# On Complexity of the Quantum Ising Model

@article{Bravyi2014OnCO, title={On Complexity of the Quantum Ising Model}, author={Sergey Bravyi and Matthew B. Hastings}, journal={Communications in Mathematical Physics}, year={2014}, volume={349}, pages={1-45} }

We study complexity of several problems related to the Transverse field Ising Model (TIM). First, we consider the problem of estimating the ground state energy known as the Local Hamiltonian Problem (LHP). It is shown that the LHP for TIM on degree-3 graphs is equivalent modulo polynomial reductions to the LHP for general k-local ‘stoquastic’ Hamiltonians with any constant $${k \ge 2}$$k≥2. This result implies that estimating the ground state energy of TIM on degree-3 graphs is a complete…

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## 51 Citations

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- Mathematics, Computer Science2014 IEEE 55th Annual Symposium on Foundations of Computer Science
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This work characterises the complexity of the k-local Hamiltonian problem for all 2-local qubit Hamiltonians and proves for the first time QMA-completeness of the Heisenberg and XY interactions in this setting.

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A formalized version of this problem called Guided Stoquastic Hamiltonian is shown to be complete for the complexity class MA (a probabilistic analogue of NP) and the ground state and thermal equilibrium properties of the ferromagnetic TIM can be simulated in polynomial time on a classical Probabilistic computer.

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