions give QMA-completeness of SIH3 and of geometrically local SIH8 with bounded degree and interaction range. In the stoquastic setting, we obtain StoqMA-completeness of Stoq-SIH3 and QMA-completeness of 1-Pinned Stoq-SIH4. 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.
We introduce a classical algorithm to approximate the free energy of local, translation-invariant, one-dimensional quantum systems in the thermodynamic limit of infinite chain size. While the ground state problem (i.e., the free energy at temperature T=0) for these systems is expected to be computationally hard even for quantum computers, our algorithm runs for any fixed temperature T>0 in subpolynomial time, i.e., in time O((ε1)c) for any constant c>0 where ε is the additive approximation error. Previously, the best known algorithm had a runtime that is polynomial in ε1. Our algorithm is also particularly simple as it reduces to the computation of the spectral radius of a linear map. This linear map has an interpretation as a noncommutative transfer matrix and has been studied previously to prove results on the analyticity of the free energy and the decay of correlations. We also show that the corresponding eigenvector of this map gives an approximation of the marginal of the Gibbs state and thereby allows for the computation of various thermodynamic properties of the quantum system.
We introduce a classical algorithm to approximate the free energy of local, translation-invariant, one-dimensional quantum systems in the thermodynamic limit of infinite chain size. While the ground state problem (i.e., the free energy at temperature T=0 ) for these systems is expected to be computationally hard even for quantum computers, our algorithm runs for any fixed temperature T>0 in subpolynomial time, i.e., in time O((1ε)c) for any constant c>0 where ε is the additive approximation error. Previously, the best known algorithm had a runtime that is polynomial in 1ε . Our algorithm is also particularly simple as it reduces to the computation of the spectral radius of a linear map. This linear map has an interpretation as a noncommutative transfer matrix and has been studied previously to prove results on the analyticity of the free energy and the decay of correlations. We also show that the corresponding eigenvector of this map gives an approximation of the marginal of the Gibbs state and thereby allows for the computation of various thermodynamic properties of the quantum system.
Determining spectral gaps in the thermodynamic limit is a central challenge in quantum many-body physics. When a fixed boundary condition is imposed, this problem is undecidable: no algorithm, even with unlimited resources, can decide for every Hamiltonian whether it is gapped. Yet the thermodynamic limit is meant to capture the intrinsic properties of very large systems, which should not depend on how their boundaries are chosen. We therefore consider the bulk spectral gap, which concerns only the intrinsic excitations of the infinite system, and ask what can and cannot be computed about it. First, we give a general algorithm producing a hierarchy of certified upper bounds on the bulk gap, arbitrarily tight at the cost of more computation. Second, we prove that the spectral gap problem remains undecidable in the bulk, already for translationally invariant nearest-neighbour chains of fixed local dimension. Hence no general algorithm produces convergent lower bounds, and no convergent upper-bounding algorithm, ours included, admits a computable convergence rate. The bulk spectral gap is thus certifiable from above but uncomputable from below; more precisely, the bulk-gaplessness promise problem is RE-complete, exactly as hard as the halting problem. The same construction shows that no general algorithm can decide whether a given product state is a ground state, and that convergent bounds on ground-state expectation values admit no computable convergence rate in general. Finally, we obtain the first nontrivial certified upper bounds on the bulk gap of the spin-1/2 kagome Heisenberg antiferromagnet.
We consider the complexity of the local Hamiltonian problem in the context of fermionic Hamiltonians with N=2 supersymmetry and show that the problem remains QMA -complete. Our main motivation for studying this is the well-known fact that the ground state energy of a supersymmetric system is exactly zero if and only if a certain cohomology group is nontrivial. This opens the door to bringing the tools of Hamiltonian complexity to study the computational complexity of a large number of algorithmic problems that arise in homological algebra, including problems in algebraic topology, algebraic geometry, and group theory. We take the first steps in this direction by introducing the k -local Cohomology problem and showing that it is QMA1 -hard and, for a large class of instances, is contained in QMA . We then consider the complexity of estimating normalized Betti numbers and show that this problem is hard for the quantum complexity class DQC1 , and for a large class of instances is contained in BQP . In light of these results, we argue that it is natural to frame many of these homological problems in terms of finding ground states of supersymmetric fermionic systems. As an illustration of this perspective we discuss in some detail the model of Fendley, Schoutens, and de Boer consisting of hard-core fermions on a graph, whose ground state structure encodes l -dimensional holes in the independence complex of the graph. This offers a new perspective on existing quantum algorithms for topological data analysis and suggests new ones.