Computational Complexity of the Ground State Energy Density Problem

Watson, James D.; Cubitt, Toby S.

doi:10.48550/arxiv.2107.05060

Cited by 2 publications

(3 citation statements)

References 17 publications

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“…Both [AI21] and [WC21] study the complexity of computing the ground energy density of infinite TI Hamiltonians to within a desired precision making use of the technique introduced by Cubitt, Prerez-Garcia, and Wolf which embeds finite Hamiltonian constructions of exponentially increasing sizes, into the 2D infinite lattice, using Robinson tiles. Robinson tiling rules [Rob71] force an aperiodic structure on the tiling of the infinite plane, with squares of exponentially increasing size.…”

Section: Discussion Related Work and Open Problemsmentioning

confidence: 99%

“…The quantum construction of [AI21] layers a TI 1D Hamiltonian on top of one of the sides of all the squares. The classical construction of [WC21] layers a classical finite construction on each square. Neither work obtains tight results due to the same issue with invalid queries, although the two papers compromise in completely different ways.…”

Section: Discussion Related Work and Open Problemsmentioning

confidence: 99%

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Translationally Invariant Constraint Optimization Problems

Aharonov¹,

Irani²

2022

Preprint

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We study the complexity of classical constraint satisfaction problems on a 2D grid. Specifically, we consider the computational complexity of function versions of such problems, with the additional restriction that the constraints are translationally invariant, namely, the variables are located at the vertices of a 2D grid and the constraint between every pair of adjacent variables is the same in each dimension. The only input to the problem is thus the size of the grid. This problem is equivalent to one of the most interesting problems in classical physics, namely, computing the lowest energy of a classical system of particles on the grid. We provide a tight characterization of the complexity of this problem, and show that it is complete for the class FP NEXP . Gottesman and Irani (FOCS 2009) also studied classical constraint satisfaction problems using this strong notion of translational-invariance; they show that the problem of deciding whether the cost of the optimal assignment is below a given threshold is NEXP-complete. Our result is thus a strengthening of their result from the decision version to the function version of the problem. Our result can also be viewed as a generalization to the translationally invariant setting, of Krentel's famous result from 1988, showing that the function version of SAT is complete for the class FP NP .An essential ingredient in the proof is a study of the computational complexity of a gapped variant of the problem. We show that it is NEXP-hard to approximate the cost of the optimal assignment to within an additive error of Ω(N 1/4 ), where the grid size is N × N . To the best of our knowledge, no gapped result is known for CSPs on the grid, even in the non-translationally invariant case. This might be of independent interest. As a byproduct of our results, we also show that a decision version of the optimization problem which asks whether the cost of the optimal assignment is odd or even is also complete for P NEXP .

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Section: Discussion Related Work and Open Problemsmentioning

confidence: 99%

Section: Discussion Related Work and Open Problemsmentioning

confidence: 99%

Translationally Invariant Constraint Optimization Problems

Aharonov¹,

Irani²

2022

Preprint

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“…It would also be interesting to connect our results to existing efforts in the study of complexity theory. Various results are known on the computational resources required to, e.g., calculate the ground state energy density of local Hamiltonians [42,43]. What can be said for the calculation of hydrodynamical quantities?…”

Section: Discussionmentioning

confidence: 99%

Operator backflow and the classical simulation of quantum transport

von Keyserlingk,

Pollmann,

Rakovszky

2021

Preprint

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Tensor product states have proved extremely powerful for simulating the area-law entangled states of many-body systems, such as the ground states of gapped Hamiltonians in one dimension. The applicability of such methods to the dynamics of many-body systems is less clear: the memory required grows exponentially in time in most cases, quickly becoming unmanageable. New methods reduce the memory required by selectively discarding/dissipating parts of the many-body wavefunction which are expected to have little effect on the hydrodynamic observables typically of interest: for example, some methods discard fine-grained correlations associated with n-point functions, with n exceeding some cutoff * . In this work, we present a theory for the sizes of 'backflow corrections', i.e., systematic errors due to discarding this fine-grained information. In particular, we focus on their effect on transport coefficients. Our results suggest that backflow corrections are exponentially suppressed in the size of the cutoff * . Moreover, the backflow errors themselves have a hydrodynamical expansion, which we elucidate. We test our predictions against numerical simulations run on random unitary circuits and ergodic spin-chains. These results lead to the conjecture that transport coefficients in ergodic diffusive systems can be captured to a given precision with an amount of memory scaling as exp î O(log( ) 2 ) ó , significantly better than the naive estimate of memory exp O(poly( −1 )) required by more brute-force methods.

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Computational Complexity of the Ground State Energy Density Problem

Cited by 2 publications

References 17 publications

Translationally Invariant Constraint Optimization Problems

Translationally Invariant Constraint Optimization Problems

Operator backflow and the classical simulation of quantum transport

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