Computational complexity of the ground state energy density problem

@article{Watson2022ComputationalCO,
  title={Computational complexity of the ground state energy density problem},
  author={James D Watson and Toby S. Cubitt},
  journal={Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing},
  year={2022}
}
  • J. Watson, T. Cubitt
  • Published 11 July 2021
  • Physics
  • Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing
We study the complexity of finding the ground state energy density of a local Hamiltonian on a lattice in the thermodynamic limit of infinite lattice size. We formulate this rigorously as a function problem, in which we request an estimate of the ground state energy density to some specified precision; and as an equivalent promise problem, GSED, in which we ask whether the ground state energy density is above or below specified thresholds. The ground state energy density problem is unusual, in… 

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References

SHOWING 1-10 OF 39 REFERENCES
Hamiltonian complexity in the thermodynamic limit
TLDR
This work studies the complexity of estimating the ground energy of a fixed, translationally invariant Hamiltonian in the thermodynamic limit, to within a given precision; the number of bits $n$ for the precision is the sole input to the problem.
Ground State Connectivity of Local Hamiltonians
TLDR
This article introduces the physically motivated notion of “ground state connectivity” of local Hamiltonians, which captures problems in areas ranging from quantum stabilizer codes to quantum memories, and obtains a natural QCMA-complete problem.
Undecidability of the spectral gap
TLDR
This work constructs families of quantum spin systems on a two-dimensional lattice with translationally invariant, nearest-neighbour interactions, for which the spectral gap problem is undecidable, and implies that there exists no algorithm to determine whether an arbitrary model is gapped or gapless.
Computational Difficulty of Computing the Density of States
TLDR
The difficulty of both problems is exactly captured by a class which is the counting version of the quantum complexity class quantum Merlin Arthur, which implies that computing the ground state degeneracy or the density of states for classical Hamiltonians is just as hard as it is for quantum Hamiltonians.
The Quantum and Classical Complexity of Translationally Invariant Tiling and Hamiltonian Problems
  • D. GottesmanS. Irani
  • Mathematics
    2009 50th Annual IEEE Symposium on Foundations of Computer Science
  • 2009
TLDR
This paper shows hardness of a classical tiling problem on an (N x N) 2-dimensional grid and a quantum problem involving finding the ground state energy of a 1-dimensional quantum system of N particles.
Size-driven quantum phase transitions
TLDR
This work constructs simple examples of 2D quantum spin-lattice models with small (≤10) local state spaces which exhibit very unusual finite-size effects that they term “size-driven phase transitions”, and proves that the construction is thermally robust, showing that these effects are in principle accessible to experimental observation.
The Power of Quantum Systems on a Line
TLDR
The proof of the QMA-completeness result requires an additional idea beyond the usual techniques in the area: Some illegal configurations cannot be ruled out by local checks, and are instead ruled out because they would, in the future, evolve into a state which can be seen locally to be illegal.
General nonperturbative estimate of the energy density of lattice Hamiltonians.
Employing a theorem on lower bounds on the zeros of orthogonal polynomials, the plaquette expansion to order 1/[ital N][sub [ital p]] of the tridiagonal Lanczos matrix elements is solved for the
On the computational complexity of Ising spin glass models
TLDR
In a spin glass with Ising spins, the problems of computing the magnetic partition function and finding a ground state are studied and are shown to belong to the class of NP-hard problems, both in the two-dimensional case within a magnetic field, and in the three-dimensional cases.
Quantum Hamiltonian Complexity
TLDR
This survey provides an introduction to the rapidly growing field of Quantum Hamiltonian Complexity, which includes the study of quantum constraint satisfaction problems, and provides a novel information theoretic presentation of Bravyi's polynomial time algorithm for Quantum 2-SAT.
...
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