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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 $kge 2$. This result implies that estimating the ground state energy of TIM on degree-3 graphs is a complete problem for the complexity class StoqMA - an extension of the classical class MA. As a corollary, we complete the complexity classification of 2-local Hamiltonians with a fixed set of interactions proposed recently by Cubitt and Montanaro. Secondly, we study quantum annealing algorithms for finding ground states of classical spin Hamiltonians associated with hard optimization problems. We prove that the quantum annealing with TIM Hamiltonians is equivalent modulo polynomial reductions to the quantum annealing with a certain subclass of k-local stoquastic Hamiltonians. This subclass includes all Hamiltonians representable as a sum of a k-local diagonal Hamiltonian and a 2-local stoquastic Hamiltonian.
We show that Gibbs states of non-homogeneous transverse Ising chains satisfy a emph{shielding} property. Namely, whatever the fields on each spin and exchange couplings between neighboring spins are, if the field in one particular site is null, the r
We establish a set of nonequilibrium quantum phase transitions in the Ising model driven under monochromatic nonadiabatic modulation of the transverse field. We show that besides the Ising-like critical behavior, the system exhibits an anisotropic tr
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There is great interest in using near-term quantum computers to simulate and study foundational problems in quantum mechanics and quantum information science, such as the scrambling measured by an out-of-time-ordered correlator (OTOC). Here we use an
Frustration represents an essential feature in the behavior of magnetic materials when constraints on the microscopic Hamiltonian cannot be satisfied simultaneously. This gives rise to exotic phases of matter including spin liquids, spin ices, and st